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This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]

Based on the feature of piecewise linear (PWL) functions, a nonlinear feedback of a higher-order system can be transformed to an approximately equivalent PWL function so as to ease system implementation in engineering. As an example, a cubic feedback term can be approximately equivalently transformed to be a PWL function. Since the PWL function can be expressed by many simple functions such as signum function and absolute-valued function, the cubic term can be approximately equivalently replaced with these functions. Consequently, the method of approximate equivalence is employed in the JCS-08-13-2022 (JCS) chaotic system for simple circuit design and implementation. In this approach, the widely used multipliers are avoided and the circuits become more economical and also more robust. In this paper, the cubic Chua's resistor is equivalently approximately replaced by a PWL function. To show the effectiveness of the approximate equivalence, numerical simulations are demonstrated and verified by circuit implementation. [ABSTRACT FROM AUTHOR]

A liquid bath vibrating vertically can lead to the emergence of a self-propelled walking droplet on its free surface, which can exhibit chaotic motion. It is well-known that trajectories of a chaotic system are sensitive to its initial condition, known as the "butterfly-effect", while its statistics normally remain stable to small disturbances: this type of chaos is called "normal-chaos". However, a concept called "ultra-chaos" has been recently introduced, whose statistical features are unstable, i.e. extremely sensitive to small disturbances. Up to now, a few examples of ultra-chaos have been reported. In this paper, the influence of tiny disturbances on the motion of walking droplet is investigated. It is found that both normal-chaos and ultra-chaos exist in the motion of the walking droplet. Different from the normal-chaotic motion, even the statistical properties of the droplet's ultra-chaotic motion are sensitive to tiny disturbances. Therefore, this illustrates once again that ultra-chaos indeed exists widely and represents a higher disorder compared with normal-chaos. The ultra-chaos as a new concept can widen our knowledge about chaos and provide us with a new point of view to study chaotic properties. It should be emphasized that, for an ultra-chaos, it is impossible to repeat any results of its physical experiments or numerical simulations even in the meaning of statistics! Unfortunately, reproducibility is a corner stone of modern science. Thus, the paradigm of modern scientific research might be invalid for an ultra-chaotic system. [ABSTRACT FROM AUTHOR]

