638 results
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2. The interplay of rock-paper-scissors competition and environments mediates species coexistence and intriguing dynamics.
- Author
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Mohd, Mohd Hafiz and Park, Junpyo
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COMPETITION (Biology) , *COEXISTENCE of species , *ABIOTIC environment , *LIMIT cycles , *ECOLOGICAL models , *HOPF bifurcations , *DYNAMICAL systems - Abstract
• We introduce the effect of changing environmental carrying capacity on evolution of asymmetric rock-paper-scissors game. • According to assumptions on environmental gradients, the system can exhibit various survival states including multistability of single species survival. • Symmetry-breaking of competition rates and environmental carrying capacity can be significant factors to yield rich behavior of species survival in systems of cyclic competition. • Considering ecological factors is found to be an important issue on understanding mechanisms of evolution among cyclically competing species in the perspective of maintaining coexistence and promoting biodiversity. Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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3. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations.
- Author
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Verma, Tina and Gupta, Arvind Kumar
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COEXISTENCE of species , *LIMIT cycles , *BIODIVERSITY conservation , *RANDOM walks , *HOPF bifurcations , *HABITATS - Abstract
• The rock-paper-scissors model is presented for two-person non-zero sum game in which the strategies of rock mutate to scissors and paper. • The evolutionary dynamics of the population of species rock, paper and scissors is studied when the patches are connected through random walk. • When the patches are coupled, the state of synchronization and stability is observed. • When mutation is allowed, the limit cycle converges to stable state and the transition from one phase to another phase is observed. • The replicator-mutator equations are solved analytically as well as numerically. Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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4. Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology.
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He, Ke, Shi, Jianping, and Fang, Hui
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DYNAMICAL systems , *FINANCIAL risk , *HOPF bifurcations , *FINANCIAL risk management , *PHASE space , *POLYNOMIAL chaos - Abstract
A comprehensive theoretical and numerical analysis of the dynamical features of a fractional-order delay financial risk system(FDRS) is presented in this paper. Applying the linearization method and Laplace transform, the critical value of delay when Hopf bifurcation first appears near the equilibrium is firstly derived in an explicit formula. Comparison simulations clarify the reasonableness of fractional-order derivative and delay in describing the financial risk management processes. Then we employ persistent homology and six topological indicators to reveal the geometric and topological structures of FDRS in delay interval. Persistence barcodes, diagrams, and landscapes are utilized for visualizing the simplicial complex's information. The approximate values of delay when FDRS undergoes different periodic oscillations and even chaos are determined. The existence of periodic windows within the chaotic interval is correctly decided. The results of this paper contribute to capturing intricate information of underlying financial activities and detecting the critical transition of FDRS, which has promising and reliable implications for a deeper comprehension of complex behaviors in financial markets. • Determine delay τ 0 when Hopf bifurcation appears. • The effects of fractional orders and parameters on τ 0 are elucidated. • Topological features are visualized by simplicial complex in phase space. • Six indicators based on persistent homology identify varied oscillations. • A fractional-order delay system is reasonable to describe financial activities. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population flow.
- Author
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Park, Junpyo
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ECOLOGY , *ECOSYSTEMS , *COMMUNITY change , *SYSTEM dynamics , *HOPF bifurcations - Abstract
• We introduce population flow and its effect on biodiversity in evolutionary dynamics of rock-paper-scissors game. • According to assumptions on population flows, the system can exhibit various survival states including persistent coexistence and multistability of single group survival. • Population flow can change the carrying simplex for evolutions of the system. • The basin structure for multistability may be spirally entangled and discontinuous. • The coexistence state can exhibit oscillatory dynamics according to the magnitude of population flow. Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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6. A novel dimensionality reduction approach by integrating dynamics theory and machine learning.
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Chen, Xiyuan and Wang, Qiubao
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MACHINE learning , *MACHINE theory , *MACHINE dynamics , *HOPF bifurcations , *BIFURCATION theory , *DYNAMICAL systems , *MOTION - Abstract
This paper aims to introduce a technique that utilizes both dynamical mechanisms and machine learning to reduce dimensionality in high-dimensional complex systems. Specifically, the method employs Hopf bifurcation theory to establish a model paradigm and use machine learning to train location parameters. The effectiveness of the proposed method is evaluated by testing the Van Der Pol equation and it is found that it possesses good predictive ability. In addition, simulation experiments are conducted using a hunting motion model, which is a well-known practice in high-speed rail, demonstrating positive results. To ensure the robustness of the proposed method, we tested it on noisy data. We introduced simulated Gaussian noise into the original dataset at different signal-to-noise ratios (SNRs) of 10 db, 20 db, 30 db, and 40 db. All data and codes used in this paper have been uploaded to GitHub. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays.
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Gao, Yuequn and Li, Ning
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HOPF bifurcations ,TIME delay systems ,HEPATITIS B virus ,HEPATITIS B - Abstract
This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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8. Spatiotemporal patterns in a diffusive resource–consumer model with distributed memory and maturation delay.
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Shen, Hao and Song, Yongli
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HOPF bifurcations , *MEMORY , *DIFFUSION coefficients - Abstract
In this paper, we propose a diffusive consumer–resource model in which the consumer is involved with spatiotemporal memory and the resource has maturation delay. Firstly, for the case without maturation delay, the influence of the spatiotemporal distributed memory on the stability of the positive steady state is investigated. It has been shown that memory delay can induce Turing bifurcation for the negative memory-based diffusion coefficient and induce Hopf bifurcation for the positive memory-based diffusion coefficient. Then the joint effect of the memory delay and the maturation delay on the stability of the positive steady state is investigated and it has been shown that the joint effect of the memory delay and the maturation delay can induce more complicated spatiotemporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our theoretical analysis results to a consumer–resource model with Holling-II type functional response to illustrate the existence of the different spatiotemporal patterns. • Propose a diffusive resource-consumer model with distributed memory and maturation delay. • Investigate the joint effect of the memory delay and the maturation delay on the spatiotemporal dynamics. • Find the spatially inhomogeneous spatial patterns and spatially inhomogeneous periodic patterns. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Emergence of spiral and antispiral patterns and its CGLE analysis in leech-heart interneuron model with electromagnetic induction.
