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Crystallographic and structural anisotropies are essential in governing the direction of crack propagation, particularly for brittle materials and their composites. However, capturing their combined effects and relative influence on crack-path formation at atomistic-continuum scales remains challenging. This paper presents a multiscale framework to determine the role of crystallographic anisotropy in controlling fracture in 3C-SiC and its composites. This framework decomposes the continuum media into a collection of "crystal-symmetry preserved sub-domains" (CSPS) before finite element discretization. Interactions and continuum scale behavior of the CSPS are described by continuum scale parameters determined from atomistic simulations. The framework reproduces all essential features of the atomic scale fracture, including bifurcation, arrest, renucleation, deflection, and penetration. Results reveal that "crystallographic anisotropy" controls the local anisotropy in the propagation pathway, whereas "structural anisotropy" controls the path deviation from the symmetry plane. The fracture pattern emerges from a competition between structural and crystallographic anisotropy effects and long-range elastic interactions among the stress-concentration sites. The underlying physics in high-symmetry configurations is well-explainable using "bifurcation diagrams." [ABSTRACT FROM AUTHOR]

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]

Inspired by the behaviour exhibited by certain biological systems, we demonstrate, through analytical solutions, a new coupling phenomenon in fibre-reinforced materials, which we term the dual Poynting effect. Specifically, we show that, in a transversely isotropic, nonlinear elastic prismatic body, constituted by a soft matrix embedded with sufficiently stiff fibres aligned along the prism axis, compression can induce a supercritical bifurcation at a specific stretch threshold. Both in the shear of a rectangular block and in the torsion of a cylindrical body, compressing beyond a bifurcation stretch threshold results in combined shear-extension or torsion-extension deformed configurations, where the object prefers to accommodate to minimize the total potential energy. Here, we rigorously analyse this effect within the framework of the nonlinear theory of hyperelastic fibre-reinforced materials, determine the constitutive requirements that make the dual solution energetically favoured, describe the (supercritical) bifurcation diagrams, and establish the corresponding stretch thresholds. [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]

This paper investigates two new Filippov-Gause predator-prey models that show the transition between individual and cooperative hunting dynamics among predators with respect to the critical prey population size. By using Filippov systems, this research offers a more realistic and complete representation of the nonlinear dynamics and discontinuities inherent in this complex ecological system so that the proposed models provide a more complete view of predator-prey interactions in different realistic environmental contexts. For the case where predators only cooperate in hunting when the prey population size is larger than its critical value, we could have at least one asymptotically stable limit cycle around a single positive inner equilibrium, or two limit cycles around the pseudo-stable equilibrium. However, in the case where predators only cooperate in hunting when the prey population size is less than its critical value, we could have one limit cycle around two inner equilibria and one pseudo-equilibrium that is unstable. In general, as the two proposed models differ in the presence of a sliding or escaping segment, the overall dynamics and bifurcation diagrams of both models change significantly. [ABSTRACT FROM AUTHOR]

Introducing dynamical systems that involve a variable in the trace of the Jacobian matrix is a difficult challenge due to it is difficult to determine whether a system is dissipative or conservative. This paper introduces a new 3D chaotic nonuniformly conservative system with a variable in the trace of the Jacobian matrix through the Hamiltonian form. The proposed system is without equilibrium points (hidden attractors) and satisfies categories C and D. The system exhibits three distinct behaviors (chaotic, quasi-periodic, periodic) under the same parameters with varying initial conditions. The characteristics system are investigated through a combination of theoretical analysis and numerical simulations, including dissipative and conservative behavior, equilibrium points, bifurcation diagrams, Lyapunov exponents, and multistability. Finally, two applications are implemented: an electronic circuit and image encryption based on the proposed system. These outcomes substantiate the sufficiency and viability of this system, demonstrating its effective performance. [ABSTRACT FROM AUTHOR]

