Narrow Results

In this paper, we discuss the stability and pattern formation issues of a spatiotemporal discrete system based on the modified Klausmeier model. We begin by constructing the corresponding coupled map lattices model. Then the existence and stability analysis is employed to derive the prerequisites for a stable homogeneous stationary state. Through the center manifold theorem and bifurcation theory, the threshold parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are individually determined. Based on the analysis of bifurcation, four pattern formation mechanisms are presented. Finally, we simulate the corresponding results numerically. The simulations exhibit rich dynamical behaviors, such as period-doubling cascades, invariant cycles, periodic windows, chaos, and rich Turing patterns. Four pattern formation mechanisms give rise to rich and complex patterns, including mosaics, spots, circles, spirals and cyclic fragmentation. The analysis and findings from this study enhance our comprehension of the intricate relationships among bifurcation, chaos and pattern formation for the spatiotemporal discrete Klausmeier model.

Local bifurcation control designs have been addressed in the literature for stationary, Hopf, and period doubling bifurcations. This paper addresses the local feedback control of the Neimark–Sacker bifurcation, in which an invariant closed curve emerges from a nominal fixed point of a discrete-time system as a parameter is slowly varied. The analysis of this bifurcation is more involved than for previously considered bifurcations. The paper develops the stability and amplitude equations for the bifurcated invariant curves of the Neimark–Sacker bifurcation, and then proceeds to apply these relationships in the design of nonlinear feedbacks. The feedback controllers are applied to two examples: the delayed logistic map and a model reference adaptive control system model.

In an already classical paper, May [1976] pointed out that simple nonlinear models in ecology can possess complicated dynamics. A long debate concerning the possible existence of chaotic ecologies has followed the appearance of this paper. Morris [1990] stressed that several problems exist when making conclusions regarding chaotic or nonchaotic dynamics as models are fitted to ecological data. Takens' theorem [1981] should, in principle, provide tools for solving some of these problems, but in reality essential assertions are most often violated. First, limits regarding the length of the used time-series must be assumed. Second, the theorem can be applied for evaluating positive Lyapunov exponents only, and is hence not applicable for distinguishing nonchaotic dynamics from chaotic.

We use a deterministic food-chain model that possesses a wide range of different dynamical patterns to demonstrate the existence of cases that are misclassified with respect to chaotic or nonchaotic motion as models are fitted to data in the various dynamical regimes. Our results are valid regardless of what finite data size is assumed. The results are best understood for high-periodic cases in the noise-free limit. If the relevant phase-space is not sufficiently well-populated with data in the vicinity of the periodic orbit, then sufficient complexity of the models are not supported by standard model selection criteria like AIC or GCV. Thus, the fitted models only in the best cases contain information concerning both the location of the periodic orbit and the eigenvalues of the Jacobians evaluated along the periodic orbit. If the space is moderately well populated with data, then the data is often described by an unstable periodic orbit of the fitted model. The attractor that was to be described by the fitted model was destabilized and is now a repeller. We call this phenomenon stability switches. It reminds about the noise-induced phenomenon reported by Rand and Wilson [1992], but we point out that the problem reported here is caused by the fitting procedure itself, not by the added noise.

The repeller that has been created by the model-fitting procedure can be located in the basins of attraction of some fixed point, the infinity, or some periodic or chaotic attractors of the fitted models. The situation seems similar when periodic attractors are not well enough populated with data for describing their location and when chaotic attractors are to be described by data. Similar stability switches occur and the dynamics of models fitted to data may differ or coincide with the dynamics of the attractor that generated the data in an unpredictable manner.

In this paper, we analyze a rather simple system in which some substance is being stored, released, and replenished simultaneously in some interdependent way. We investigate the dynamic behavior of such a system, using a two-dimensional map-based discrete-time model, and derive an integrated dynamical scene for this model. More specifically, we show the existence of an invariant curve induced by the well-known Neimark–Sacker bifurcation corresponding to the presence of a periodically oscillating behavior in this model.

This paper reports bifurcation dynamics of a discrete-time Kaldor model of business cycle. By using center manifold theorem and bifurcation theory, it is shown that the model not only undergoes flip bifurcation and Neimark–Sacker bifurcation, but also 1 : 1 resonance of codimension two bifurcation occurs. Some numerical examples are given to support the analytic results.

Using a discrete-delay nonlinear dynamic system, we model the time evolution of a stock market price index and net stock of savings in mutual funds. The proposed deterministic model has a unique steady-state, so its time evolution is determined by nonlinear effects acting out of equilibrium. For this model, we find the local stability properties and the local bifurcations conditions, given the parameter space. Specifically, we find that in both versions of the model (with and without delay) a Neimark–Sacker bifurcation can occur. Moreover, we show that the system without delay has a chaotic behavior. Finally, we formulate the associated discrete stochastic model and establish the conditions for asymptotic stability. Several numerical simulations are finally performed for both the deterministic and the stochastic model to justify the theoretical results.

