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A predator–prey model with Holling type-I and type-III functional responses, where the disease spreads between the prey, is considered in this paper. In consideration of the ecological balance, a harvest term is added to the predator. The positivity and boundedness of the solutions are discussed. Then, the conditions of the equilibrium points are analysed. According to the Routh–Hurwitz criterion, the local stability of equilibrium points can be analysed. For the disease-free equilibrium point, harvest rate h is selected as the bifurcation parameter. For the positive equilibrium point of the system, we choose infection rate b as the bifurcation parameter. By calculating and analysing the corresponding characteristic equations, the existence of Hopf bifurcation at equilibrium points is investigated. On the basis of high-dimensional bifurcation theory, we can obtain formulas which can decide the direction, period and stability of Hopf bifurcation of the system. To substantiate the theory, time history, bifurcation diagram and phase diagrams at different equilibrium points are drawn. In a disease-free environment, it may occur that the predator will prey on the prey in large numbers and eventually leads to the death of the prey. According to the numerical results, it can be seen that proper harvesting of predators is conducive to the stable development of the population. In a diseased ecology, when the infection rate experiences b ∗ , the stability of the system changes and the prey population can adapt to such changes better. It helps to eliminate some old and weak species to reduce the consumption of resources. [ABSTRACT FROM AUTHOR]

Currently, despite advances in the analysis of dynamical systems, there are still doubts about the transition between both stable and chaotic behaviors. In this research, we will explain the transition of a system that develops between two dynamic systems that have already been studied: the classical logistic model and a new chaotic system. This research addresses the study of the transition of both the system and its behaviors using computational techniques, where cobweb diagrams, time series, bifurcation diagrams, and even a graphical visualization for the maximum Lyapunov exponent will be visualized. Using a graphical and numerical methodology, bifurcation points were identified that revealed the transition of behaviors at different points. This resulted in a deep understanding of the dynamics of the system, thus highlighting the importance of incorporating computational analysis in dynamic systems, which greatly contributes to the efficient modeling of natural phenomena. [ABSTRACT FROM AUTHOR]

This article uses an approach based on the triad model–algorithm–program. The model is a nonlinear dynamic Selkov system with non-constant coefficients and fractional derivatives of the Gerasimov–Caputo type. The Adams–Bashforth–Multon numerical method from the predictor–corrector family of methods is selected as an algorithm for studying this system. The ABMSelkovFracSim 1.0 software package acts as a program, in which a numerical algorithm with the ability to visualize the research results is implemented to build oscillograms and phase trajectories. Examples of the ABMSelkovFracSim 1.0 software package operation for various values of the model parameters are given. It is shown that with an increase in the values of the parameter responsible for the characteristic time scale, regular and chaotic modes are observed. Further in this work, bifurcation diagrams are constructed, which confirm this. Aperiodic modes are also detected and a singularity is revealed. [ABSTRACT FROM AUTHOR]

In this paper, we study the shapes of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator { − ( u ′ (x) 1 − (u ′ (x)) 2 ) ′ = λ exp ⁡ (a u a + u) , − L < x < L , u (− L) = u (L) = 0 , where λ > 0 is a bifurcation parameter and a , L > 0 are evolution parameters. We determine the shapes of the bifurcation curves for different positive values a and L. [ABSTRACT FROM AUTHOR]

This study presents a novel approach by implementing an active memristor in a hyperchaotic discrete system, based on a cubic map, which is implemented by using two different microcontrollers. The key contributions of this work are threefold. The use of two microcontrollers with improved characteristics, such as speed and memory, for faster and more accurate computations significantly improves upon previous systems. Also, for the first time, an active memristor is used in a discrete-time system, which is implemented by using a microcontroller. Furthermore, the system is compared with two different types of microcontrollers regarding the execution time and the quality of the produced bifurcation diagrams. The proposed memristive cubic map uses computationally efficient polynomial functions, which are well suited to microcontroller-based systems, in contrast to more resource-intensive trigonometric and exponential functions. Bifurcation diagrams and a Lyapunov exponent analysis from simulating the system in Mathematica revealed hyperchaotic behavior, along with other significant dynamical phenomena, such as regular orbits, chaotic trajectories, and transitions to chaos through mechanisms like period doubling and crisis phenomena. Experimental verification confirmed the consistency of the results across microcontroller platforms, underscoring the practicality and potential applications of active memristor-based chaotic systems. [ABSTRACT FROM AUTHOR]

The isolated boost full-bridge converter (IBFBC) is a DC–DC conversion topology that achieves a high boost ratio and provides electrical isolation, making it suitable for applications requiring both. Its operational dynamics are often intricate due to its inherent characteristics and the prevalent usage of nonlinear switching elements, leading to bifurcation and chaos. Chaos theory was employed to investigate the impact of changes in the voltage feedback coefficient K and input voltage E on the dynamic behavior of the IBFBC when operating in discontinuous conduction mode (DCM). Based on an analysis of its operating principles, a discrete iterative mapping model of the system in DCM is constructed using the stroboscopic mapping method. The effects of two control parameters, namely the proportional coefficient and input voltage, on system performance are studied using bifurcation diagrams, fold diagrams, Poincaré sections, and Lyapunov exponents. Simulation experiments are conducted using time-domain and waveform diagrams to validate the discrete mapping model and confirm the correctness of the theoretical analysis. The results indicate that when the IBFBC operates in DCM, its operating state is influenced by the voltage feedback coefficient K and input voltage E . Under varying values of K and E , the system may operate in a single-period stable state, multi-period bifurcation, or chaotic state, exhibiting typical nonlinear behavior. [ABSTRACT FROM AUTHOR]