- Author
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Upadhyay, Ranjit Kumar, Pradhan, Debasish, Sharma, Sanjeev Kumar, and Mondal, Arnab
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ELECTROMAGNETIC induction , *INTERNEURONS , *MEMBRANE potential , *COMPUTATIONAL electromagnetics , *MAGNETIC flux , *HOPF bifurcations - Abstract
Neurons can exhibit various rhythmic activities such as bursting, spiking, and quiescent states when exposed to external input current stimulus. In this paper, a model of medicinal leech's heart (LH) interneuron is considered to describe the dynamics of neurons with a varied range of electrical activities. The crucial insights into the model's dynamics are explored in three different parameter regimes: phasic spiking, regular spiking, and bursting, based upon the codimension-one bifurcation of the model by considering V K 2 s h i f t as a bifurcation parameter. The spatiotemporal dynamics of the model are explored by allowing 1D and 2D diffusion in the membrane voltage. The 1D diffusive system produces irregular bursting dynamics for the intermediate value of diffusion coefficients, whereas, at higher values, it shows synchronized oscillations. In the presence of 2D diffusion, the emergence of different types of spiral patterns is observed in the system. Furthermore, the system is extended by incorporating electromagnetic induction in the membrane voltage to explore the effect of induction on the various dynamics of neural model. By varying its intensities, the membrane voltage in the extended model produces a variety of discharge modes, such as periodic spiking, fast-spiking, resting, and spike-adding phenomena. In addition, the emergence of anti-spiral patterns in the extended model near subcritical Hopf bifurcation is analytically verified using the complex Ginzburg-Landau equation (CGLE). These findings demonstrate that the firing patterns vary based on the control parameters, and these variations contribute to our understanding of how the brain system transmits and processes the signals. • A slow-fast neuron model of Leech's heart interneuron is considered. • The model is improved by adding EMI to study the impact of magnetic flux. • Bifurcation scenarios and different firing activities of both models are performed. • These systems with diffusion display irregular firing and multi-arm spiral patterns. • CGLE analysis in the system with EMI confirms anti-spiral patterns occurrence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. Pattern dynamics analysis of a time-space discrete FitzHugh-Nagumo (FHN) model based on coupled map lattices.
- Author
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Zhang, Xuetian, Zhang, Chunrui, and Zhang, Yazhuo
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LYAPUNOV exponents , *BIFURCATION diagrams , *DIFFUSION coefficients , *HOPF bifurcations , *DISCRETE systems , *TWO-dimensional models , *SYSTEM dynamics - Abstract
This paper investigates the dynamics of a discrete FitzHugh-Nagumo (FHN) model with self-diffusion on two-dimensional coupled map lattices. The primary objective is to analyze the complex dynamics of neuronal systems in a discrete setting. Through the application of central manifold and normal form analysis, it has been demonstrated that the system is capable of undergoing Neimark-Sacker and flip bifurcations even in the absence of diffusion. Additionally, when influenced by diffusion, the system can manifest pure Turing instability, Neimark-Sacker-Turing instability, and Flip-Turing instability. In the numerical simulation section, the path from bifurcation to chaos is explored by calculating the maximum Lyapunov exponent and drawing the bifurcation diagram. The interconversion between various Turing instabilities is simulated by varying the values of the time step and the self-diffusion coefficient. This study contributes to a deeper understanding of the complexity of neural network systems. • A time-space discrete FitzHugh-Nagumo (FHN) model base on coupled map lattices is considered. • Conditions of Neimark-Sacker-Turing instability and Flip-Turing instability are obtained. • The path from bifurcation to chaos is given by calculating the maximum Lyapunov exponent. • The mutual transformation in various Turing patterns can be realized by changing some parameters. [ABSTRACT FROM AUTHOR]
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- 2024
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11. Complex dynamics of a four-species food web model with nonlinear top predator harvesting and fear effect.
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Shang, Zuchong and Qiao, Yuanhua
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FOOD chains , *TOP predators , *NONLINEAR dynamical systems , *GEOMETRIC approach , *HOPF bifurcations , *LIMIT cycles , *POPULATION density - Abstract
In this paper, a four-species food web model is formulated to investigate the influence of fear effect and nonlinear top predator harvesting on the dynamical behaviors. The global stability of the system at the interior equilibrium is explored by Li–Muldowney geometric approach. By applying the Sotomayor's theorem, it is shown that the system undergoes transcritical bifurcation and pitchfork bifurcation. The conditions for the occurrence of Hopf bifurcation are established, and the stability of the bifurcating limit cycle is discussed by normal form theory. Finally, the numerical simulations are carried out. It is observed that the system presents chaotic dynamics, periodic window and chaotic attractors. Two distinct routes to chaos are discovered, one is period doubling cascade and the other is the generation and destruction of quasi-periodic states. Moreover, it is found that the fear effect can suppress fluctuations in population density to stabilize the system. [ABSTRACT FROM AUTHOR]
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- 2024
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12. Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and Logistic growth.
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Chen, Wei, Zhang, Long, Wang, Ning, and Teng, Zhidong
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BASIC reproduction number , *MIXED infections , *HOPF bifurcations , *VIRAL load , *HIV infection transmission , *HIV - Abstract
In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R 0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV. [ABSTRACT FROM AUTHOR]
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- 2024
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13. Hopf bifurcation and fixed-time stability of a reaction–diffusion echinococcosis model with mixed delays.
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Chen, Weixin, Xu, Xinzhong, and Zhang, Qimin
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HOPF bifurcations , *ECHINOCOCCOSIS , *NONLINEAR systems , *COMPUTER simulation - Abstract
In this paper, a model with spatial diffusion and mixed delays is presented to describe the spread of echinococcosis between dogs and livestock. Firstly, the local stability is investigated using the Routh–Hurwitz criterion. Furthermore, when considering time delays as bifurcation parameters, the conditions for the occurrence of Hopf bifurcation are discussed based on the linear approximation of the nonlinear system. Lastly, the fixed-time stability is studied under the implementation of appropriate control measures. Numerical simulations are provided to give a better understanding of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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14. Qualitative behavior in a fractional order IS-LM-AS macroeconomic model with stability analysis.
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Bazán Navarro, Ciro Eduardo and Benazic Tomé, Renato Mario
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MACROECONOMIC models , *GLOBAL asymptotic stability , *HOPF bifurcations , *STRUCTURAL stability , *ECONOMIC systems , *STRUCTURAL models - Abstract
In this article, we analyze the conditions for the structural stability of a fractional order IS-LM-AS dynamic model with adaptive expectations. It is a generalization of our previous research lately published in the literature. We also present the conditions that the structural parameters of the model must meet for the economic system to present a periodic movement when the critical value of the fractional order of the system, q * , guarantees the presence of a Hopf bifurcation of degenerate type. The theoretical analysis is complemented with numerical simulations of the phase portraits in R 3 and of the temporal trajectories of the solutions of the model in MATLAB software. Finally, it is important to highlight that unlike the results of our previous research, the qualitative results found in this paper show that all the structural parameters of the model are essential in determining its global asymptotic stability and Hopf bifurcation. • A fractional economic model generalizes an integer-order model recently reported. • The memory effect in numerical simulations of economic systems is analyzed. • The fractional order when a Hopf bifurcation occurs is determined. • All the parameters of the model are essential in determining its global stability. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method.
