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

A newly disclosed nonstandard finite difference method has been used to discretize a Lotka–Volterra model to investigate the critical normal form coefficients of bifurcations for both one-parameter and two-parameter bifurcations. The discrete-time prey–predator model exhibits a variety of local bifurcations such as period-doubling, Neimark–Sacker, and strong resonances. Critical normal form coefficients are determined to reveal dynamical scenarios corresponding to each bifurcation point. We also investigate the complex dynamics of the model numerically by Matlab package using MatcotM based on numerical continuation technique. The numerical continuation validates the theoretical analysis, which is discussed from an ecological perspective. 

Dynamics and bifurcations of a discrete‐time Lotka–Volterra model using nonstandard finite difference discretization method

Bifurcation analysis and complex dynamics of a Kopel triopoly model

Analytically and numerically, the study examines the stability and local bifurcations of a discrete-time SIR epidemic model. For this model, a number of bifurcations are studied, including the transcritical, flip bifurcations, Neimark–Sacker bifurcations, and strong resonances. These bifurcations are checked, and their non-degeneracy conditions are determined by using the normal form technique (computing of critical normal form coefficients). We use the MATLAB toolbox MatcontM, which is based on the numerical continuation method, to confirm the obtained analytical results and specify more complex behaviors of the model. Numerical simulation is employed to present a closed invariant curve emerging from a Neimark–Sacker point and its breaking down to several closed invariant curves and eventually giving rise to a chaotic strange attractor by increasing the bifurcation parameter.

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Dynamical Behaviors of an SIR Epidemic Model with Discrete Time

This work investigated the stochastic P-bifurcation (SPB) of a new class of unilateral impact vibration (VI) system under random parametric excitations, in which the elastoplastic stage of impact process is described by the restitution coefficient dependent with velocity model. By dint of the transformation of non-smooth, a new system without impact could be transformed from the original VI system. Subsequently, the stationary probability density functions (PDFs) of new system could be calculated by the associated equivalent nonlinear system. And then the critical parameters condition of SPB could be derived by means of the catastrophe theory. In the course of the study, it is verified that the new parameters induce the bifurcation of the system, which can be used to accurately measure the bifurcation behavior of the system quantitatively. And the system will tend to be stable under the precondition of enhancing the noise intensity. In addition, the validation of analytical results is examined by relative figures.

Stochastic P-bifurcation analysis of a novel type of unilateral vibro-impact vibration system

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.

Qizhi He

Papers

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