Purpose: The structure having negative Poisson’s ratio (NPR) is termed as auxetic structures which is mainly used for energy absorption that is a crucial parameter to prevent structures under loading. The new auxetic sandwich nanocomposite with a core of carbon nanotube-reinforced polymer composite having negative Poisson’s ratio (CNTRC-NPR) has been proposed for energy absorption. The present work aims to obtain the nonlinear vibration behaviours of the proposed auxetic sandwich nanocomposite structures for studying the energy absorption and dissipation capacities. Method: Based on the classically laminated theory (CLT), a particular staking sequence of composite laminate has been achieved to obtain negative Poisson’s ratio of sandwich nanocomposite laminate. The present formulation for nonlinear vibration analysis of such auxetic structures is based on the Reddy’s third-order shear deformation theory and von-Karman nonlinear strain–displacement relations. The governing equations of motion of such auxetic structures are derived using the Hamilton’s principle. The obtained governing coupled nonlinear partial differential equations are converted to an ordinary nonlinear differential equation using the Galerkin’s method for the analysis of the out-plane displacement. Results: The present work addresses an in-depth study of various responses of nonlinear vibrations by obtaining time series responses, Poincaré maps, bifurcation diagrams, frequency responses and load–deflection curves. The effective out of plane NPR has been determined and analysed for different auxetic sandwich nanocomposite laminates. The jump-up ad jump-down phenomenon associated with the dynamics of such auxetic structures have been determined and discussed. The energy absorption and dissipation capacities of such auxetic structures have been determined from the obtained load–deflection curves. Conclusion: From the nonlinear vibration analysis, it is found that the auxetic sandwich structure is more flexible but produces less deflection than that of non-auxetic structures. It can be concluded from the bifurcation diagrams analyses that the period to chaotic vibration in the auxetic sandwich nanocomposite structures may occur for the variation of the amplitude of excitation. It is also observed that the auxetic sandwich nanocomposite structure has more energy-absorbing and dissipation capacity compared to the non-auxetic sandwich nanocomposite structures. [ABSTRACT FROM AUTHOR]

Analysis of a dynamic system helps scientists understand its properties and utilize it properly in different applications. This study analyzes the effects of various external excitements on a recently proposed mathematical neuron model derived from the original Fitzhugh–Nagumo model. Different bifurcation analyses on this system are conducted to detect chaotic behaviors that are common and of great importance in biological systems, considering the effects of different types of external excitements. Lyapunov exponents (LEs) confirm the existence of chaotic patterns. Furthermore, a bifurcation diagram that looks into the changes in the system dynamics caused by the simultaneous application of the external stimulants is represented. Neurons are bound to play a role in a network in which synchrony is an analytical quality. Therefore, the potential of a network of this model in showing synchronization is examined using the master stability function (MSF) technique. Ultimately, it is concluded that this neural model can produce chaotic behaviors and synchronous networks. [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 paper deals with a model of chemostat with two cross‐feeding species and involving two inhibitions. The excess of hydrogen inhibits the growth of acetogenic bacteria which, in turn, inhibit the growth of methanogenic hydrogenotrophic bacteria. The model is described by a system of four ordinary differential equations. We established the conditions of existence and stability of equilibria with respect to the operating parameters which are the dilution rate and the input substrate concentrations of the species of the model, then we illustrate the operating and bifurcation diagrams. This technique makes it easy to interpret different regions of operating diagrams. Inhibitions let some regions of stability rise and others vanish. [ABSTRACT FROM AUTHOR]

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel n-dimensional discrete chaotic map. First, the chaotic behaviors of the proposed map are studied. Next, to illustrate the effectiveness of our map, the 4D hyperchaotic map as an example is analyzed using the phase diagram, Lyapunov exponents, bifurcation diagrams and different types of entropy, etc. Moreover, the chaotic signals generated by the proposed map passed the NIST test and were implemented on the hardware DSP. Finally, we apply our chaotic map in image encryption using true numbers and extended XOR. The experimental results express that the presented encryption algorithm has a higher level of security than same other sophisticated methods. [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]

Bifurcations are one of the most remarkable features of dynamical systems. Corral et al. [Sci. Rep. 8(11783), 2018] showed the existence of scaling laws describing the transient (finite-time) dynamics in discrete dynamical systems close to a bifurcation point, following an approach that was valid for the transcritical as well as for the saddle-node bifurcations. We reformulate those previous results and extend them to other discrete and continuous bifurcations, remarkably the pitchfork bifurcation. In contrast to the previous work, we obtain a finite-time bifurcation diagram directly from the scaling law, without a necessary knowledge of the stable fixed point. The derived scaling laws provide a very good and universal description of the transient behavior of the systems for long times and close to the bifurcation points. [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]