A new conservative chaotic system with a cubic root is constructed in this paper. The system has no equilibria. An offset parameter is introduced into the system. By modulating different offset parameters and corresponding initial conditions, multiple coexisting conservative chaotic attractors can be generated consequently. By replacing the offset parameter with an appropriate sign function, multiscroll conservative chaotic attractors can be generated effectively by connecting multiple coexisting chaotic attractors based on offset boosting. Moreover, the number of scrolls tend to be infinite. Finally, a novel chaotic circuit with a cubic root is designed successfully by connecting two multipliers in opposite directions. The hardware implementation of the chaotic circuit is also completed to further verify the existence of conservative chaos and the consistency with the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complex dynamics of a mapping derived from a differential equation with simple time-periodic delay. Firstly, we calculate the truncated normal form of 1:1 resonance of the mapping at a degenerate fixed point and obtain an approximating system of the mapping by using Picard iteration. By analyzing the approximate system, we find that the mapping will undergo a 1:1 resonance at the degenerate fixed point. Secondly, the qualitative property and the stability of the degenerate fixed point are determined, which provide a new view to understand the dynamic of differential equation with simple time-periodic delay. However, the approximate system does not have the versal unfolding of the Bogdanov–Takens singularity of codimension 2. These phenomena show that simple time-periodic delay can support complex dynamics. Finally, a numerical simulation is carried out to verify the analytic results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a diffusive templator model in chemical self-replication, which describes the process of an individual molecule duplicating itself. Firstly, the stability of non-negative constant equilibrium solution is introduced. Then the existence of Hopf bifurcation is proved. Particularly, the stability and the direction of Hopf bifurcation for the spatially homogeneous model are discussed. Furthermore, by space decomposition and implicit function theorem, it is shown that the system may undergo a steady-state bifurcation with a two-dimensional kernel. Finally, several numerical simulations are completed to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit cycles from perturbed center (trivial and nontrivial equilibria) on the invariant algebraic surface of the unified Lorenz-type system (ULTS), which contains some common chaotic systems as its particular cases. First, based on the parameter-dependent center manifold, we obtain the approximate two-dimensional center manifold from the perturbation of invariant algebraic surface, as well as the two-dimensional system on this center manifold. Second, by applying the averaging method of third order to the above two-dimensional system, we show that under suitable perturbation of parameters of the ULTS, there is one limit cycle bifurcating from the perturbed center on the invariant algebraic surface of the ULTS, and the stability of this limit cycle is determined as well. By using the averaging method of fourth order, we show the same results with the averaging method of third order. Finally, numerical simulation is used to verify the theoretical analyses. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Brusselator system with cross-diffusion-like coupling and study Turing bifurcation of its stable Hopf bifurcating periodic solutions. According to the Brusselator system's diffusivity, a formula is established to determine the Turing bifurcation of periodic solutions. Additionally, for the Brusselator system with cross-diffusion-like coupling, a proof method for its periodic solutions to generate Turing bifurcation is given. Finally, the theoretical results are further verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study the spatiotemporal dynamics in a diffusive cell polarization model with source and loss terms under zero flux boundary conditions. Firstly, we investigate the stability of the positive equilibrium and the existence of Hopf bifurcation for the system without diffusion. Secondly, Turing instability and Turing bifurcation near the positive constant equilibrium are discussed for the diffusive system. Thirdly, the normal forms on the center manifold are given to determine the properties of Turing and Hopf bifurcations. Finally, numerical simulations are performed to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results. [ABSTRACT FROM AUTHOR]

At present, stochastic resonance (SR) based on Duffing and Langevin equations has been widely studied, but there is little research on SR phenomenon in higher-dimensional fractional-order systems. Based on the fractional-order Lorenz-like system, the SR phenomenon is studied from the perspective of equilibrium point in this paper. The dynamic process of SR phenomenon is presented, and several key parameters are derived. The results of numerical simulation show that the boundary of SR and chaos system will be broken if the order of the fractional-order system or internal parameters is properly adjusted, and the effect of SR phenomenon will also be influenced. This paper connects two important nonlinear phenomena (chaos and SR) based on the fractional-order system, which provides a new idea for the conversion between chaos and SR systems. [ABSTRACT FROM AUTHOR]

A power system shows chaotic oscillation when it is subjected to periodic load disturbance, which makes it a challenging and interesting problem for stability control of chaotic power system. In this paper, a unified controller is designed to solve the fixed-time and preassigned-time stability problems of fourth-order chaotic power systems. In addition, based on Lyapunov stability theory, sufficient conditions are established for the fixed-time and preassigned-time stability of the underlying chaotic power systems, and a more accurate estimation of the settling time is also derived. Finally, according to numerical simulations, we validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study one type of predator–prey model with simplified Holling type IV functional response by incorporating the fear effect into prey species. The existence and stability of all equilibria of the system are studied. And bifurcation behaviors including saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation of the system are completely explored. Numerical simulation is carried out to illustrate the theoretical analysis. It is shown that the fear effect does affect some dynamic behaviors of the system. Finally, we summarize the findings in a conclusion. [ABSTRACT FROM AUTHOR]

A delay partial difference equation with a delay controller is studied in this paper. We show that it can lead to both Devaney chaos and Li–Yorke chaos by applying the modified Marotto's theorem, with three criteria established for generating chaos. Computer simulations of chaotic behaviors and spatiotemporal graphs are performed with three examples. [ABSTRACT FROM AUTHOR]