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Li, Jia-Xuan, Yan, Yan, and Wang, Wen-Quan
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CANTILEVERS , *TIME delay systems , *HOPF bifurcations - Abstract
• Frequency response of a strongly nonlinear time delay system is obtained. • Optimal time delays of control system are given. • First statement about the feedback control relationship between displacement and velocity. • Appropriate time delay parameters can make the system softening or hardening. In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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16. A delayed deterministic and stochastic [formula omitted] model: Hopf bifurcation and stochastic analysis.
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Hajri, Youssra, Allali, Amina, and Amine, Saida
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STOCHASTIC analysis , *HOPF bifurcations , *BASIC reproduction number , *WHITE noise , *STOCHASTIC models - Abstract
In this paper, we present a delayed deterministic and stochastic S I R I C V models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic S I R I C V model, the basic reproduction number R 0 and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when R 0 < 1 the disease-free equilibrium is locally stable for any positive value of τ. Furthermore, when R 0 > 1 , the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay τ as a bifurcation parameter. For the stochastic S I R I C V model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers R 0 s and R 0 s ∗. Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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17. Spatial movement with memory-induced cross-diffusion effect and toxin effect in predator.
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Ye, Luhong, Zhao, Hongyong, and Wu, Daiyong
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SPATIAL memory , *HOPF bifurcations , *PREDATOR management , *ANIMAL mechanics , *ANIMAL memory , *POISONS - Abstract
It is well known that spatial memory exists in animal movement modelling. Since spatial memory is related to the past information, then delay arises. In view of the protection of ecological environment, more and more scholars are concerned about the effects of toxic substances on it. A prey–predator system with memory-induced cross-diffusion and toxin in predator is considered in this paper. First, we discuss the fundamental dynamics in detail. Then considering the memory-induced cross-diffusion coefficient and the averaged memory period delay in predator as the controlling parameters, we get that n -mode Hopf bifurcations exist at the positive steady state. Stability switches are generated, and there are spatially nonhomogeneous periodic solutions. Namely if the cross-diffusion coefficient is small, the populations always coexist. If the cross-diffusion coefficient is moderate, two kinds of critical values of the averaged memory period of Hopf bifurcations occur and stability switches may be induced by the delay. If the cross-diffusion coefficient becomes larger, one kind of critical value of the averaged memory period of Hopf bifurcation occur. From simulations, the memory-based cross-diffusion and toxin have vital influences on stability. The moderate toxin can be good for population coexistence. When the averaged memory period is small, the toxin does not change the stability. Once the averaged memory period is bigger, the toxin can change the stability. Moreover, it shows that the averaged memory delay could switch the stability of the system. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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18. Stability and Hopf bifurcation for a quaternion-valued three-neuron neural network with leakage delay and communication delay.
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Zhu, Mengfan, Wang, Baoxian, and Wu, Yihong
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HOPF bifurcations , *LEAKAGE - Abstract
This paper proposes the stability and Hopf bifurcation for a quaternion-valued neural network(QVNN) with leakage delay and communication delay. Due to the higher order of the characteristic equation, the order of characteristic equation is reduced via the matrix block theory. Simultaneously, taking the leakage delay and communication delay as bifurcation parameter respectively, a number of conditions ensuring the stability of Hopf bifurcation periodic solutions are established. It lavishly illustrates that the stability of QVNN can be demolished more when considering the leakage delay as bifurcation parameter. Finally one simulations example is given to validate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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19. Bifurcations and multistability in a virotherapy model with two time delays.
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Dai, Qinrui, Rong, Mengjie, and Zhang, Ren
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HOPF bifurcations , *VIROTHERAPY , *FUNCTIONAL differential equations - Abstract
In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior. • A delayed virotherapy model including infected and uninfected tumor cells is established. • We mainly analyze the stability and Hopf bifurcation of the model under different time delays. The system also undergoes Bogdanov–Takens and Zero-Hopf bifurcation. • In this paper, some interesting dynamic phenomena are simulated, such as bistability, periodic coexistence and chaotic behavior. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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20. Nonlinear modeling and stability analysis of asymmetric hydro-turbine governing system.
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Lai, Xinjie, Huang, Huimin, Zheng, Bo, Li, Dedi, and Zong, Yue
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WATER diversion , *BIFURCATION theory , *HOPF bifurcations , *MATHEMATICAL optimization , *NONLINEAR systems - Abstract
• A novel full nonlinear mathematical model of asymmetric hydro turbine governing system established. • The multi-stability characteristics of as well as its generation mechanism are revealed. • The effect mechanism of water diversion system topology on the system stability is revealed. • The recommended setting values of system parameters under different topologies are provided. This paper aims to study the stability and nonlinear dynamics of hydro-turbine governing system with asymmetric water diversion system, i.e. asymmetric hydro turbine governing system, by using Hopf bifurcation theory. Firstly, the full nonlinear mathematical model of asymmetric hydro turbine governing system is established by all system components and nonlinear head loss. This model contains two units with different capacities which share a common pipeline. Based on the nonlinear mathematical model, the multi-stability characteristics of asymmetric hydro turbine governing system under load disturbance is studied by using stable domain and verified by numerical simulation. Moreover, the multi-time scale oscillation is revealed and its relationship to system multi-stability is investigated. Furthermore, the effect of system parameters and topological parameters on system stability is analyzed. Results indicate that: two stable domains are emerged under load disturbance, the overall stability of asymmetric hydro turbine governing system is determined by the intersection area of the two stable domains. In addition, the system parameters and topological parameters both have obvious effect on system stability, which can be significantly improved by reasonable tuning of system parameters and optimization of water diversion system layout. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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21. Multiscale stabilized finite element computation of the non-Newtonian Casson fluid flowing in double lid-driven rectangular cavities.
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Kumar, B.V. Rathish and Chowdhury, Manisha
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FLUID flow , *REYNOLDS number , *DEPENDENCY (Psychology) , *FINITE element method , *HOPF bifurcations , *NON-Newtonian fluids - Abstract
This article presents a detailed study of the non-Newtonian transient Casson fluid flow behavior in three different rectangular domains with their upper and bottom parallel walls moving horizontally either in same direction or in opposite direction to each other. The numerical computations are carried out applying the dynamic subscales approach of the Algebraic Sub-Grid multi-Scale (ASGS) stabilized finite element method upto a sufficiently high Reynolds number, Re =10000. The flow patterns, which include the generation of the primary and secondary vortices, are highly impacted by the Re values, the structures of the domains and the directions in which the parallel walls are moving. We have observed the appearances of both the symmetric and stable as well as unstable asymmetric types of primary vortices in the domains of aspect ratios ≤ 1, with walls moving in the same direction. On the other hand, we notice the existence of multiple phases of the solutions for the Casson fluid flowing in the rectangular domains of aspect ratio > 1, with walls moving in the opposite direction. Besides, the evolution of the solutions with time are also carried out here. In most of the cases, the solutions achieve the steady state after a certain point of time, except for a particular type of domain, where the fluid flow solutions show time dependent behavior after reaching a critical value of Re. This paper has reported the ranges, where the critical Reynolds number (R e c) lies, indicating the appearance of the first Hopf bifurcation for the fluid flow problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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22. Multi-scale oscillation characteristics and stability analysis of pumped-storage unit under primary frequency regulation condition with low water head grid-connected.