This paper skillfully incorporates the memristor model into a chaotic system, creating a two-dimensional (2D) hyperchaotic map. The system's exceptional chaotic performance is verified through methods such as phase diagrams, bifurcation diagrams, and Lyapunov exponential spectrum. Additionally, a universal framework corresponding to the chaotic system is proposed. To enhance encryption security, the pixel values of the image are preprocessed, and a hash function is used to generate a hash value, which is then incorporated into the secret keys generation process. Existing algorithms typically encrypt the three channels of a color image separately or perform encryption only at the pixel level, resulting in certain limitations in encryption effectiveness. To address this, this paper proposes a novel encryption algorithm based on 2D hyperchaotic maps that extends from single-channel encryption to multi-channel encryption (SEME-TDHM). The SEME-TDHM algorithm combines single-channel and multi-channel random scrambling, followed by local cross-diffusion of pixel values across different planes. By integrating both pixel-level and bit-level diffusion, the randomness of the image information distribution is significantly increased. Finally, the diffusion matrix is decomposed and restored to generate the encrypted color image. Simulation results and comparative analyses demonstrate that the SEME-TDHM algorithm outperforms existing algorithms in terms of encryption effectiveness. The encrypted image maintains a stable information entropy around 7.999, with average NPCR and UACI values close to the ideal benchmarks of 99.6169% and 33.4623%, respectively, further affirming its outstanding encryption effectiveness. Additionally, the histogram of the encrypted image shows a uniform distribution, and the correlation coefficient is nearly zero. These findings indicate that the SEME-TDHM algorithm successfully encrypts color images, providing strong security and practical utility. [ABSTRACT FROM AUTHOR]

To explore intermittency in discrete systems with two or more degrees of freedom, we analyze the general characteristics of type I intermittency within a two-dimensional map. This investigation is carried out numerically, concentrating on the system's attractors, bifurcation diagrams, and the characteristic relation associated with type I intermittency. We present two methods for determining the laminar interval and the channel structure. Our computations yield numerical results for the average laminar length as a function of the control parameter, which we then compare with findings from intermittency in one-dimensional maps. We observe a strong agreement between the numerical data and the theoretical predictions. [ABSTRACT FROM AUTHOR]

An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He's frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system's behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior. [ABSTRACT FROM AUTHOR]

Host–parasitoid systems are an essential area of research in ecology and evolutionary biology due to their widespread occurrence in nature and significant impact on species evolution, population dynamics, and ecosystem stability. In such systems, the host is the organism being attacked by the parasitoid, while the parasitoid depends on the host to complete its life cycle. This paper investigates the effect of parasitoid aggregation attacks on a host in a host–parasitoid model with self-diffusion on two-dimensional coupled map lattices. We assume that in the simulation of biological populations on a plane, the interactions between individuals follow periodic boundary conditions. The primary objective is to analyze the complex dynamics of the host–parasitoid interaction model induced by an aggregation effect and diffusion in a discrete setting. Using the aggregation coefficient k as the bifurcating parameter and applying central manifold and normal form analysis, it has been shown that the system is capable of Neimark–Sacker and flip bifurcations even without diffusion. Furthermore, with the influence of diffusion, the system exhibits pure Turing instability, Neimark–Sacker–Turing instability, and Flip–Turing instability. The numerical simulation section explores the path from bifurcation to chaos through calculations of the maximum Lyapunov exponent and the construction of a bifurcation diagram. The interconversion between different Turing instabilities is simulated by adjusting the timestep and self-diffusion coefficient values, which is based on pattern dynamics in ecological modeling. This contributes to a deeper understanding of the dynamic behaviors driven by aggregation effects in the host–parasitoid model. [ABSTRACT FROM AUTHOR]

This study examines a Caputo-type fractional-order food chain model, considering the Holling type II functional response with the vigilance effect. The model explores the interaction dynamics of the food chain model, which consists of prey, middle predators, and top predators. Additionally, habitat complexity is integrated into the model, which is assumed to reduce predation rates by lowering the encounter rates between predators and prey. All possible feasible equilibrium points are determined and the stability of our proposed model is explored near the equilibrium points. To support the analytical findings, numerical simulation results are given in terms of time series, phase portraits, and bifurcation diagrams. It is discovered that the proposed model can become more stable under a fractional-order derivative. Moreover, the interplay between the vigilance effect and habitat complexity is shown to influence the existence of stable and periodic dynamics. [ABSTRACT FROM AUTHOR]

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law's outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. [ABSTRACT FROM AUTHOR]

This paper uses the unified solver process to acquire soliton outcomes for the fractional Phi-4 model. The dynamic characteristic of the governing model is investigated for its planar dynamical system by applying the bifurcation method. Under the given parameters, 2D and 3D phase portraits, time series, return map, Lyapunov exponent, recurrence plot, strange attractor, bifurcation diagram, and fractal dimension plot are provided. These plots show the periodic, quasi-periodic, and chaotic nature of the suggested nonlinear problem. Moreover, the sensitivity and multistability assessments of the stated model are studied for a clear understanding of chaotic behavior. To understand the system's long-term behavior, we also test the stability of our results. Our results agree with previous results and may help researchers better understand the behavior of nonlinear systems. Furthermore, other fields such as biology, economics, and engineering can apply our results. [ABSTRACT FROM AUTHOR]