In this paper, a four-dimensional chaotic system based on a flux-controlled memristor with cosine function is constructed. It has infinitely many equilibria. By changing the initial values x (0) , z (0) and u (0) of the system and keeping the parameters constant, we obtained the distribution of infinitely many single-wing and double-wing attractors along the u -coordinate, which verifies the initial-offset boosting behavior of the system. Then the complex dynamical behavior of the system is studied in detail through the phase portraits of coexisting attractors, the average value of state variables, Lyapunov exponent spectrum, bifurcation diagram, attraction basin and the complexity of spectral entropy (SE). In addition, the simulation of the Multisim circuit is also carried out, and the results of numerical simulation and analog circuit simulation are consistent. Finally, the chaotic sequence generated by the system is applied to image encryption, and according to the performance analysis, the proposed chaotic system has good security performance. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed host-generalist parasitoid diffusion model subject to homogeneous Dirichlet boundary conditions, where generalist parasitoids are introduced to control the invasion of the hosts. We construct an explicit expression of positive steady-state solution using the implicit function theorem and prove its linear stability. Moreover, by applying feedback time delay τ as the bifurcation parameter, spatially inhomogeneous Hopf bifurcation near the positive steady-state solution is proved when τ is varied through a sequence of critical values. This finding implies that feedback time delay can induce spatially inhomogeneous periodic oscillatory patterns. The direction of spatially inhomogeneous Hopf bifurcation is forward when parameter m is sufficiently large. We present numerical simulations and solutions to further illustrate our main theoretical results. Numerical simulations show that the period and amplitude of the inhomogeneous periodic solution increase with increasing feedback time delay. Our theoretical analysis results only hold for parameter k when it is sufficiently close to 1, whereas numerical simulations suggest that spatially inhomogeneous Hopf bifurcation still occurs when k is larger than 1 but not sufficiently close to 1. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction–diffusion–advection chemostat model with two species growing and competing for a single-limited resource. By taking the growth rates of the two species as variable parameters, we study the effect of growth rates on the dynamics of this system. It is found that there exist several critical curves, which may classify the dynamics of this system into three scenarios: (1) extinction of both species; (2) competitive exclusion; (3) coexistence. Moreover, we take numerical approaches to further understand the potential behaviors of the above critical curves and observe that the bistable phenomenon can occur, besides competitive exclusion and coexistence. To further study the effect of advection and diffusion on the dynamics of this system, we present the bifurcation diagrams of positive equilibrium solutions of the single species model and the two-species model with the advection rates and the diffusion rates increasing, respectively. These numerical results indicate that advection and diffusion play a key role in determining the dynamics of two species competing in a flow reactor. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear dynamical behaviors of a time-delayed supercavitating vehicle system varying with the system parameters, and proposes control strategies to stabilize the trajectory. A simplified dynamical model of the supercavitating vehicle is firstly constructed and then analyzed on its local and global stability in terms of different parameter values and initial conditions. Due to the forces resulting from the fins and the afterbody plane causing one of the system parameters to become a piecewise constant function of position, the system is described as a linear switched one with time delay. Both theoretical analysis and numerical simulations are adopted to describe the system behaviors. Several special phenomena such as bifurcations, deviation behaviors and period-doubling processes emerging from the system for a given set of parameters are discussed in detail in this paper. Besides, the equilibrium and chaotic solutions are accurately located by plotting the Lyapunov exponent map. Chaotic attractors present different strange forms with the variation of the parameters. Finally, based on the system's global property, the control and anti-control strategies are proposed to stabilize the system or generate desirable chaos and it is shown that the system can be controlled by selecting appropriate control parameters. [ABSTRACT FROM AUTHOR]

This paper presents a study of the existence conditions of limit cycles and mechanisms when losing their stability in an open-loop DC–DC boost converter loaded with a constant power load and operating in discontinuous conduction mode. First, the existence conditions are determined. Then, boundaries associated with different kinds of instabilities such as the classical smooth fold and period-doubling bifurcations and nonsmooth border-collision bifurcations such as fold border-collision bifurcations are elucidated. To perform the stability analysis, an implicit 1D map for the system is derived. From this model, it is obtained that smooth period-doubling and fold bifurcations may occur at certain values of the circuit parameters. Some results from numerical simulations performed on the circuit-based switched model are provided showing consistency with the theoretical analysis from the derived model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results. [ABSTRACT FROM AUTHOR]

