Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

This paper studies the Neimark–Sacker bifurcation of a diffusive food‐limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank‐Nicolson difference scheme, and nonstandard finite‐difference scheme. The existence of Neimark‐Sacker bifurcation at the equilibrium is obtained. Our results show that Crank‐Nicolson and nonstandard finite‐difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Complex dynamics of a discrete predator–prey model with the prey subject to the Allee effect

In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.     

Neimark–Sacker bifurcation of a third-order rational difference equation

In this paper, we investigate the existence and direction of the Neimark–Sacker bifurcation of a third-order rational difference equation with positive parameters. Firstly, it is found that there exists a Neimark–Sacker bifurcation when the parameter passes a critical value by analysing the characteristic equation. Secondly, the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived by using the normal form theory. Finally, computer simulations are performed to illustrate the analytical results found.     

Bifurcation analysis of the flour beetle population growth equations

In this article, we study a discrete delayed flour beetle population equation. Firstly, we study the existence of period-doubling bifurcation and Neimark–Sacker bifurcations for the system by analysing its characteristic equations. Secondly, we investigate the direction of the two bifurcations and the stability of the bifurcation periodic solutions by using normal form theory. Finally, some numerical simulations are carried out to support the analytical results.     

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Delayed feedback on the dynamical model of a financial system

We investigate the effect of delayed feedbacks on the financial model, which describes the time variation of the interest rate, the investment demand, and the price index, for establishing the fiscal policy. By local stability analysis, we theoretically prove the occurrences of Hopf bifurcation. Through numerical bifurcation analysis, we obtain the supercritical and subcritical Hopf bifurcation curves which support the theoretical predictions. Moreover, the fold limit cycle and Neimark–Sacker bifurcation curves are detected. We also confirm that the double Hopf and generalized Hopf codimension-2 bifurcation points exist.     

Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay

In this paper, a discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark–Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.     

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant . It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Neimark‐Sacker bifurcation of a fourth order difference equation

In this article, we study stability and bifurcation of a fourth order rational difference equation. We give condition for local stability, and we show that the equation undergoes a Neimark‐Sacker bifurcation. Moreover, we consider the direction of the Neimark‐Sacker bifurcation. Finally, we numerically validate our analytical results.     

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.     

Bifurcation analysis in a discrete differential-algebraic predator–prey system

The discrete-time predator–prey biological economic system obtained by Euler method is investigated. Some conditions for the system to undergo flip bifurcation and Neimark–Sacker bifurcation are derived by using new normal form of differential-algebraic system, center mainfold theorem and bifurcation theory. Numerical simulations are given to show the effectiveness of our results and also to exhibit period-doubling bifurcation in orbits of period 2, 4, 8 and chaotic sets. The results obtained here reveal far richer dynamics in discrete differential-algebraic biological economic system. The contents are interesting in mathematics and biology.     

Bifurcation analysis on a discrete model of Nicholson's blowflies

In this paper, the discrete Nicholson's blowflies model with delay is considered. First, the stability of the equilibria of the system is investigated by analyzing the characteristic equation and then the existence of fold and Neimark–Sacker bifurcations are verified. Subsequent to that, the direction and stability of the bifurcation are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out in order to support the results of mathematical analysis.     

Box dimension of Neimark–Sacker bifurcation

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case.     

Neimark–Sacker bifurcation analysis in an intraguild predation model with general functional responses

We analyse the dynamics of a discrete system coming from an intraguild food web model by using the average method. The intraguild predation model is formed by three populations corresponding to prey (P), mesopredator (MP) and superpredator (SP), where these last two populations are specialist. We give sufficient condition to guarantee the existence of a coexistence point at which the intraguild predation discrete model undergoes a Neimark–Sacker bifurcation independently of the functional responses that govern the interactions. We show numerical applications that consist in to assume that P has logistic growth and that the relation of MP–P is through a Holling type II functional response. Besides, we will consider that the interaction of MP–P is such that population MP has defense. The interaction of SP–P will be through a Holling functional response type III or IV. In particular, we give sufficient conditions to guarantee that the three species coexist. The techniques used to obtain the results can be applied to other models with different functional responses.     

Bifurcation analysis of discrete survival red blood cells model

In this paper, a discrete survival red blood cells model with delay is considered. Firstly, the stability of the equilibria of the system is investigated by analyzing the characteristic equation and then the existence of Neimark–Sacker and flip bifurcations are verified. Subsequent to that, the direction and stability of the Neimark–Sacker and flip bifurcations are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to support the results of mathematical analysis.     

Global bifurcations in a piecewise-smooth Cournot duopoly game

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu [2]. The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark–Sacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties differ significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.     

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population 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.