- Author
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Zhao, Kunjie, Xu, Yanhe, Guo, Pengcheng, Qian, Zhongdong, Zhang, Yongchuan, and Liu, Wei
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SYNCHRONOUS generators , *STABILITY of nonlinear systems , *WATER hammer , *OSCILLATIONS , *HOPF bifurcations , *NONLINEAR equations - Abstract
This paper aims to study the multi-scale oscillation characteristics and stability of pumped storage unit(PSU) under primary frequency regulation(PFR) condition with low water head grid-connected. Firstly, a novel nonlinear mathematical model of pumped storage unit governing system(PSUGS) considering the nonlinear characteristics of diversion tunnel is established, and the nonlinear state equations under power control mode is deduced. On this basis, the bifurcation characteristics and dynamic response process of PSUGS system with coupled first-order, third-order and fifth-order synchronous generator are studied respectively, and the effects of different order synchronous generator on the nonlinear PSUGS system are compared. In addition, the transient characteristics of the nonlinear PSUGS system considering the fifth order synchronous generator under different water heads are studied. Finally, the sensitivity of the regulation system parameters of PSU under PFR with low water head is analyzed, and the effects of hydraulic, mechanical and electrical factors on the multi-scale oscillation characteristics and stability of the nonlinear PSUGS system are revealed. The results show that the stability of PSU considering the first-order synchronous generator is less than that considering the third-order and fifth-order synchronous generator. The stability of PSU considering the third-order synchronous generator is close to that considering the fifth-order synchronous generator. In the operation of PSU, the lower the water head, the smaller the stability of the system. Under PFR condition, the nonlinear PSUGS system meets the supercritical Hopf bifurcation. During PFR oscillation of the unit, the high-frequency wavelet is excited by the generator excitation system and is sensitive to the q-axis parameters of the generator. The low-frequency wavelet is mainly excited by the power grid and mapped with the water hammer effect in the dynamic process. The research results of this paper provide a theoretical basis for the stability analysis and optimal regulation of PSU under PFR condition with low water head. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. Hopf bifurcation and global exponential stability of an epidemiological smoking model with time delay.
- Author
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Hu, Xiaomei, Pratap, A., Zhang, Zizhen, and Wan, Aying
- Subjects
EXPONENTIAL stability ,HOPF bifurcations ,EPIDEMIOLOGICAL models - Abstract
There are many harmful effects of smoking. It not only engulfs the health and life of smokers, but also pollutes the air, endangers the life and health of others, and brings a heavy burden to public health. To this end, a delayed smoking model including potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters and smokers with some illness, is investigated in the present paper. Firstly, local stability and existence of Hopf bifurcation of the model is conducted. Secondly, global exponential stability is explored. Lastly, we numerically simulate the correctness of the obtained theoretical results in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
24. Dynamics of a plankton community with delay and herd-taxis.
- Author
-
Ding, Linglong, Zhang, Xuebing, and Lv, Guangying
- Subjects
- *
NEUMANN boundary conditions , *HOPF bifurcations , *PLANKTON , *JUDGMENT (Psychology) - Abstract
The movements of the plankton in the ocean are driven by random diffusion and cognitive judgement with herd-taxis. In this paper, we formulate a phytoplankton–zooplankton model with time delay in the herd-taxis effect diffusion and homogeneous Neumann boundary conditions. The conditions to guarantee the existence of the coexistence equilibrium of the model are given. By analyzing the distribution of the eigenvalues of the characteristic equation, the local asymptotic stability of the coexistence equilibrium is achieved under certain condition. When there is no time delay in the herd-taxis effect, the model can possess the Turing bifurcation when we consider the nonlinear diffusion term, which leads to instability. When taking the time delay into account, the Hopf bifurcation occurs instead as the time delay varies. Furthermore, we investigate the situation without the fact of time, that is the steady-state bifurcation and the stability of bifurcating solution. Finally, the stability of the coexistence equilibrium, the Turing bifurcation and the Hopf bifurcation of the system are modeled by numerical simulation. The simulations shown are coordinated with the theoretical results which we arrive at in the former part of the paper. The results illustrate that the time delay in the herd-taxis effect of the zooplankton influence the dynamics of the plankton system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Stability of hydropower units under full operating conditions considering nonlinear coupling of turbine characteristics.
- Author
-
Xu, Rongli, Tan, Xiaoqiang, Wang, He, Zhu, Zhiwei, Lu, Xueding, and Li, Chaoshun
- Subjects
- *
WATER power , *HOPF bifurcations , *DYNAMIC stability , *TURBINES , *UNITS of time , *ELECTRIC power distribution grids - Abstract
The nonlinearity of the hydro-turbine governing system (HTGS) will affect the regulation stability. This paper investigates the stability of HTGS considering the nonlinear coupling of hydro-turbine characteristics (NCTC) under full operating conditions. Firstly, the HTGS models considering the NCTC and power grid conditions are established. Then, the Hopf bifurcation analysis is carried out to comprehensively explain the mechanism of HTGS stability. Finally, the effect regularity of the NCTC on the stability of HTGS is summarized under the full operating conditions. The results indicate that the NCTC includes the coupling of nonlinear speed characteristic and linear head characteristic (NSHC), the coupling of nonlinear head characteristic and linear speed characteristic (NHSC), and the coupling of nonlinear head characteristic and linear opening characteristic (NHOC). The effect of NCTC on stability and dynamic response is mainly realized through the NHOC. The NHOC is favorable for stability under negative load disturbances and unfavorable for stability under positive load disturbances. A smaller unit inertia time constant or a greater flow inertia reduce the effect of NHOC on stable domain. The NHOC has different effects on stability under different operating conditions with a certain regularity. This paper provides potential support for optimizing control parameters of the HTGS. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Bifurcations in a fractional-order BAM neural network with four different delays.