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey. [ABSTRACT FROM AUTHOR]

In this paper, the Melnikov method for an abstract stochastic nonsmooth hybrid system is derived in detail and employed to study the homoclinic bifurcations and induced chaos in a bistable vibro-impact SD oscillator with bounded noise first proposed by Li et al. 2021, in which the geometric nonlinearity is used to produce the irrational restoring force and an impact map is introduced to describe energy loss during collisions. The initial Poincaré section is selected on the right rigid constraint so that rigorous perturbation analysis can be carried out to avoid the extension of trajectories near the switching manifold. The stochastic Melnikov function with geometrical intuition is obtained to detect the threshold of parameters for homocinic tangency of the perturbed stochastic stable and unstable manifolds by calculating the energy difference of their initial points on the Poincaré section. Finally, the analytical Melnikov analysis combined with numerical simulations is carried out to validate the developed Melnikov method for analyzing the global dynamics of the bistable vibro-impact SD oscillator subject to bounded noise. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics of the Sel'kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel'kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the hybrid-scale bifurcation evolutionary phenomena in one-cycle controlled (OCC) single-inductor dual-output (SIDO) Buck DC–DC converter are investigated thoroughly. First, the single-cycle and crossing-cycle iterative mapping models of the OCC SIDO converter in four operation modes are derived to describe the dynamical information under different topology switching sequences. Second, the evolutionary trend of hybrid-scale bifurcation behaviors is identified by numerical simulation. Third, bifurcation analysis based on the proposed models is performed by the trajectories of the eigenvalues, and significantly, the occurrence mechanism of hybrid-scale bifurcations in the evolutionary phenomena is explained in detail. And then, the participation factor is used to analyze the relationship between the variables and the eigenvalues. Especially, the relative parameter sensitivity of the iterative mapping model is proposed to evaluate the influence of such key parameters as load resistors on the eigenvalues so that plenty of design-oriented information will be provided to avoid the hybrid-scale bifurcation behaviors. Finally, PSpice circuit simulation and hardware experiments are carried out to verify the analysis results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Allee effect and predator competition in the Bazykin's model with Holling I functional response. Theoretically, we analyze the existence and stability of equilibria, and derive the existence conditions of saddle-node bifurcation and Hopf bifurcation. In addition, in order to determine the stability of limit cycles, we explicitly calculate the first Lyapunov coefficient and prove that the positive equilibrium is not a center, but a weak focus with a multiplicity of at least two. Therefore, the system has Hopf bifurcation and Bautin bifurcation with two limit cycles. Our results indicate that the Allee effect and predator competition lead to a series of complex dynamic phenomena. Finally, numerical simulation verifies the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a commensal model with Allee effect and herd behavior. Firstly, the stability of all possible equilibria are investigated. Secondly, using the Sotomayor's theorem, we prove the existence of saddle-node bifurcation and pitchfork bifurcation. Finally, all the theoretical predictions on the bifurcation are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

Memristive synaptic weight is a changeable connection synaptic weight. It reflects the self-adaption physical processing in biological neurons. To study its dynamical effect, this paper presents a memristive synaptic weight-based tabu learning neuron model. It is constructed by replacing the resistive self-connection synaptic weight in the tabu learning neuron with a memristive self-connection synaptic weight. The equilibrium point of the memristive tabu learning model is time-varying and switches between no equilibrium state and line equilibrium state with the change of the external current. Particularly, the stability of the line equilibrium state closely relies on the initial state of the memristor, resulting in the emergence of coexisting infinitely many nonchaotic attractors. By employing the bifurcation plots, Lyapunov exponents, and phase plots, this paper numerically reveals the initial state-switched coexisting bifurcation behaviors and initial state-relied extreme multistability, and thereby discloses the coexisting infinitely many nonchaotic attractors composed of mono-periodic, multiperiodic, and quasi-periodic orbits. In addition, PSIM circuit simulations and printed-circuit board-based experiments are executed and the coexisting infinitely many nonchaotic attractors are realized physically. The results well verify the numerical simulations. [ABSTRACT FROM AUTHOR]