In this paper, we use the idea of dynamic coupling to describe the effect of drugs or chemical substances on the electrophysiological properties of the inferior olive neuron (ION). Therefore, a six-dimensional dynamically coupled Kazantsev master–slave configuration of the ION is obtained. In this configuration, the master and slave subsystems have an indirect interaction, and they are not connected through common signals; instead, the slave subsystem receives a coupling signal, which is dynamically generated by a second order linear system. This type of coupling is called dynamic coupling and enables us to take into consideration the state of the medium through which the master and slave are interconnected. The dynamical behavior of the new model is analyzed analytically using limit cycle prediction and numerically via the two-dimensional bifurcation diagrams with respect to two essential bifurcation parameters of the model. Taking the nonlinear function parameter a and two parameters from the coupling subsystem, the adaptive coupling parameter γ2 and the master–slave coupling strength b, as essential bifurcation parameters, the results show that a change of one of these parameters gives rise to complex dynamics such as periodic oscillations, period doubling scenarios, and chaotic states characterized by spike-bursting. Assuming two coupled neurons with parameter mismatch, it is observed that an increase in the external coupling strength ɛ1 favors their synchronization. Furthermore, the analog circuit of the complete new model confirms the burst analysis and the existence of chaos in the model. [ABSTRACT FROM AUTHOR]

The main objective of this study is to investigate the chaotic dynamics and analog circuit implementation of a nonautonomous single inertial neuron. First, the dimensionless mathematical model of such neuron is established. The equilibria with three kinds of stabilities are then depicted, and system symmetry and Hamilton energy function are analyzed, respectively. In numerical simulations, by using the two-dimensional/one-dimensional bifurcation diagrams, Lyapunov exponent spectra, phase plots, and basins of attraction, bifurcation of coexisting bubbles as well as bi-stable patterns are revealed, where the generated coexisting attractors are symmetric about the origin. In addition, we confirm that the complexity of the basins of attraction deteriorates with the increase of the frequency parameter. Finally, analog circuit experiments with few analog circuit elements are deployed, via which the bi-stable patterns of the model are well verified. [ABSTRACT FROM AUTHOR]

The Basener and Ross mathematical model is widely recognized for its ability to characterize the interaction between the population dynamics and resource utilization of Easter Island. In this study, we develop and investigate a discrete-time version of the Basener and Ross model. First, the existence and the stability of fixed points for the present model are investigated. Next, we investigated various bifurcation scenarios by establishing criteria for the occurrence of different types of codimension-one bifurcations, including flip and Neimark–Sacker bifurcations. These criteria are derived using the center manifold theorem and bifurcation theory. Furthermore, we demonstrated the existence of codimension-two bifurcations characterized by 1:2, 1:3, and 1:4 resonances, emphasizing the model's complex dynamical structure. Numerical simulations are employed to validate and illustrate the theoretical predictions. Finally, through bifurcation diagrams, maximal Lyapunov exponents, and phase portraits, we uncover a diversity of dynamical characteristics, including limit cycles, periodic solutions, and chaotic attractors. [ABSTRACT FROM AUTHOR]

В статье исследуется динамические режимы дробной системы Селькова с переменной наследственностью (памятью). Эффект переменной наследственности означает, что наследственность изменяется во времени, т.е. зависимость текущего состояния системы от предыдущих также зависит от времени. Переменная наследственность в дробной системе Селькова с точки зрения математики описываеься с помощью производных дробных переменных порядков типа Герасимова-Капуто. Дробная динамическая система Селькова исследуется с помощью численного метода Адамса-Башфорта-Мултона из семейства предиктор-корректор. С помощью численного алгоритма строятся различные бифуркационные диаграммы — зависимости полученного численного решения от различных значений параметров модельных уравнений. Численный алгоритм Адамса-Башфорта-Мултона и построение бифуркационных диаграмм были реализованы на языке Python в среде PyCharm 2024.1. Исследование бифуркационных диаграмм показало наличие не только регулярных режимов: предельных циклов и затухающих колебаний и хаотических колебаний, но и выявило сингулярность — неограниченный рост решения при изменении значений порядков дробных производных в модельном уравнении. Биффуркационные диаграммы могут содержат участки кривой со всплесками и без. Всплески могут указывать на релаксационные колебания или хаотические режимы, отсутствие всплесков соответвует затухающим колебаниям или апериодическим режимам.

