In this paper, the qualitative behavior of two discrete-time glycolysis models is discussed. The discrete-time models are obtained by implementing forward Euler’s scheme and nonstandard finite difference method. The parametric conditions for local asymptotic stability of positive steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time glycolysis model due to emergence of period-doubling and Neimark–Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.

Bifurcation analysis and chaos control in discrete-time glycolysis models

In this paper, complex dynamics of Kopel model with nonsymmetric response between oligopolists are discussed. The research show that for the nonsymmetric model fixed point may undergo fold, transcritical, pitchfork and Neimark–Sacker bifurcation when satisfying some specific parameter combinations. The research also show us that effects from the rivals may incur more complex dynamics than self-adjust. During the process of numerical simulation, different combinations of parameter are chosen to show multiple-period phenomenon, the change of stability and the transition of different dynamical behaviors. 

Complex dynamics of Kopel model with nonsymmetric response between oligopolists

In this article multiple and generic bifurcations of planar discrete-time Hindmarsh-Rose oscillator are investigated in detail by bifurcation theory and numerical continuation techniques. Three kinds of one-parameter bifurcation and five kinds of two-parameter bifurcation are studied. Different kinds of critical cases of each bifurcation are computed by inner product method and their corresponding scenario are presented, such as possible transitions between different one-parameter bifurcation points derived from two-parameter bifurcation point. Especially, the bifurcations of higher iterations and the bifurcation distributions of two-parameter bifurcation point are observed.

Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model

In this paper, a discrete-time prey-predator model with Allee effect is proposed. The parametric conditions for local asymptotic stability of steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark-Sacker bifurcations with the of help center manifold theorem and bifurcation theory. In order to control chaos due to emergence of Neimar-Sacker bifurcation, we apply OGY feedback control method and the hybrid control methodology is also implemented. Numerical simulations are provided to illustrate theoretical discussion.

Controlling chaos in a discrete-time prey-predator model with Allee effects

We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

In this paper, the two-parameter bifurcations of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes 1:2 and 1:4 resonances by using a series of affine transformations and bifurcation theory. The numerical simulations including phase portraits, two-parameter bifurcation diagrams, and maximum Lyapunov exponents diagrams for two different varying parameters in a three-dimensional space, not only illustrate the theoretical analysis, but also display the interesting and complex dynamical behaviors.

Bo Li

Papers

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh–Rose model