This paper deals with two types of bifurcation behaviors of charged particles moving on rough surface under different damping effects. Based on the derived models, the stability of the particle system is judged by eigenvalue analysis and then the eigenvalue movement of the Jacobian matrix is analyzed to reveal the underlying mechanism of the dynamical evolution. It is shown that in the particle system with constant damping force, the system loses stability via Neimark–Sacker bifurcation, whereas in the system with time-dependent damping force, the stability is lost by way of period-doubling bifurcation. In addition, a powerful tool called manifold is employed to meticulously characterize the phase space so as to clearly describe the process of energy evolution, which leads to the inherent understanding of the complex behaviors and particularly the global dynamical properties in the particle system. Finally, some bifurcation diagrams are obtained to give a more evident explanation of complex behaviors. These results are very useful for the entire transport knowledge of charged particle system.

Three kinds of bifurcations of two coupled Rulkov neurons with electrical synapses are investigated in this paper. The critical normal forms are derived based on the center manifold theorem and the normal form theory. For the flip and the Neimark–Sacker bifurcation, the quartic terms and above in the normal forms are defined as higher order terms, which originate from the Taylor expansion of the original system. Then the effects of the quartic and quintic terms on the flip and the Neimark–Sacker bifurcation structure are discussed, which verifies that the normal form is locally topologically equivalent to the original system for the infinitesimal 4-sphere of initial conditions and tiny perturbation on the bifurcation curve. By the flip-Neimark–Sacker bifurcation analysis, a novel firing pattern can be found which is that the orbit oscillates between two invariant cycles. Two disconnected cardioid cycles also appear, which makes one, two, three, four, etc. turns happen before closure. Finally, we present a global bifurcation structure in the parameter space and exhibit the distribution of the periodic, quasi-periodic and chaotic firing patterns of the coupled neuron model.

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value ${\gamma }_0$ is predicted of the treatment parameter $\gamma$. We can see Neimark–Sacker bifurcation of the system when $\gamma ={\gamma }_0$, and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when $\gamma >{\gamma }_0$.

In this paper, we report results related with the dynamics of two discrete-time mathematical models, which are obtained from a same continuous-time Brusselator model consisting of two nonlinear first-order ordinary differential equations. Both discrete-time mathematical models are derived by integrating the set of ordinary differential equations, but using different methods. Such results are related, in each case, with parameter-spaces of the two-dimensional map which results from the respective discretization process. The parameter-spaces obtained using both maps are then compared, and we show that the occurrence of organized periodic structures embedded in a quasiperiodic region is verified in only one of the two cases. Bifurcation diagrams, Lyapunov exponents plots, and phase-space portraits are also used, to illustrate different dynamical behaviors in both discrete-time mathematical models.

In this paper, we consider the dynamics of a certain class of host-parasitoid models, where some hosts are completely free from parasitism either with or without a spatial refuge and the host population is governed by the Beverton–Holt equation. We assume that, in each generation, a constant portion of the host population may find a refuge and be safe from the attack by parasitoids. We derive some criteria for the Neimark–Sacker bifurcation. Then, we apply the developed theory to the three well-known cases: $(S)$ model, Hassel and Varley model, and parasitoid–parasitoid model. Intensive numerical calculations suggest that the last two models undergo a supercritical Neimark–Sacker bifurcation.

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.

The dynamics of a discrete nonlinear prey–predator model is studied. The local dynamical results are obtained on asymptotic properties of fixed points and Neimark–Sacker bifurcations. Then global dynamics is studied by finding invariant sets. Also some achievements on attraction are shown for certain trivial invariant sets including a shadowing type result, and estimates on boundedness of orbits. Some ergodic type results are also derived. Certain issues are extended to systems with impulses by showing the influence of impulses on dynamics. Moreover, backward dynamics is investigated as well. All these results are derived analytically and numerical computations are presented to support them.

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

In this paper, we explore the dynamics of a certain class of Beddington host-parasitoid models, where in each generation a constant portion of hosts is safe from attack by parasitoids, and the Ricker equation governs the host population. Using the intrinsic growth rate of the host population that is not safe from parasitoids as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability. Then, we apply the new theory to the following well-known cases: May’s model, $(S)$-model, Hassel and Varley (HV)-model, parasitoid-parasitoid (PP) model and $(H)$ model. We use numerical simulations to confirm our theoretical results.

In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system.

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.

In this paper, by averaging the growth rate on each state, we analyze the dynamics of a discrete dynamical system coming from a system of ODEs. This differential system corresponds to a tritrophic Leslie type model which is formed by three populations (prey (P), mesopredator (MP) and superpredator (SP)), where the last two populations are generalist predators. We give sufficient conditions where the discrete model undergoes a Neimark–Sacker bifurcation at a coexistence point. This analysis is independent of the functional responses that govern the interactions. To illustrate our results, several applications are given, under the assumptions that the population P has logistic growth and that the relations MP–P and SP–MP are carried out through Holling type functional responses. From these applications, we conclude that there are sufficient conditions to guarantee that the three species coexist by means of a supercritical Neimark–Sacker bifurcation. Moreover, numerically we can detect that the discrete system exhibits a chaotic behavior.