- Author
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Huang, Chengdai, Wang, Juan, Chen, Xiaoping, and Cao, Jinde
- Subjects
- *
BIDIRECTIONAL associative memories (Computer science) , *HOPF bifurcations - Abstract
This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
27. Analysis and simulation of a delayed HIV model with reaction–diffusion and sliding control.
- Author
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Pei, Yongzhen, Shen, Na, Zhao, Jingjing, Yu, Yuping, and Chen, Yasong
- Subjects
- *
NEUMANN boundary conditions , *TREATMENT effectiveness , *HOPF bifurcations , *HIV - Abstract
Selection of an appropriate therapy threshold to restrain the virus load is still challenging on structured treatment interruptions (STIs) for HIV. In this paper, we ponder that how the sliding control and multistability to regulate the treatment effect through comprehensive dynamics of a virus-immune model. Firstly, based on piecewise therapy, we propose a delayed reaction–diffusion virus-immune model under the homogeneous Neumann boundary condition. Secondly, the existence and stabilities of five kinds of equilibria as well as the direction and stability of spatial Hopf bifurcation at regular equilibrium are investigated. Thirdly, the sliding domain and the boundary node bifurcations are addressed by theoretical analysis. Finally, we appraise the effects of therapy threshold, sliding domain and multistability on HIV therapy by simulations, and further seek out the appropriate therapy threshold for infected patients with given physiological parameters and present the corresponding principles. Our explorations will provide evidence for HIV and other disease therapies. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. Chaos, Hopf bifurcation and control of a fractional-order delay financial system.
- Author
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Shi, Jianping, He, Ke, and Fang, Hui
- Subjects
- *
HOPF bifurcations , *LAPLACE transformation , *BUSINESS forecasting , *LONG-term memory , *FINANCIAL risk - Abstract
The evolution of financial system depends not only on the current state, but also on the previous state. Due to "long-term memory" and "non-locality" of the fractional derivative, fractional-order model can effectively characterize the dynamic features of financial process. An incommensurate fractional-order delay financial system (FDFS) is considered in this paper. Based on linearization and Laplace transformation, the characteristic equation of linearized system of FDFS is obtained. The critical value of the time delay for the occurrence of Hopf bifurcation is determined through the discussions of the eigenvalues of the characteristic equation and the transversality condition. A periodic pulse delay feedback controller is added to the FDFS to control the Hopf bifurcation and to regulate the stability domain of the system. Two illustrative examples are provided to validate our theoretical results. Moreover, numerical simulations demonstrate that the increase of the fractional-order can induce chaos in FDFS, which is detected by 0 − 1 test for chaos. This paper contributes to a better understanding of the dynamic behavior of financial market, forecasting financial risk and implementing effective financial regulation. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
29. Stability of spatial patterns in a diffusive oxygen–plankton model with time lag effect.
- Author
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Gökçe, Aytül, Yazar, Samire, and Sekerci, Yadigar
- Subjects
- *
TIME delay systems , *PLANKTON , *CLIMATE change , *HOPF bifurcations , *MARINE resources conservation , *OXYGEN , *ENVIRONMENTAL protection - Abstract
Although marine ecosystem is a highly complex phenomenon with many non-linearly interacting species, dissolved oxygen and plankton among these have perhaps the most fundamental relationship not only for the protection of marine environment but also for continuation of life on Earth. This paper deals with a generic diffusive model of dissolved oxygen, phytoplankton and zooplankton species, for which constant time delays are incorporated in growth response of phytoplankton and in the gestation time of zooplankton. We mainly focus on the stability analysis of the coexisting states and the existence of Hopf bifurcation through the characteristic equation, where time delay and oxygen production rate are considered as control parameters for all cases. Studying the effect of both time delays on a stable system, we show destabilisation of the system and irregularity in the spatio-temporal dynamical regimes, leading to chaotic oscillations. Although both delay terms have a destabilising effect, our findings indicate that time delay in zooplankton gestation may induce sharp strongly irregular pattern, whereas time delay in phytoplankton growth gives rise to more regular but higher frequency oscillations for oxygen–plankton interactions. The findings of this paper may provide new insights into main environmental issues including global warming and climate change. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
30. Robust tracking-surveillance and landing over a mobile target by quasi-integral-sliding mode and Hopf bifurcation.
- Author
-
Ibarra Jiménez, Efraín and Jiménez-Lizárraga, Manuel
- Subjects
- *
HOPF bifurcations , *SLIDING mode control , *CIRCULAR motion , *COMPUTER-generated imagery , *LANDING (Aeronautics) - Abstract
This paper addresses the problem of a robust UAV tracking, surveillance and landing of a mobile ground target. The translational and angular dynamics of the vehicle are affected by bounded uncertainties; a Quasi-Integral Sliding Mode control is designed to obtain robustness from nearly the initial time. The flying mission considers three different dynamics of movement: the take-off to the desired altitude, the relative circular surveillance motion around the mobile ground target and eventually precise landing over the ground vehicle. This paper introduces a novel dynamic motion planning generator to perform such tracking maneuvers. It is based on the solution of a second order nonlinear differential equation, whose solution is force to move in a set of new parameterized 'Bifurcation Sliding Mode Surfaces' that exploit the Hopf Bifurcation properties to change the dynamic around the equilibrium point. A temporal switching technique is introduced for changing between three different bifurcation sliding surfaces at different time intervals. To illustrate that the quadcopter effectively performs the desired maneuvers, a computer animation is provided at the end of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
31. Self-swaying of oblique bending cantilevers under steady illumination.
- Author
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Li, Kai, Wu, Haiyang, Liu, Yufeng, Dai, Yuntong, and Yu, Yong
- Subjects
- *
CRYSTAL whiskers , *CANTILEVERS , *LIQUID crystals , *HOPF bifurcations , *LIGHTING , *AUTONOMOUS robots , *SOFT robotics - Abstract
Self-sustained systems are able to absorb energy from steady external environment to maintain their own motion without additional energy or control. This feature has facilitated their widespread use in micromachines, actuatosr and soft robotics. To the address the relative complexity and difficulty in fabrication of the current self-sustained systems, this paper constructs a novel self-oscillating liquid crystal elastomer (LCE) fiber-beam system, which can sway continuously and periodically under steady illumination. It consists of a LCE fiber, two oblique bending cantilevers and two masses. In conjunction with the well-established LCE dynamic model and beam theory, the governing equations of the self-swaying oblique bending cantilevers system are established in this paper. The self-swaying process of the oblique bending cantilevers system under steady illumination is described and its motion mechanism is explained in detail. Numerical results show that the system can undergo supercritical Hopf bifurcation between the static regime and self-swaying regime. The effects of system parameters on the self-swaying amplitude and frequency are discussed quantitatively. The results of this paper can deepen the understanding of self-swaying and provide guidance for autonomous robots, energy harvesters, sensors and bionic instruments. • A new self-swaying oblique bending cantilevers based on liquid crystal elastomer fiber is established. • The system has two typical motion regimes, namely the static regime and self-swaying regime. • The triggering conditions, frequency and amplitude of self-swaying can be modulated. • The amplitude and frequency of self-swaying can be designed by several parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Dynamical aspects of a tuberculosis transmission model incorporating vaccination and time delay.
