Dynamics of a neuron model in different two-dimensional parameter-spaces

We report some two-dimensional parameter-space diagrams numerically obtained for the multi-parameter Hindmarsh-Rose neuron model. Several different parameter planes are considered, and we show that regardless of the combination of parameters, a typical scenario is preserved: for all choice of two parameters, the parameter-space presents a comb-shaped chaotic region immersed in a large periodic region. We also show that exist regions close these chaotic region, separated by the comb teeth, organized themselves in period-adding bifurcation cascades.     

Changes in the dynamics of two-dimensional maps by external forcing

We investigate changes in periodicity, and even its suppression, by external periodic forcing in different two-dimensional maps, namely the Hénon map and the sine square map. By varying the amplitude of a periodic forcing with a fixed angular frequency, we show through numerical simulations in parameter-spaces that changes in periodicity may take place. We also show that windows of periodicity embedded in a chaotic region may be totally suppressed.     

Hyperchaotic states in the parameter-space

In this paper we propose a numerical method to characterize hyperchaotic points in the parameter-space of continuous-time dynamical systems. The method considers the second largest Lyapunov exponent value as a measure of hyperchaotic motion, to construct two-dimensional parameter-space color plots. Different levels of hyperchaos in these plots are represented by a continuously changing yellow-red scale. As an example, a particular system modeled by a set of four nonlinear autonomous first-order ordinary differential equations is considered. Practical applications of these plots include, by instance, walking in the parameter-space of hyperchaotic systems along desirable paths.