Cellular automaton (CA) as a discrete and deterministic dynamic system has been widely used in cryptography. In this paper, a novel triple-coupling CA model is proposed. We first apply the triple-coupling model to 1D CA to design a high-quality pseudo-random number generator (PRNG). Numerical simulation and analyses show that triple-coupling CA exhibits more complex evolutionary behavior than uncoupling CA. The pseudo-random sequence generated by triple-coupling 1D CA can pass all NIST SP800-22 tests. Subsequently, the triple-coupling model is also applied to 2D CA, and a novel image encryption scheme based on this model is proposed. Thanks to the triple-coupling model, the diffusion operation can be completed between different color channels of the color image. Experimental simulations and extensive cryptanalysis show that the image encryption algorithm using the triple-coupling model has superior security and high efficiency. [ABSTRACT FROM AUTHOR]

Filippov systems have found applications in various fields. This paper mainly studies five Lotka–Volterra models of Filippov type, including a competitive system with linear interaction between two species, a competitive system with Holling type II or type III functional response and a symbiosis system with Holling type II or type III functional response. We investigate the stability of all equilibria and the boundary equilibrium bifurcations of these systems, either a persistence bifurcation or a nonsmooth fold bifurcation. We present the numerical simulation results for each case. Consequently, based on both theory and simulation, we analyze the ecological aspects under the intervention of harvesting at different prescribed thresholds. [ABSTRACT FROM AUTHOR]

Phytoplankton and zooplankton's interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplankton should be in steady state but in previous eras, there have been a universal explosion in destructive plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So in this paper, we studied dynamical characteristics of a discrete-time phytoplankton–zooplankton model with Holling type-II predational functional response and toxic substance distribution. More specifically, we studied local stability with topological classification at equilibrium solutions, periodic points and possible bifurcation analysis of the model under consideration. It is proved that at trivial and semitrivial equilibrium solutions the phytoplankton–zooplankton model does not undergo flip bifurcation but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution. It is also proved that the interior equilibrium solution model does not undergo the fold bifurcation. We also studied chaos in the sense of feedback control strategy. Finally, extensive numerical simulations are provided to demonstrate theoretical results. [ABSTRACT FROM AUTHOR]

A fractional-order three-species food chain ecosystem with prey refuge and Holling-II type functional response for predation is proposed and studied. Several sufficient conditions for the existence and uniqueness of the solution of the fractional-order system are obtained. The boundedness of the solution of the system is proven. We investigate the asymptotic behavior of the model by using eigenvalue analysis, and some sufficient conditions on local asymptotic stability of the equilibrium points are given. Furthermore, the conditions for the occurrence of bifurcation at some equilibrium points are presented. We find that the order of the proposed fractional-order ecosystem is one of the parameters for its bifurcation. Several numerical simulations are provided to show the effectiveness of our findings in this paper. Lastly, some new numerical simulations are given to discuss the influence of the half-saturation constant, prey refuge coefficient and the order of fractional-order derivative on the stability of the discussed fractional-order system. [ABSTRACT FROM AUTHOR]

Initial condition-dominated offset boosting provides a special channel for coexisting orbits. Due to the nonlinearity and inherent periodicity, sinusoidal function is often introduced into a dynamical system for multistability design. Typically, the distance between two attractors or two petals of an attractor is fixed. Moreover, any chaotic signal and sequence need to be modified with amplitude and offset for a real application. In this paper, an initially-controlled double-scroll hyperchaotic map is constructed based on two sine functions. Four patterns of the double-scroll hyperchaotic orbits are found as 0-degree, 90-degree, 45-degree and 135-degree. Consequently, different modes for attractor growing are demonstrated. In this case, all the coexisting attractors are arranged in phase space in a direction defined by the initial value and the distance between two petals of any double-scroll orbit is adjusted. Finally, hardware experiments based on STM32 are carried out to verify the theoretical analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

