Integral Step Size Makes A Difference To Bifurcations Of A Discrete-Time Hindmarsh-Rose Model

A three-dimensional discrete-time Hindmarsh-Rose model obtained by the forward Euler scheme is investigated in this thesis. When the integral step size is chosen as a bifurcation parameter, conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Great difficulties appear during the discussion of the Hopf bifurcation because the common center manifold theorem can not clearly present the conditions of existence in our discrete-time Hindmarsh-Rose model. So we find a useful criterion of Hopf bifurcation that can solve the problems. Besides, this criterion does not need any eigenvalue properties. Sufficient conditions for the existence of Hopf bifurcation of n-dimensional systems have been given just by simple calculations of parameters and coefficients. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Research results in this paper 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 an analytical basis to the study of discrete-time Hindmarsh-Rose model in future, 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.