Narrow Results

To comprehend the dynamics of disease propagation within a society, mathematical formulation is a crucial tool to understand the complex dynamics. In order to transform the mathematical model with the objective of bolstering the immune system into a fractional-order model, we use the definition of Fractal-Fractional with Mittag-Leffler kernel. For an assessment of the stable position of a recently modified system, qualitative as well as quantitative assessments are carried out. We validate the property positivity and reliability of the developed system by evaluating its boundedness and uniqueness, which are important features of an epidemic model. The positive solutions with linear growth have been verified by the global derivative, and the level of effects of different parameters in each sub-section is determined through employing Lipschitz criteria. By employing Lyapunov’s first and second derivatives of the function, the framework is examined on a global scale to evaluate the overall effect with symptomatic and asymptomatic measures. Bifurcation analysis was performed to check the behavior of each sub-compartment under different parameters effects. The Mittag-Leffler kernel is used to obtain a robust solution via Fractal-Fractional operator for continuous monitoring of spread and control of cholera disease under different dimensions. Simulations are carried out to observe both the symptomatic and asymptomatic consequences of cholera globally, also to observe the actual behavior of cholera disease for control measures, and it has been confirmed that those with strong immune systems individuals recover early due to early detection measures. The actual state of cholera disease can be controlled by taking the following measures: early detection of disease for both individuals receiving medication and those who do not require medication because of their robust immune systems. This kind of research will be beneficial in determining how diseases spread and in developing effective control plans based on our validated findings.

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.

In this work, we build a two-dimensional dynamical fishery model in which the total harvest is obtained by a multiagent game with best reply strategy and naive expectations, i.e. each agent decides the harvest quantity by solving a profit maximization problem. Special attention is paid to the global dynamic analysis in the light of feasible domains (initial conditions giving non-negative trajectories converging to an equilibrium), which is related to the crisis of extinction. We also study the existence and stability of non-negative equilibria for models through mathematical analysis and numerical simulations. We discover the increase in the margin price of fish stock may lead to instability of the fixed point and make the system sink into chaotic attractors. Thus the fishery resource may fluctuate in a stochastic form.

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

In this paper, the dynamical behavior of a Hamiltonian system under impulsive control is discussed by both theoretical and numerical analyses. The existence and stability of its period-one and period-three solutions are obtained in virtue of a discrete map. The conditions of existence for flip bifurcation and Hopf bifurcation are further derived by using center manifold theorem and bifurcation theory. Three attracting invariant closed curves, surrounding three fixed points respectively, are investigated. Moreover, chaos in the sense of Marotto is rigorously proven. Finally, some detailed numerical results including periodic solutions, bifurcation diagrams, and chaotic attractors, are illustrated by examples, which are in good agreement with the theoretical analysis.

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.

We study the impact of the Allee effect and prey refuge on the stability of a discrete time predator–prey system. We focus on the stability behavior of the system with the Allee effect in predator, prey and both populations. Based on the combination of analytical and numerical results, we observe that: (1) the Allee effect stabilizes the systems dynamics in a moderate value of prey refuge. (2) For a large fraction of prey refuge no significant improvement in stability is observed due to Allee effect. (3) Refuge may play an important role in managing the populations which are subject to the Allee effect. The population remains stable at an intermediate level of refuge parameter, whereas at relatively low and high refuge effect, prey exhibits chaotic oscillation. Such chaotic behavior is suppressed in the presence of Allee effect. The Allee mechanism and refuge are considered simultaneously on the populations and is shown to have a significant impact on the predator–prey dynamics that may be helpful in the conservation of endangered species.

A three-dimensional discrete-time Hindmarsh–Rose model obtained by the forward Euler scheme is investigated in this paper. When the integral step size is chosen as a bifurcation parameter, conditions of existence for the fold bifurcation, the flip bifurcation, and the Hopf bifurcation are derived by using the center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Our research results demonstrate that the integral step size makes a difference corresponding to local and global bifurcations of the three-dimensional discrete-time Hindmarsh–Rose model. These results can supply a solid analytical basis to the study of Hindmarsh–Rose model, and it is necessary to illustrate how much the integral step size is adopted in advance when numerical solutions or approximate solutions of the original continuous-time model is concerned.

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

We study the intersection of double-flip (period-doubling) bifurcations in a parameter plane. We derive normal forms for discrete-time and continuous-time systems. Using these normal forms, we clarify the bifurcation structure around the flip-flip bifurcation point. We apply these analytical results to a system of coupled ventricular cell models. We determine the coexistence of in-phase and anti-phase two-periodic solutions. We make the simplest model for generating discordant alternans and clarify that two parameters (free concentration of potassium ions in the extracellular compartment and the conductance of the gap junction) play key roles in generating discordant alternans.

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.

A deceptively simple difference system of a bouncing ball is investigated. Applying the center manifold theorem and the normal form analysis, we first give the parameter conditions for the flip bifurcation. Then we discuss the 1:2 resonance. Because the nondegeneracy conditions are not satisfied, we use the approximation of mappings by a flow and change this difference system into an ordinary differential system with dimension two. With the aid of the Melnikov method, the homoclinic bifurcation and the Poincaré bifurcation are analyzed, which imply the existence of the invariant circles for the difference system. Furthermore, we compute the normal forms to provide the parameter conditions for the Chenciner bifurcation and also obtain the stability of the fixed point. Finally, some numerical simulations are presented to verify the obtained results.

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.

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, the one- and two-parameter bifurcations of a discrete-time prey–predator model with a mixed functional response are investigated by computing their critical normal form coefficients. The flip, Neimark–Sacker and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complex dynamical behavior of the model up to the 16th iteration is investigated.

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua’s circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds.

By using model discretization of the piecewise constant argument method, a discrete amensalism model with nonselective harvesting and Allee effect is formulated. The dynamic analysis of the model is studied and the existence and stability of the equilibrium point are discussed. The fold bifurcation and flip bifurcation at the equilibrium point of the system are proved by using the bifurcation theory and the center manifold theorem. In order to control flip bifurcation and restore the system to a stable state, a hybrid control strategy of parameter perturbation and state feedback is adopted. Finally, the effectiveness of the theoretical results and the control strategy is verified by numerical simulations.

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter $v_1$. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter $c_s$. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.