There is a problem of capacity mismatch between different companies in the current market. In this paper, we propose a dual-channel capacity sharing supply chain consisting of two competing manufacturers (i.e. a capacity seller and a capacity buyer) and one platform. Considering the gap of matching success rates, we investigate whether the buyer and seller can benefit from capacity sharing and how the different matching success rates influence the manufacturers' decisions on spontaneously match through a platform. Through numerical simulation, we find that (1) the seller can always benefit from capacity sharing even when the match fails, while the buyer benefits from capacity sharing only when the capacity constraint is relatively severe; (2) both the buyer and seller prefer to participate through the platform when the platform has a sufficiently high advantage in matching success rates; (3) if the capacity constraint is much severe, the capacity buyer may turn to self-match rather than capacity sharing. Besides, we also study the sustainability of the capacity-sharing system and show how the platform can improve sustainability by forgoing accumulated profits. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter p = p ∗ , the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when p = p ∗ , the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014]. [ABSTRACT FROM AUTHOR]

This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed. [ABSTRACT FROM AUTHOR]

In this paper, we present a study on a mutual interference host-parasitoid model with Beverton–Holt growth. It is well known that, mutual interference of parasites has a stabilizing influence on the dynamics of the host-parasitoid model since the variance in searching efficiency, with parasite density, significantly depends on parasites' mutual interference. Thus, we have incorporated a mutual interference functional response into a host-parasitoid model to characterize such a phenomenon. The qualitative behaviors of the present model is investigated in this paper. Firstly, the existence and local stability of the model fixed points are discussed. Then, using perturbation method and normal form theory, we derived the emergence conditions of Neimark–Sacker bifurcation of the model. Furthermore, chaotic behavior of the model in the sense of Marotto is proved. In order to control chaotic behavior of the present model, we apply OGY feedback control strategy. Finally, numerical simulations are provided to support our theoretical discussion. [ABSTRACT FROM AUTHOR]

This paper studies chaos and chaotification for a class of first-order partial difference equations with a finite or infinite system size by using the theory of heteroclinic cycles connecting repellers. Three criteria of chaos induced by regular and nondegenerate heteroclinic cycles connecting repellers or regular heteroclinic cycles connecting repellers are established, respectively. Especially, one chaotification theorem for general discrete systems in the finite-dimensional space Y k or the infinite-dimensional space l ∞ is established, which is also based on heteroclinic cycles connecting repellers. Then, by using this result, two chaotification schemes for first-order partial difference equations are established. For illustrating the existence of chaos and the validity of chaotification schemes, three examples are provided with computer simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the evolution of an ensemble of noninteracting classical particles confined in a closed region. Each one of these particles experiences several elastic collisions with a rigid-smooth oval-like boundary. As known in the literature, numerical simulations for this kind of system demand too much time, mostly because of the time spent in solving transcendental equations. To deal with that, we highlight an alternative approach to speed up the search of solutions for the transcendental equations via the tangent method, which has proved to be a faster way to solve this problem when compared with other methods, like the bisection one. The methods applied here are generic and can be used or adapted for other billiard boundaries. [ABSTRACT FROM AUTHOR]