These days, keeping information safe from people who should not have access to it is very important. Chaos maps are a critical component of encryption and security systems. The classical one-dimensional maps, such as logistic, sine, and tent, have many weaknesses. For example, these classical maps may exhibit chaotic behavior within the narrow range of the rate variable between 0 and 1and the small interval's rate variable. In recent years, several researchers have tried to overcome these problems. In this paper, we propose a new one-dimensional chaotic map that improves the sine map. We introduce an additional parameter and modify the mathematical structure to enhance the chaotic behavior and expand the interval's rate variable. We evaluate the effectiveness of our map using specific tests, including fixed points and stability analysis, Lyapunov exponent analysis, diagram bifurcation, sensitivity to initial conditions, the cobweb diagram, sample entropy and the 0-1 test. [ABSTRACT FROM AUTHOR]

In the present paper, by the forward Euler scheme, we propose a discrete-time Leslie-Gower type predator-prey model of ratio-dependent functional response and Michalis-Menten function prey harvesting. We not only investigate the existence and stability of equilibria of the model, but also derive codimension-one and codimension-two bifurcations of interior equilibria, including fold, flip, Hopf, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations by the center manifold method and bifurcation theorem. Moreover, we provide numerical simulations including bifurcation diagrams and phase portraits to illustrate the correctness of the obtained theoretical results and reveal the complex dynamic behaviours of discrete models. [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.

Full ceramic bearings provide exceptional characteristics, including elevated accuracy and substantial load-bearing capability, rendering them indispensable in high-end rotating machinery. Still, the long-term use of the bearing will lead to the gradual wear and defects of the cage pocket, which may affect the unstable operation of the rotor system. This study examines the vibration characteristics of cage failure through simulation and experiments. To study the dynamic characteristics of the bearing-rotor system, a dimensionless model is established by using the lumped mass method, and a cage failure model is added to the model, which can study the influence of cage failure on the system stability. Phase trajectories, Poincaré plots, and bifurcation diagrams are employed to analyze the impact of individual parameters on the nonlinear vibration of the system. Ultimately, to gather cage vibration data from two different types of faulty, the experimental approaches are used. The findings indicate that in the event of cage failure, the system's amplitude amplifies and the time-domain vibration signal exhibits distinct periodicity. Then, the vibration energy gradually focuses on low-frequency vibrations. Additionally, the spectrograms allow for easier identification of frequency components associated with the cage frequency, such as nfc and mfs ± nfc, the largest error is 3.32% when comparing the two models' simulated results with the experimental data. The model accurately depicts the frequency characteristics of failures resulting from wear and defects in full ceramic bearing cages. It offers crucial information for detecting failures and monitoring the health of full ceramic bearings. [ABSTRACT FROM AUTHOR]

This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models. [ABSTRACT FROM AUTHOR]

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]

This study is concerned with the influence of astronomical forcing and stochastic disturbances on non-linear dynamics of the Earth's climate. As a starting point, we take the system of climate equations derived by Saltzman and Maasch for late Cenozoic climate changes. This system contains variations of three prognostic variables: the global ice mass, carbon dioxide concentration, and deep ocean temperature. The bifurcation diagram of deterministic system shows possible existence/coexistence of stable equilibria and limit cycle leading either to monostability or bistability. Fitting the astronomical forcing by an oscillatory function and representing the deep ocean temperature deviations by means of white Gaussian noise of various intensities, we analyze the corresponding stochastic system of Saltzman and Maasch equations for the deviations of prognostic variables from their average values (equilibrium state). The main conclusions of our study are as follows: (i) astronomical forcing causes the climate system transitions from large-amplitude oscillations to small-amplitude ones and vice versa; (ii) astronomical and stochastic forcings together cause the mixed-mode climate oscillations with intermittent large and small amplitudes. In this case, the Earth's climate would be shifting from one stable equilibrium with a warmer climate to another stable equilibrium with a colder climate and back again. [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]