In a recent paper [Belykh et al., 2024], we proved that the bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional Henon map. In this paper, we consider a similar problem for a generalized map which combines the Henon-type maps, the Poincaré return map for Shilnikov bifurcation of saddle-focus homoclinic orbit, the Lurie discrete time system, etc. We have obtained the expected result about the persistence of periodic orbits and their bifurcations when passing from a One-Dimensional (1D) endomorphism to the generalized map as a small parameter becomes nonzero. We have revealed the precise mechanism of change of homoclinic orbits and splitting of unstable manifolds as a result of the transition of 1D endomorphism to multidimensional map. Thereby we have derived the reconstruction rules of nonwandering set of orbits and bifurcations of the generalized map from those of 1D endomorphism. [ABSTRACT FROM AUTHOR]

In this paper, a novel locally active memristor is constructed. Based on the 3D Hindmarsh–Rose (HR) and 2D FitzHugh–Nagumo (FHN) neuron, a heterogeneous neuronal system is constructed by connecting the two neurons with the locally active memristor. The equilibrium point and stability of the system are investigated. The dynamic behavior of the system is numerically and experimentally revealed by utilizing dynamic analyses in terms of interspike interval bifurcation diagram, two-parameter bifurcation diagram and so on. The unique and abundant dynamic behavior is found in the proposed neuronal system by varying the coupling strength, external stimulus current, memristive parameter and other system parameters. Multiple bursting firing groups in the tongue-shaped domains have been discovered for the first time. Finally, in order to validate the numerical simulation, an analog equivalent circuit of the heterogeneous neuronal system is devised, which demonstrates that the system is physically realizable. [ABSTRACT FROM AUTHOR]

This paper proposed a brand‐new multidegree model of a single‐pair gear system with a bearing stiffness control module. By establishing the bending‐torsional coupled governing equation, the bifurcation characteristics, initial parameter attractors, solution domain structures, and double‐parameter stability were analyzed systematically and several conclusions were obtained. First, bifurcation diagrams show colorful characteristics, and the critical switching points and the routes between different motion states were investigated, and the bifurcation behavior under different parameters was examined which provided a comprehensive cognition of the gear system. Second, the dynamic initial motion attractors at different parameter groups were studied and their attractor structure diagrams were obtained, which offer an intuitive understanding of the motion state under different initial values, and were utilized as the theoretical basis for practical adjustment at different working conditions. Third, the solution domain structural diagrams were obtained at different parameters and the diverse characteristics could be observed, which could be utilized as a visualized tool for the design of the gear system and is referred as guiding for field operators to control the system away from unstable state as well. Finally, the double‐parameter stability of the gear system under different input power and damping was researched and its characteristics were analyzed. This stability region offered a direct tool to help both the designer and the operator to find the fastest and shortest route to make the system enter into a stable motion state quickly and effectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigated a three-dimensional chaotic system that models key aspects of a monetary system, including interest rates, investment demand, and price levels. The proposed system is described by a set of autonomous quadratic ordinary differential equations. We analyze the dynamic behavior of this system through equilibrium points and their stability, Lyapunov exponents (LEs), and bifurcation diagrams. The system demonstrates a variety of behaviors, including chaotic, periodic, and equilibrium states depending on parameter values. Additionally, we explore the multistability of the system and present a radial basis function neural network (RBFNN) controller design to stabilize the chaotic behavior. The effectiveness of the controller is validated through numerical simulations, highlighting its potential applications in economic and financial modeling. [ABSTRACT FROM AUTHOR]

The study of chaotic systems with special properties addresses one of the ongoing research topics. In this paper, we introduce a new class of third-order chaotic systems possessing a cyclic symmetry obeying unidirectional or mutual coupling. These new models are built by considering nonlinearities with three, four, or five segments and are distinguished from each other by the topological structure of the associated chaotic attractor. The projection of the novel chaotic attractors on certain planes reveals their multiscroll or multiwing character. For illustration, one of these models is studied in detail using analytical and numerical methods. The mechanism giving rise to the establishment of the chaotic regime starting from the state of equilibrium under the variation of a parameter is described using bifurcation diagrams, phase portraits, basins of attraction as well as the maximum Lyapunov exponent. Several zones of parameters are identified where the model experiences two or more attractors (i.e. multistability). In order to demonstrate the practical feasibility of the proposed circulant models, a series of experimental measurements are carried out using a physical implementation with an Arduino microcontroller. [ABSTRACT FROM AUTHOR]

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

In this study, we establish a fractional‐order direct drive permanent magnet synchronous wind turbines (DPMSG) model defined by Caputo based on the fractional calculus theory to overcome the singularity and limitations of integer‐order DPMSG models. The path and characteristics of the DPMSG system entering the bifurcation and chaos caused by the internal parameter changes and external disturbances were analyzed. First, we established a nonlinear fractional‐order mathematical model of a DPMSG system. Second, a bifurcation diagram was drawn using the maximum algorithm, and the path to chaos of the system at different orders was analyzed by combining its chaotic phase portrait and temporal sequence diagram. Subsequently, the impact of variations in the system order on the chaotic features of the original system was analyzed. The internal parameter adjustments of the system and changes in the system stability under external disturbances and other external excitations were analyzed. The influence of the system on its bifurcation phenomenon and chaotic behavior under multidimensional orders was determined, and it was observed that its path into chaos was opened by period‐doubling bifurcation. Lastly, the dual‐parameter stability domain of the system order corresponding to the internal parameters of the system was obtained by determining the parameter conditions for the critical stability of the system. [ABSTRACT FROM AUTHOR]