One type of tree-grass competition model with cross-diffusion is analyzed in this paper. Firstly, according to the distribution theory of the roots of the characteristic equation, the existence and stability conditions of the positive equilibrium point are obtained, and the global stability of the system is proved by using the Bendixson–Dulac theorem. Secondly, we select the fire factor as the control parameter, and obtain the critical value of Turing bifurcation and the conditions of Turing pattern. Then, we derive the amplitude equation near the Turing bifurcation point by using the standard multiscale analysis method. Finally, we do a sensitivity analysis on the tree-grass system with the fire factor, and then a series of numerical simulations based on these analysis results are carried out to verify our theoretical analysis and Turing pattern structure, which is consistent with the biological significance of the tree-grass model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Leslie–Gower predator–prey model with Allee effect on the prey and a linear functional response. Here the Allee effect impacts the birth rate of the prey, which is different from the common multiplicative and additive Allee effects. The model is well-posed, that is, all solutions are bounded. Applying the blow-up method indirectly, we prove that the origin which is not an equilibrium of the system is an attractor. Then we study the existence and stability of equilibria, which indicate that the system undergoes bifurcations. With the help of Sotomayor's theorem, we show the occurrence of saddle-node bifurcation. Moreover, there is degenerate Hopf bifurcation of codimension at least three. By choosing two (three) parameters of the system as bifurcation parameters and calculating a versal unfolding near the cusp, we demonstrate that the system undergoes Bogdanov–Takens bifurcation of codimension two (three). These theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a boost converter emulator with a memristive load instead of a resistive load is proposed, and a fractional-order model of the memristive boost converter is created. Firstly, the fractional-order models of the memristor and the memristive boost converter are established respectively, and then based on different switching states, the circuit equations of the fractional-order memristive boost converter operating in continuous conduction mode are derived. Secondly, the dynamic characteristics of the fractional-order system are analyzed by numerical simulations and are compared with the integer-order memristive boost converter. Furthermore, the effects of the memristive load on the system is studied by comparing it to the traditional boost converter with resistive load. The results indicate that the fractional-order memristive boost converter can exhibit rich dynamic behavior by adjusting bifurcation parameters, and the fractional-order circuit system expands the stable working regions of the integer-order circuit system. Meanwhile, the addition of the memristive load significantly widens the normal working regions. Finally, simulations of the fractional-order system circuit based on PSIM further verify the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system. [ABSTRACT FROM AUTHOR]

In this paper, we study a dual-channel closed-loop supply chain (CLSC) consisting of one manufacturer, one retailer and one third-party firm or platform (3P). The manufacturer wholesales new products through the traditional retail channel and distributes remanufactured products via 3P. We focus on establishing the dynamic Stackelberg game models for nondelayed and delayed cases, respectively. The existence and local stability of Nash equilibrium are examined as well as the complex dynamical behaviors of each model under various scenarios are investigated by numerical simulations, such as stability region, bifurcations, chaos, strange attractors, and so on. Moreover, the impacts of some key parameters on the performance of chain members are analyzed. In addition, the variable feedback control method is utilized to eliminate the system chaos. The results reveal that the high value of the consumer discount perception for remanufactured products and excessively fast price adjustment speed have a destabilization effect on the Nash equilibrium point. In addition, adopting delay decisions by manufacturer does not always make the system more stable because it can exert either positive or negative effect on the system's stability, while an intermediate delay weight is conducive to the system have a higher chance to stay stable. Furthermore, the manufacturer's profits will be declined significantly while the profits of retailer and 3P will be elevated to some extent when the system falls into periodic and chaotic motions, so chaos is not always necessarily detrimental to all the decision makers in the dual-channel CLSC. [ABSTRACT FROM AUTHOR]

In this paper, a sliding-mode-based controller is designed for finite-time synchronization of Memristor Chaotic Systems (MCSs). It demonstrates that the synchronization errors of the MCSs reach the designed sliding-mode surface within a finite time and the finite-time stability is achieved on the surface, which implies that the finite-time synchronization for MCSs is achieved by employing the proposed sliding-mode-based controller. Furthermore, a new image encryption algorithm is proposed and implemented based on the results of finite-time synchronization. Finally, the numerical simulation and the corresponding statistical performance analysis are presented to verify the practicability, effectiveness and superiority of the presented sliding-mode-based controller and encryption algorithm, especially their potential applications in secure communication. [ABSTRACT FROM AUTHOR]