- Author
-
Zhang, Zizhen, Zhang, Weishi, Nisar, Kottakkaran Sooppy, Gul, Nadia, Zeb, Anwar, and Vijayakumar, V.
- Subjects
LATENT tuberculosis ,TUBERCULOSIS ,HOPF bifurcations ,VACCINATION ,INFECTIOUS disease transmission - Abstract
To explore transmission dynamics of tuberculosis, a tuberculosis transmission model with vaccination and time delay is developed in current paper. Positivity and boundedness are analyzed. Local stability of tuberculosis-free equilibrium in respect of the time delay due to latent period of tuberculosis is analyzed and we have found threshold value of the time delay for the local stability of tuberculosis-free equilibrium. Then, local stability of tuberculosis-existence equilibrium following exhibition of Hopf bifurcation at the crucial value of the time delay due to latent period of tuberculosis is derived. It is shown that the developed model undergoes a Hopf bifurcation around the tuberculosis-existence when the time delay due to latent period of tuberculosis passes through the threshold value. Direction and stability of the Hopf bifurcation are investigated with the help of the normal form method and center manifold theory. Finally, numerical simulations are carried out in the justification of obtained analytical findings. The results obtained provide significant information for tuberculosis disease controlling. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. Bifurcation analysis of an SIR model considering hospital resources and vaccination.
- Author
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Zhang, Jiajia and Qiao, Yuanhua
- Subjects
- *
HOPF bifurcations , *VACCINATION , *HOSPITALS , *ORBITS (Astronomy) , *RATINGS of hospitals , *CLINICS , *EQUILIBRIUM - Abstract
In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Bifurcations in a fractional-order neural network with multiple leakage delays.
- Author
-
Huang, Chengdai, Liu, Heng, Shi, Xiangyun, Chen, Xiaoping, Xiao, Min, Wang, Zhengxin, and Cao, Jinde
- Subjects
- *
LEAKAGE , *HOPF bifurcations - Abstract
This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
35. Turing-Turing bifurcation and multi-stable patterns in a Gierer-Meinhardt system.
- Author
-
Zhao, Shuangrui and Wang, Hongbin
- Subjects
- *
NORMAL forms (Mathematics) , *HOPF bifurcations , *COMPUTER simulation - Abstract
• The conditions for the occurrence of Turing-Turing bifurcation are established. • The normal forms near Turing-Turing bifurcation are derived. • Multi-stable and superimposed patterns are revealed. • Some vitro experimental patterns of vascular mesenchymal cells are interpreted. The classical Gierer-Meinhardt system portrays the formation process of a self-organizing pattern of vascular mesenchymal cells. In this paper, the coexistence of multi-stable patterns with different spatial responses and the superposition for patterns have been explored in theory from the perspective of Turing-Turing bifurcation. On the one-dimensional region, the system is simplified near the Turing-Turing singularity to obtain a third-order ordinary differential equation employing center manifold and normal form theory, which is locally topologically equivalent to the primitive system and its coefficients can be represented by the parameters of original equation. Especially, considering the simplified system, it is theoretically revealed that the system supports semi-stable patterns superimposed by two different spatial resonances and the coexistence of four stable steady states with different single characteristic wavelengths, indicating that different initial conditions may tent to completely different spatial patterns. Finally, some numerical simulations are given, which are consistent with the theoretical analysis. The multi-stable and superimposed modes of the system are also studied on a two-dimensional region, which shows that some experimental patterns of vascular mesenchymal cells in vitro can be interpreted as the superposition of different spatial modal patterns to a certain extent. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
36. Dynamics of a time-delay differential model for tumour-immune interactions with random noise.
- Author
-
Rihan, F.A., Alsakaji, H.J., Kundu, S., and Mohamed, O.
- Subjects
WHITE noise ,HOPF bifurcations ,LYAPUNOV functions ,CELL growth ,SYSTEM dynamics - Abstract
This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and activated immune system cells. We discuss stability and Hopf bifurcation of the deterministic system. We then explore stochastic stability, and the dynamics of the system in view of environmental fluctuations. Criteria for persistence and sustainability are discussed. Using multiple Lyapunov functions, some sufficient criteria for tumour cell persistence and extinction are obtained. Under certain circumstances, stochastic noise can suppress tumour cell growth completely. In contrast to the deterministic model which shows no stable tumour-free state, the white noise can either lead to tumour dormancy or tumour elimination. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
37. Dynamical analysis of tumor model with obesity and immunosuppression.
- Author
-
Abd-Rabo, Mahmoud A., Zakarya, Mohammed, Alderremy, A.A., and Aly, Shaban
- Subjects
FAT cells ,HOPF bifurcations ,IMMUNOSUPPRESSION ,TUMORS ,OBESITY - Abstract
In this paper, we have studied a three-dimensional model to describe the behavior of tumor cells in the absence of the immune response, which is composed of tumor, normal and fat cells (TNF) under time delay effect. The center manifold theory has been used to study the bifurcation behavior. It is proven that TNF model undergoes codimension-1 bifurcation, while Hopf bifurcation does not occur in the non-delay model. For delayed TNF, we take into the late response of fat cells to tumor cells. Consequently, the delay factor is a considerable parameter in TNF model. Hence, we presented a formula of the time delay value for the occurrence of periodic solutions. Additionally, the stability of Hopf bifurcation was analysed. Numerical simulations are provided to illustrate and extend the theoretical results, we provide our study with numerical simulations inclusive families of periodic solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. Emergence of hidden dynamics in different neuronal network architecture with injected electromagnetic induction.
