Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-cha^tic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-cha^tic system is useful in the secure communication.

DYNAMICAL BEHAVIOR OF AN INNOVATION DIFFUSION MODEL WITH INTRA-SPECIFIC COMPETITION BETWEEN COMPETING ADOPTERS

In this paper,we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation(product)in a particular region.The model exhibits two equilibria,namely,the adopter-free and an interior equilibrium.The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number(BIN)R_(A).It is investigated that the adopter free steady-state is stable if R_(A)<1.By consideringτ(the adoption experience of the adopters)as the bifurcation parameter,we have been able to obtain the critical value ofτresponsible for the periodic solutions due to Hopf bifurcation.The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem.Exhaustive numerical simulations in the support of analytical results have been presented.

STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage （time delay） is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period （time delay, T） passes through a critical value. Applying the normal form theory and the center manifold argument, we de- rive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Traveling Waves for an Epidemic Model with Spatio-Temporal Delays

This paper is concerned with traveling waves for an epidemic model with spatio-temporal delays. It is shown that the traveling wave solutions are still persistently existing in the model when the non-locality is caused by small average time delays via the geometric singular perturbation theory and the center manifold theorems.

Bifurcations and chaos control in a discrete-time biological model

In this papcr,bifurcations and chaos control in a discrete-time Lotka-Volterra predator-prey model have been studied in quadrant-I.It is shown that for all parametric values,model hus boundary equilibria:P00(0,0),Px0(1,0),and the unique positive equilibrium point:P^+xy(d/c,r(c-d)/bc) if c>d.By Linearization method,we explored the local dynamics along with different topological classifications about equilibria.We also explored the boundedness of positive solution,global dynamics,and existence of prime-period and periodic points of the model.It is explored that flip bifurcation occurs about boundary equilibria:Poo(0,0),P.o(1,0),and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of P^+xy(d/c,r(c-d)/bc).Further,it is also explored that about P^+xy(d/c,r(c-d)/bc) the model undergoes a N-S bifurcation,and meanwhile a stable close invariant curves appears.From the perspective of biology,these curves imply that betwecn predator and prey populations,there exist periodic or quasi-periodic oscillations.Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23.The Maximum Lyapunov exponents as well as fractal dimension are computed numeri-cally to justify the chaotic behaviors in the model.Finally,feedback control method is applied to stabilize chaos existing in the model.

Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and her- bivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilib- rium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.