With the widespread application of chaotic systems in many fields, research on chaotic systems is becoming increasingly in-depth. This article first proposes a new dynamic model of financial risk contagion based on financial principles and discusses some basic dynamic characteristics of the new chaotic system, such as equilibrium points, dissipativity, Poincaré diagrams, bifurcation diagrams, etc. Secondly, with the consideration of privacy during data transmission, the method was designed to protect the privacy of controlled systems in finite time based on perturbation. A controller designed for finite time was developed based on Lyapunov stability principles, which achieves system synchronization within a finite time and protects the privacy of the controlled system. The effectiveness was also verified by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes an image encryption scheme based on an improved four-dimensional chaotic system. First, a 4D chaotic system is constructed by introducing new state variables based on the Chen chaotic system, and its chaotic behavior is verified by phase diagrams, bifurcation diagrams, Lyapunov exponents, NIST tests, etc. Second, the initial chaotic key is generated using the hash function SHA-512 and plain image information. Parity scrambling is performed on the plain image using the chaotic sequence generated by the chaotic system. The image is then converted into a hexadecimal character matrix, divided into two planes according to the high and low bits of the characters and scrambled by generating two position index matrices using chaotic sequences. The two planes are then restored to a hexadecimal character matrix, which is further converted into the form of an image matrix. Finally, different combined operation diffusion formulas are selected for diffusion according to the chaotic sequence to obtain the encrypted image. Based on simulation experiments and security evaluations, the scheme effectively encrypts gray images and shows strong security against various types of attacks. [ABSTRACT FROM AUTHOR]

This article explores the nonlinear dynamical analysis and control of a 2 − DOF system that emulates the lateral vibration of an asymmetric rotor model under external, multi-parametric, and mixed excitation. The linear integral resonant controller (L I R C) has been coupled to the rotor as an active damper through a magnetic actuator. The complete mathematical model, governing the nonlinear interaction among the rotor, controller, and actuator, is derived based on electromagnetic theory and the principle of solid mechanics. This results in a discontinuous 2 − DOF system coupled with two 1 / 2 − DOF systems, incorporating the rub-impact effect between the rotor and stator. The complicated mathematical model is investigated using analytical techniques, employing the perturbation method, and validated numerically through time response, basins of attraction, bifurcation diagrams, 0 − 1 chaotic test, and Poincaré return map. The main findings indicate that the asymmetric system model may exhibit nonzero bistable forward whirling motion under external excitation. Additionally, it can whirl either forward or backward under multi-parametric excitation, besides the trivial stable solution. Furthermore, in the case of mixed excitation, the rotor displays nontrivial tristable solutions, with two corresponding to forward whirling orbits and the other one corresponding to backward whirling oscillation. These findings are validated through the establishment of different basins of attraction. Finally, the performance of the L I R C in mitigating rotor vibrations and averting nonlinear catastrophic bifurcations under various excitation conditions. Furthermore, the rotor's dynamical behavior and stability are explored in the event of an abrupt failure of one of the connected controllers. The outcomes demonstrate that the proposed L I R C effectively eliminates dangerous nonlinearities, steering the system to respond akin to a linear system with controllable oscillation amplitudes. However, the sudden controller failure induces a local rub-impact effect, leading to a nonlocal quasiperiodic oscillation and restoring the dominance of the nonlinearities on the system's response. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical behavior of a discrete fractional-order modified Leslie–Gower model with a Michaelis–Menten-type harvesting mechanism and a Holling-II functional response. We analyze the existence and stability of the nonnegative equilibrium points. For the interior equilibrium points, we study the conditions for period-doubling and Neimark–Sacker bifurcations using the center manifold theorem and bifurcation theory. To control the chaos arising from these bifurcations, two chaos control strategies are proposed. Numerical simulations are performed to validate the theoretical results. The findings provide valuable insights into the sustainable management and conservation of ecological systems. [ABSTRACT FROM AUTHOR]

This study investigates inventory management systems using a sample of listed manufacturing companies in China from 2019 to 2023. By constructing a static mathematical model, the impact of inventory management on corporate performance was empirically tested. Additionally, based on a classical inventory management dynamical model and considering inventory delay characteristics, a new class of two-dimensional inventory management systems was reconstructed. The system's periodic and chaotic nonlinear characteristics were verified using 0-1 tests, bifurcation diagrams, Lyapunov exponents, and system eigenvalue plots. Furthermore, MATLAB simulations were employed to examine the effect of resource transfer rates on the nonlinear dynamic behavior of the inventory management system. The results from both static mathematical models and dynamical models provide a theoretical basis for inventory management and safety stock level predictions in the manufacturing industry. [ABSTRACT FROM AUTHOR]

We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair (G , B) , which consists of a cuspidal edge G in R 3 that contains a distinguishing regular curve B , are classified. This classification was used as a means to investigate the contact that a general cuspidal edge G equipped with a regular curve B ⊂ G has with planes. The singularities of the height functions on (G , B) are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge. [ABSTRACT FROM AUTHOR]

Rotor–stator whirling is a critical malfunction frequently encountered in rotating machinery, often resulting in severe damages. This study investigates the nonlinear dynamics of a whirling rotor interacting with a snubber ring through numerical simulations that account for the stiffness asymmetries of the snubber ring. A two-degrees-of-freedom (DOF) model is employed to analyse the contact interactions that occurred between the rotor and the snubber ring, assuming a linear elastic contact model. The analysis also incorporates the static offset between the centers of the rotor and the snubber ring. The dynamic behaviour of the whirling system is characterised by pronounced nonlinearity due to transitions between contact and non-contact states. The model is first validated against our prior theoretical and experimental studies. The nonlinear responses of the rotor are analysed to evaluate the effects of stator asymmetry through various techniques, including time-domain waveforms, frequency spectra, rotor orbits, and bifurcation diagrams. Furthermore, the influence of varying system parameters, such as rotational speed and the damping ratio, both with and without stator asymmetry, are systematically analysed. The results demonstrate that the rubbing response is highly sensitive to small variations in system parameters, with stator asymmetry significantly affecting system behaviour, even at low asymmetry levels. Direct stiffness asymmetry is shown to have a more pronounced effect than cross-coupling stiffness. The system exhibits a range of dynamics, including periodic, quasi-periodic, and chaotic responses, with regions of periodic orbits coexisting with chaotic ones. Complex phenomena such as period doubling, period halving, and jump bifurcations are identified, alongside quasi-periodic and period doubling routes to chaos. These findings contribute to a deeper understanding of the nonlinear phenomena associated with rotor–stator whirling and provide valuable insights into the unique characteristics of rubbing faults, which could facilitate fault diagnosis. [ABSTRACT FROM AUTHOR]