- Author
-
Upadhyay, Ranjit Kumar, Sharma, Sanjeev Kumar, Mondal, Arnab, and Mondal, Argha
- Subjects
- *
ELECTROMAGNETIC induction , *NEURAL circuitry , *HOPF bifurcations , *REDUCED-order models , *PHASE velocity , *GROUP velocity , *FUZZY neural networks , *HOPFIELD networks - Abstract
• An attempt has been made to understand the effect of electromagnetic induction on the network of neurons. • An excitable slow-fast memristive model has been studied with its various intrinsic dynamics. • We analytically establish the existence and stability of Hopf bifurcation with bifurcating periodic solutions. • We investigate the emergence of multi-arm antispiral waves in the system with proper analytical justification. • A random network architecture is used to study the dynamics of coupled network that generates MMOs and bursting patterns. The diverse firing responses in a single neuron model as well as in a neuronal network play a major role in understanding the collective neuronal dynamics. In this paper, we consider an excitable slow-fast memristive model and study its various intrinsic dynamics by allowing a periodic external stimulus. The single model exhibits various types of spiking, bursting, mixed-mode oscillations (MMOs), and mixed-mode bursting oscillations (MMBOs) depending on the amplitude and frequency of the periodic injected current. Corresponding bifurcation analysis reveals the existence of supercritical and subcritical Hopf bifurcations in the system depending on the major predominant parameters that establish the scenarios of the neuronal responses. We have verified analytically the existence and stability of Hopf bifurcations. The memristive system can produce cascade of period doubling bifurcations for particular parameter regimes. Next, we investigate the emergence of multi-arm antispiral waves in the diffusively coupled system with proper analytical justification. We have also computed the group and phase velocities to discern the spiral and antispiral waves near the Hopf instability. A transition from target wave to multi-arm antispiral wave is observed in the diffusive system. Moreover, we use a random network architecture different from the diffusive network to study the dynamics of the coupled network for certain firing activities. It is observed that the network of heterogeneous desynchronized neurons with higher electrical couplings, can generate MMOs, MMBOs, and bursting for different external stimuli. However, the uncoupled systems cannot reveal such typical dynamics for particular parameters. Further, we introduce a reduced-order model to summarize the complete dynamics of the larger random network and report the findings. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
39. Dynamic behaviors and optimal control of a new delayed epidemic model.
- Author
-
Liu, Qixuan, Xiang, Huili, and Zhou, Min
- Subjects
- *
PONTRYAGIN'S minimum principle , *OPTIMAL control theory , *GLOBAL analysis (Mathematics) , *HOPF bifurcations , *EPIDEMICS , *INFECTIOUS disease transmission - Abstract
We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart.
- Author
-
Sk, Nazmul, Mondal, Bapin, Thirthar, Ashraf Adnan, Alqudah, Manar A., and Abdeljawad, Thabet
- Subjects
- *
ECOLOGICAL disturbances , *PREDATION , *STOCHASTIC systems , *HARVESTING , *BIOLOGICAL extinction , *HOPF bifurcations - Abstract
The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability. • Study on a cooperative prey-predator model with harvesting. • Bifurcations like saddle-node, Hopf, and transcritical, have been explored. • Bistability and tristability are observed in the proposed model system. • Critical radius of transition is determined by evaluating the confidence ellipses. • Noise in the stochastic system leads to transitions between different equilibria. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Environmental regulation and economic cycles.
- Author
-
George, Halkos E., George, Papageorgiou J., Emmanuel, Halkos G., and John, Papageorgiou G.
- Subjects
BUSINESS cycles ,ENVIRONMENTAL regulations ,HOPF bifurcations ,FISCAL policy ,BUDGET surpluses - Abstract
This paper examines economic cycles that do not depend on exogenous economic actions. More precisely, the paper develops a positive model of government behavior in order to define the intertemporal fiscal policies that are optimal for a country and which, determine both the optimal budget level and the optimal level of environmental quality. For this purpose, we establish an optimal control model involving intertemporal subsidy strategies characteristically used by an authoritarian government similar to those found in central Europe. It will be shown that by applying the Hopf bifurcation theorem,a cyclical strategy – that is, waves of regulation, environmental subsidies alternating with deregulation and cuts in social programs – may represent an optimal policy. In this paper, we propose an extremely simple optimal control model which is applied to budget surpluses and environmental subsidies. We investigate the cyclical environmental policies as applied through the bifurcation theorem. A number of propositions are stated during the solution process. The first proposition sets the rules in the model parameters in order to establish the cyclical policies. A second proposition sets the relationship between the discount rate and the opportunity cost of capital, which is used to determine a taxing strategy which becomes the future optimal subsidy policy. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
42. Complex dynamic analysis of a reaction-diffusion network information propagation model with non-smooth control.
- Author
-
Ma, Xuerong, Shen, Shuling, and Zhu, Linhe
- Subjects
- *
RUMOR , *INFORMATION networks , *INFORMATION modeling , *BASIC reproduction number , *SOCIAL perception , *HOPF bifurcations - Abstract
Rumors do the social's perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction–diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Fractional-order PD control at Hopf bifurcation in a delayed predator–prey system with trans-species infectious diseases.
- Author
-
Du, Wentong, Xiao, Min, Ding, Jie, Yao, Yi, Wang, Zhengxin, and Yang, Xinsong
- Subjects
- *
PREDATION , *HOPF bifurcations , *COMMUNICABLE diseases , *FAMILY stability - Abstract
In this paper, a delayed fractional-order predator–prey system with trans-species infectious diseases is proposed and the corresponding control strategy is implemented via fractional-order proportional-derivative (PD) control. Firstly, for the uncontrolled fractional-order predator–prey system, explicit conditions of stability and Hopf bifurcation are established by selecting time delay as the bifurcation parameter. The predator–prey system will lose its stability and a family of oscillations will emerge when the time delay passes through the critical value. Secondly, under the fractional-order PD control, the influences of the controller on the system stability and bifurcation are investigated. It is demonstrated that the Hopf bifurcation can be postponed or advanced, and the desired dynamic can be achieved by choosing appropriate control gain parameters. In addition, the impacts of fractional order and control parameters on dynamics are explored. Finally, some numerical simulations are depicted to validate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
44. Bifurcation control analysis based on continuum model with lateral offset compensation.
- Author
-
Ai, WenHuan, Zhu, JiuNiu, Zhang, Yifan, Wang, Mingming, and Liu, DaWei
- Subjects
- *
TRAFFIC congestion , *TRAFFIC signs & signals , *TRAFFIC flow , *LIMIT cycles , *MATHEMATICAL continuum , *TRAFFIC engineering , *HOPF bifurcations , *TRAFFIC regulations , *LANE changing - Abstract
The study of traffic flow bifurcation control analysis is of great significance for understanding the essential evolution law of traffic flow, controlling traffic flow with practical means and alleviating traffic congestion. This paper describes the influence of lateral distance compensation in the process of lane change, and adjusts the traffic flow from the perspective of bifurcation control, which can capture the characteristics of traffic flow better, so as to describe the practical significance of the actual traffic phenomenon better. For example, the green ratio of traffic lights in the road, the planning and design of traffic signs, and the guidance of traffic condition prediction in ITS. In addition, the traffic flow is introduced and applied to the traffic model from the perspective of random function, and the bifurcation control is carried out according to the bifurcation behavior in traffic. That is, the bifurcation characteristics of the system are modified by the designed feedback controller, and the appearance of the equilibrium point of the system is adjusted to make it move forward, backward or disappear, so as to prevent or alleviate traffic congestion. There are few researches on bifurcation control of traffic flow model with lateral distance compensation. This paper introduces stochastic function based on lateral compensation model to study Hopf bifurcation phenomenon and Hopf bifurcation control. Firstly, the existence condition of Hopf bifurcation at the equilibrium point in the model is proved theoretically. Then, a feedback controller is designed to control the amplitudes of the Hopf bifurcation and the limit cycles formed by the Hopf bifurcation. Finally, the theoretical results are verified by numerical simulation. The research shows that by adjusting the control parameters in the feedback controller, the influence of boundary conditions on the stability of the traffic system is fully described, the effect of unstable focus and saddle point on the system is inhibited, and the traffic flow is slowed down. In addition, the unstable bifurcation points can be eliminated, and the amplitude of the limit cycle formed by Hopf bifurcation can be adjusted, so as to achieve the stability of the control system. • A new continuum model is proposed taking into account the lateral offset compensation. • Introducing the traffic flow into the traffic model from the perspective of the random function. • The designed feedback controller corrects the bifurcation characteristics of the system and adjusts the appearance of the balance point of the system to make it forward, backward or disappear, so as to prevent or reduce traffic jams. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. Twenty Hopf-like bifurcations in piecewise-smooth dynamical systems.