The article delves into the dynamics of natural convection in a vertical channel, specifically examining the formation of convection rolls and the emergence of different flow patterns. Through numerical simulations and bifurcation theory, the study explores the competition between three and four co-rotating rolls, leading to various bifurcation scenarios and transitions between steady, periodic, and chaotic dynamics. The research sheds light on the stability properties of fixed points and periodic orbits in the system, as well as the influence of Rayleigh numbers on the stability of flow patterns. The study contributes to a deeper understanding of complex convective flows and acknowledges the support of previous researchers and funding sources. [Extracted from the article]

The article "Path instabilities of freely falling oblong cylinders" delves into numerical simulations of the behavior of falling cylinders with varying length-to-diameter and density ratios. It identifies two transitional states, fluttering and weakly oscillating, influenced by the interplay of solid and fluid modes. The research sheds light on the scatter of drag coefficient values and oscillation frequencies, proposing correlations to fit simulation results. By exploring the complexities of path instabilities and fluid dynamics, the study contributes to a deeper understanding of the behavior of oblong cylinders in transitional regimes. [Extracted from the article]

The text explores the evolution of natural convection in a vertical channel, focusing on fixed points (FP) and periodic orbits (PO) in the system. It discusses various bifurcations, including homoclinic and heteroclinic bifurcations, leading to the termination of periodic orbits. The research aims to understand the dynamics of turbulent flows and the role of periodic orbits in chaotic systems, with a focus on fluid dynamics, convection, and turbulence. The document includes investigations into periodic orbits for Navier-Stokes flows, inclined layer convection, unstable periodic orbits, and standing waves in Couette-Taylor flow, among other topics. [Extracted from the article]

In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. [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]

The effects of a rigid barrier on the stability of the structure while it is undergoing phase-locked motions due to the surrounding fluid–structure interactions, are studied. The primary structure and the near wake dynamics are modeled as a harmonic oscillator and a Van der Pol oscillator, respectively, and are weakly coupled via acceleration coupling. Qualitative changes in the dynamical behavior of this system are investigated in the context of discontinuity-induced bifurcations that result from the interaction of fluid flow and nonsmoothness from the impact of the primary structure. Phenomenological behaviors like the co-existence of attractors and period-adding cascades of limit cycles separated by chaotic orbits are observed. The behavior of orbits in the local neighborhood of the discontinuity boundary is defined using a higher-order transverse discontinuity map, and the corresponding stability analysis is carried out using Floquet theory. The derived mapping is implemented to obtain the respective Lyapunov exponents. Results obtained using the mapping are demonstrated to accurately predict both the stable and chaotic regimes, as observed from the corresponding numerical bifurcation diagrams. [ABSTRACT FROM AUTHOR]

The global bifurcation diagrams for two different one-parametric perturbations ( + λ x and + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation x ′ = f (t , x) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work. [ABSTRACT FROM AUTHOR]

We investigate the influence of the finite Larmor radius on the dynamics of guiding-center test particles subjected to an E × B drift in a large aspect-ratio tokamak. For that, we adopt the drift-wave test particle transport model presented by Horton et al. [Phys. Plasmas 5, 3910 (1998)] and introduce a second-order gyro-averaged extension, which accounts for the finite Larmor radius effect that arises from a spatially varying electric field. Using this extended model, we numerically examine the influence of the finite Larmor radius on chaotic transport and the formation of transport barriers. For non-monotonic plasma profiles, we show that the twist condition of the dynamical system, i.e., KAM theorem's non-degeneracy condition for the Hamiltonian, is violated along a special curve, which, under non-equilibrium conditions, exhibits significant resilience to destruction, thereby inhibiting chaotic transport. This curve acts as a robust barrier to transport and is usually called shearless transport barrier. While varying the amplitude of the electrostatic perturbations, we analyze bifurcation diagrams of the shearless barriers and escape rates of orbits to explore the impact of the finite Larmor radius on controlling chaotic transport. Our findings show that increasing the Larmor radius enhances the robustness of transport barriers, as larger electrostatic perturbation amplitudes are required to disrupt them. Additionally, as the Larmor radius increases, even in the absence of transport barriers, we observe a reduction in the escape rates, indicating a decrease in chaotic transport. [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]