We consider the Paneitz type equation Δ 2 u − α Δ u + β (u − u q) = 0 on a closed Riemannian manifold (M n , g) of dimension n ≥ 3. We reduce the equation to a fourth order ordinary differential equation assuming that (M , g) admits a proper isoparametric function. Assuming that q > 1 , α and β are positive and α 2 > 4 β , we prove that the global nonconstant solutions of this ordinary differential equation only have nondegenerate critical points. Applying global bifurcation theory we then prove multiplicity results for positive solutions of the equation when q < p ⁎ , where p ⁎ = n + 4 n − 4 if n > 4 and p ⁎ = ∞ if n = 3 , 4. As an application and motivation we prove multiplicity results for conformal constant Q -curvature metrics. For example, consider closed positive Einstein manifolds (M n , g) and (X m , h) of dimensions n , m ≥ 3. Assuming that M admits a proper isoparametric function (with a symmetry condition) we prove that as δ > 0 gets close to 0, the number of constant Q -curvature metrics conformal to g δ = g + δ h goes to infinity. [ABSTRACT FROM AUTHOR]

This study explores the Ivancevic Option Pricing Model, a nonlinear wave-based alternative to the Black-Scholes model, using adaptive nonlinear Schrödingerr equations to describe the option-pricing wave function influenced by stock price and time. Our focus is on a comprehensive analysis of this equation from multiple perspectives, including the study of soliton dynamics, chaotic patterns, wave structures, Poincaré maps, bifurcation diagrams, multistability, Lyapunov exponents, and an in-depth evaluation of the model's sensitivity. To begin, a wave transformation is applied to convert the partial differential equation into an ordinary differential equation, from which soliton solutions are derived using the (G′G) method. We explore various forms of the option price function at different time points, including singular-kink, periodic, hyperbolic, trigonometric, exponential, and complex solutions. Furthermore, we simulate 3D surface plots and 2D graphs for the real, imaginary, and modulus components of some of the obtained solutions, assigning specific parameter values to enhance visualization. These graphical representations offer valuable insights into the dynamics and patterns of the solutions, providing a clearer understanding of the model's behavior and potential applications. Additionally, we analyze the system's dynamic behavior when a perturbing force is introduced, identifying chaotic patterns using the Lyapunov exponent, Sensitivity, multistability analysis, RK4 method, wave structures, bifurcation diagrams, and Poincaré maps. [ABSTRACT FROM AUTHOR]

Short-term synaptic plasticity is found in many areas of the central nervous system. In the inhibitory half-center central pattern generators involved in locomotion, synaptic depression is believed to act as a burst termination mechanism, allowing networks to generate anti-phase bursting patterns of varying periods. To better understand burst generation in these central pattern generators, we study a minimal network of two neurons coupled through depressing synapses. Depending on the strength of the synaptic conductance between the two neurons, this network can produce symmetric n : n anti-phase bursts, where neurons fire n spikes in alternation, with the period of such solutions increasing with the strength of the synaptic conductance. Relying on the timescale disparity in the model, we reduce the eight-dimensional network equations to a fully-explicit scalar Poincaré burst map. This map tracks the state of synaptic depression from one burst to the next and captures the complex bursting dynamics of the network. Fixed points of this map are associated with stable burst solutions of the full network model, and are created through fold bifurcations of maps. We derive conditions that predict the bifurcations between n : n and (n + 1) : (n + 1) solutions, producing a full bifurcation diagram of the burst cycle period. Predictions of the Poincaré map fit excellently with numerical simulations of the full network model and allow the study of parameter sensitivity for rhythm generation. [ABSTRACT FROM AUTHOR]

The paper is concerned with the analysis of a six-degree-of-freedom non-linear model which describes the vibrations of a rotor. The model takes into account the impacts between the rotating element and a limiter of motion. Using numerical integration and spectrum analysis, the influence of the excitation frequency, static loads, and the position of the limiter of motion on the type of vibrations of the system was studied. A multiparametric analysis has been performed to determine the areas of influence of two system parameters on the type of excited vibrations. Different types of vibration are further illustrated by plots of time histories, frequency spectra, phase portraits, stroboscopic portraits and bifurcation diagrams. The quality index of the system has also been determined and defined as the average value of the rotor kinetic energy. Depending on the parameters of the system, periodic, quasi-periodic or chaotic oscillations take place. The article primarily focuses on the risk of chaotic vibrations occurring in the system. [ABSTRACT FROM AUTHOR]