- Author
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Simpson, D.J.W.
- Subjects
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DYNAMICAL systems , *POINCARE maps (Mathematics) , *LIMIT cycles , *HOPF bifurcations , *PREDATION , *LOTKA-Volterra equations , *BIFURCATION theory - Abstract
For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such 'Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincaré maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator–prey systems, ocean circulation, and the McKean and Wilson–Cowan neuron models. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
46. Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications.
- Author
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Wu, Shuhao, Song, Yongli, and Shi, Qingyan
- Subjects
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HOPF bifurcations , *POLLEN tube , *SPATIAL systems , *BIOLOGICAL models , *COMPUTER simulation - Abstract
In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained. • Derive the algorithm of normal form of double Hopf bifurcation for reaction-diffusion system with delay and spatial average. • Investigate the dynamics near the double Hopf bifurcation point for two biological models. • Find bistability of homogeneous and inhomogeneous periodic solutions and pattern transitions for pollen tube tip growth model. • Find stable inhomogeneous quasi-periodic solutions and pattern transitions of periodic and quasi-periodic solutions for Lotka-Volterra model. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
47. Analysis of a mathematical model arising from stage-structured predator–prey in a chemostat.
- Author
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Zhou, Hui
- Subjects
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CHEMOSTAT , *MATHEMATICAL analysis , *MATHEMATICAL models , *HOPF bifurcations , *OSCILLATIONS - Abstract
In this article, we consider a 4-dimensional predator–prey chemostat model of nitrogen-phytoplankton-rotifer interactions with staged structure proposed by Blasius et al. (2020). Although it is still difficult to prove the simulation observations in Blasius et al. (2020) by mathematical arguments, we explore the dynamics in order to better understand the dynamical mechanism of cyclic persistence for this model. We firstly investigate the corresponding system without staged structure, i.e., when the juvenile is absent, the asymptotical behavior of the solutions is given. When the juvenile is present, a threshold condition for the uniform persistence of the 4-dimensional system is provided. Finally, by choosing the life development time delay as a bifurcation parameter, we show that the system admits periodic solutions near one semi-equilibrium undergoing Hopf bifurcation. The rigorous theoretical analytic work in this paper provides some helpful transient information between coherent oscillation and non-coherent oscillation described by the experimental data of Blasius et al. (2020). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Dynamics and stability of two predators–one prey mathematical model with fading memory in one predator.
- Author
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Yılmaz, Zeynep, Maden, Selahattin, and Gökçe, Aytül
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PREDATION , *MATHEMATICAL models , *HOPF bifurcations , *PREDATORY animals , *MEMORY , *SUPERCRITICAL water , *COMPUTER simulation - Abstract
This paper concentrates on dynamics and stability analysis of two predators–one prey mathematical model with competition between predators and fading memory in one predator. The investigation of the constructed model shows that there exist five equilibria, e.g. trivial extinction state of all populations, extinction of both predators state, extinction of first or second predator state and coexisting state. Investigating the eigenvalues of characteristic polynomial, conditions for the local stability around each equilibrium are also determined depending on the parameter space. Analytical formulations are complemented with numerical simulations, where time simulations and single parameter numerical continuation of each variable are performed with respect to model parameters and multiple sub-and super-critical Hopf bifurcations, period doubling bifurcation and transcritical bifurcation are detected for different values of memory related parameter. Our results show that fading memory and competition between predators have substantial impact on the existence and dynamics of all three populations and may shed lights on further understanding of interacting species in ecology. • A model comprising one prey and two competitive predators with fading memory is constructed. • Fading memory and competition between predators have a large impact on the dynamics. • Fading memory in one predator may affect the existence and nature of various bifurcations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
49. A note on a stochastic Holling-II predator–prey model with a prey refuge.
- Author
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Zou, Xiaoling, Lv, Jingliang, and Wu, Yunpei
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HOPF bifurcations , *PREDATORY animals , *COMPUTER simulation , *STOCHASTIC analysis , *BIOLOGICAL extinction - Abstract
A correction is noted to the previously published paper (Zou and Lv, 2017). Regularity, positive recurrence and stationary distribution are considered in this paper. A complete threshold analysis of strong stochastic persistence and extinction is investigated. Stochastic Hopf bifurcation is considered from the viewpoint of numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
50. A Goodwin type cyclical growth model with two-time delays.
- Author
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Sportelli, Mario and De Cesare, Luigi
- Subjects
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REAL wages , *HOPF bifurcations , *UNEMPLOYMENT statistics , *BUSINESS cycles , *ECONOMIC equilibrium , *DIFFERENTIAL equations - Abstract
• Goodwin's model with delayed investment function. • Cyclical growth model with time delays and delay dependent coefficients. • Existence of Hopf bifurcations analyzed by choosing time delays as bifurcation parameters. • Chaotic dynamics. In this paper, we reconsider the Goodwin 1967 growth-cycle model, where the antagonistic relationship between wages and profits is assimilated to the prey-predator conflict modeled by Volterra in 1931. Here we propose an extension of Goodwin's basic model by adding two important elements of the business cycle theory: (i) a finite time delay between investment orders and deliveries of finished capital goods, as theorized by Kalecki (1935); (ii) a delayed reaction of real wages to the unemployment levels, as suggested by Chiarella (1990). Both these delays preserve the two-dimensionality of the original model, but it becomes a delayed differential equation system, with two discrete time delays and one-delay dependent parameters. The qualitative study of the system shows that without lags the economic meaningful equilibrium is structurally stable. Nevertheless, as soon the time delays become positive, that equilibrium loses its stability and, according to the combinations of parameters and length of the lags, either periodic or non-periodic (chaotic) fluctuations arise. Numerical simulations supporting the economic analysis show that, in the very long run, a "strange attractor" depicts the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
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