This paper studies the bounded confidence model on growing fully-mixed populations. In this model, in addition to the usual opinion clusters, significant secondary clusters of smaller size appear systematically, while those secondary clusters appear erratically and include much fewer agents when the population is fixed. Through simulations, we derive the bifurcation diagram of the growing population model and compare it to the diagram obtained with an evolving probability density instead of agents, and with their equivalent having a fixed population. Our tests, when changing the usual bounded confidence function into a smooth bounded confidence function, suggest that these secondary clusters are mainly generated by a different mechanism when the population is growing than when it is fixed. [ABSTRACT FROM AUTHOR]

Focusing on the nonlinear instability problem of the blended-wing-body aircraft, first, numerical bifurcation analysis is adopted to obtain catastrophe points and branches where the elevator is taken as a continuous parameter. Second, based on virtual flight tests and bifurcation theory, an experimental bifurcation analysis method is developed. Open-loop and closed-loop experiments are carried out, and the corresponding experimental bifurcation diagrams are obtained. The comparative analysis shows that the experimental bifurcation analysis enables the aircraft to make a response that is close to reality under the influence of nonlinear and unsteady aerodynamics and has a more intuitive embodiment of instability characteristics, providing an accurate prediction of departure. The open-loop experiment predicts a sudden longitudinal nose-up at an angle of attack of 7.4°. The closed-loop experiment can transform the open-loop unstable branch into a stable branch. However, roll instability still occurs at a critical angle of attack of 16.0°. These research results provide important dynamic information for the stable flight and the design of the controller. [ABSTRACT FROM AUTHOR]

This study aims to investigate a financial system consisting of four ordinary differential equations associated with the rate of interest, investment demand, price index, and the density of financial information gained by the population. The equilibrium and local stability of the system are investigated numerically. The impact of saving amounts and the rate of investment demand increases after getting financial information on the system are discussed. The findings of the study are verified graphically. It is found that the system becomes stable if the rate of investment demand increases after getting financial information kept at a certain level, such that the savings amount is maintained at a higher level. Also, the bifurcation diagrams of the system for various significant parameters that affect the system's stability have been depicted. [ABSTRACT FROM AUTHOR]

Different functional areas of the brain are composed of varying numbers of heterogeneous neurons, which transmit signals via synapses. As an electronic element, memristor can store resistance values and simulate the memory properties of synapses. At present, research on coupling heterogeneous neurons through memristive synapses is relatively scarce. A heterogeneous neural network model is proposed in this paper, where the memristor acts as a coupled synapse between a Two-Dimensional Hindmarsh–Rose (2D HR) neuron and a Three-Dimensional Hopfield Neural Network (3D HNN) neuron. The activation function of the neural network can simulate the firing behavior and nonlinearly transform the input signals. Therefore, an activation gradient is added to the 3D HNN neuron. In this paper, heterogeneous neurons and activation gradients are simultaneously introduced into the neural network. The model provides a more realistic artificial neural network, which can simulate the diversity of neurons in the brain. The firing behaviors varying with the coupling strength between the memristor and neurons are analyzed, and the influence of the activation gradient on the neural network model is studied. Various firing patterns are discovered, including chaotic/periodic spiking, chaotic/periodic bursting, stochastic bursting and transient chaotic spiking. By observing the Lyapunov Exponents (LEs) and bifurcation diagram under the initial values of two memristors, it is found that there are infinitely many attractors, indicating the extreme multistability firing in this heterogeneous neural network model. Finally, circuit design is implemented by Multisim simulation, and hardware implementation is completed on the Field-Programmable Gate Array (FPGA) platform. [ABSTRACT FROM AUTHOR]

This study focused on the dynamical behavior analysis of a discrete fractional Leslie-Gower model incorporating antipredator behavior and a Holling type Ⅱ functional response. Initially, we analyzed the existence and stability of the model's positive equilibrium points. For the interior positive equilibrium points, we investigated the parameter conditions leading to period-doubling bifurcation and Neimark-Sacker bifurcation using the center manifold theorem and bifurcation theory. To effectively control the chaos resulting from these bifurcations, we proposed two chaos control strategies. Numerical simulations were conducted to validate the theoretical results. These findings may contribute to the improved management and preservation of ecological systems. [ABSTRACT FROM AUTHOR]

Due to spatial scarcity and uncertainties in sediment data, initial and boundary conditions in deep-time climate simulations are not well constrained. On the other hand, depending on these conditions, feedback mechanisms in the climate system compete and balance differently. This opens up the possibility to obtain multiple steady states in numerical experiments. Here, we use the MIT general circulation model to explore the existence of such alternative steady states around the Permian–Triassic Boundary (PTB). We construct the corresponding bifurcation diagram, taking into account processes on a timescale of thousands of years, in order to identify the stability range of the steady states and tipping points as the atmospheric CO2 content is varied. We find three alternative steady states with a difference in global mean surface air temperature of about 10 °C. We also examine how these climatic steady states are modified when feedbacks operating on comparable or longer time scales are included, namely vegetation dynamics and air-sea carbon exchanges. Our findings on multistability provide a useful framework for explaining the climatic variations observed in the Early Triassic geological record, as well as some discrepancies between numerical simulations in the literature and geological data at PTB and its aftermath. [ABSTRACT FROM AUTHOR]

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart–Landau oscillator with time-delayed feedback. [ABSTRACT FROM AUTHOR]

We study the global bifurcation diagram of the positive solutions to the problem { Δ u + λ f (u) = 0 in B , u = 0 on ∂ B , where B is the unit ball in R N with N ≥ 3. Under general supercritical growth conditions on f (u) , we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f (u) , and we exhibit that a bifurcation curve for Δ u + λ f (u) = 0 has the same qualitative property as a classical Gel'fand problem Δ u + λ e u = 0 for N ≥ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

With the rise of the sharing economy, shared bicycles have become an important component of urban transportation. This paper explores the nonlinear dual oligopoly system for the Cournot model in the bike-sharing market; both operators have maximized profits as their competitive goals. The analysis of pivotal factors influencing passenger preferences, including pricing discounts and comfort levels, is meticulously depicted by a bifurcation diagram. A new chaotic attractor—the shared bicycle attractor—is discovered. The research results indicate that larger discounts and adjustment speeds can cause the system to be in a chaotic state, which is not conducive to the long-term development of operators, although discounts can indeed attract more passengers to a certain extent. On the other hand, the increase in the marginal cost of comfort loss can also make it difficult for enterprises to operate, which requires continuous technological innovation to improve the comfort of cycling. [ABSTRACT FROM AUTHOR]

Complex dynamical behaviors such as bifurcation and chaos exist in H-bridge inverter with RLC load, and these nonlinear behaviors will greatly increase the harmonic content of the output current and reduce the stability and reliability of the system. In this paper, a PI controller is added to widen the stable operation domain of the system. The stroboscopic mapping theory is used to model the system, the nonlinear dynamic behavior of the inverter is investigated by the bifurcation diagram, the folding diagram and the phase trajectory diagram are used for comparative verification, and the TDFC method is introduced to inhibit the chaotic behavior of the inverter, which further improves the stable range of the system operation. The fast-change stability theorem is used to analyze the stability of the system theoretically and verify the correctness of the numerical simulation. Therefore, the conclusions of the study provide a reliable theoretical basis for the design of the inverter system, which has important theoretical significance and practical value. [ABSTRACT FROM AUTHOR]

• An impact-damping model with multi-point contact and external load is proposed. • The effect of multi-point contact and external load on system impact is studied. • This impact-damping model is applied to the dynamics model of a spur gear pair. • The connection between gear system resonance and teeth impact is revealed. The damping term is an important component of the calculation model of the contact force of collision bodies because it simulates the energy dissipation of collision bodies during the contact-impact process. Therefore, an impact-damping model of a collision body with multi-point contact and external load is established by considering the restitution coefficient. The algorithm for solving this model is proposed based on iterative methods. Then, the accuracy of the established model is verified by the consistency between the calculated and tested results. The effect of multi-point contact and external load on the contact-impact process of the collision body is studied. The results show that the hysteresis damping factor increases from small to large when the collision body enters a stable state through repeated impacts. Further, the established damping model is applied to the dynamics model of a spur gear pair with backlash, extended teeth contact, and structure coupling effect of the gear body. The nonlinear dynamic characteristics, multi-state mesh, and teeth impact of the system are studied through the root-mean-square of transmission error, bifurcation diagrams with multi-mapping sections, and contact forces. The calculation results of the gear dynamics model with the impact-damping model are more consistent with the tested results than the calculation results of the gear dynamics model with the traditional damping model. Meanwhile, the connection between system resonance, response coexistence, teeth disengagement, and teeth impact is revealed. [ABSTRACT FROM AUTHOR]

This research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system's dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems. [ABSTRACT FROM AUTHOR]

Pattern formation in biological aggregations is a topic of great interest, due to the complex spatial structure of various aggregations of cells/bacteria/animals that can be observed in nature. While many such aggregations look similar at the macroscopic level, they might differ in their microscopic spatial structure. However, the complexity of the non-linear and sometimes non-local interactions among individuals inside these aggregations makes it difficult to investigate these spatial structures. In this study, we investigate numerically the transitions between different spatial patterns of animal aggregations with various symmetries (even, odd or no symmetry) that characterise the microscopic distribution of individuals inside these aggregations. To this end, we construct a bifurcation diagram starting with perturbations of spatially homogeneous solutions with low, medium, and high amplitudes. For perturbations with low amplitudes, the bifurcating structures show transitions among even-symmetric, odd-symmetric, and non-symmetric solutions. For perturbations with large amplitudes, there are wide parameter regions with non-convergent solutions, characterised by oscillatory transitions between different relatively similar solutions. These numerical results emphasize: (i) the effect of nonlinear and non-local interactions on the microscopically different symmetric/non-symmetric structures of macroscopically similar ecological aggregations; (ii) the difficulty of developing continuation algorithms for this class of non-local models. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical method for analvwing chaotic systems, focusing on applications to real-world problems. The Caputo-Fabrizio operator, a fractional derivative without a singular kernel, is used to investigate chaotic behavior. A fractional-order chaotic model is analvwed using numerical solutions derived from this operator, which captures the complexity of chaotic dynamics. In this paper, the uniqueness and boundedness of the solution are established using fixed-point theory. Due to the non-linearity of the system, an appropriate numerical scheme is developed. We further explore the model's dynamical properties through phase portraits, Lyapunov exponents, and bifurcation diagrams. These tools allow us to observe the system's sensitivity to varying parameters and derivative orders. Ultimately, this work extends the application of fractional calculus to chaotic systems and provides a robust methodology for obtaining insights into complex behaviors. [ABSTRACT FROM AUTHOR]