Hopf bifurcation, offset boosting and remerging Feigenbaum trees in an autonomous chaotic system with exponential nonlinearity

Abstract

The present paper executes an analysis of a relatively simple chaotic system with exponential nonlinearity. This dimensionless dynamic system has been obtained from the chaotic circuit introduced by Ma and collaborators (nonlinear Dyn. https://doi.org/10.1007/s11071-018-4307-x) using scale transformation. The major dynamic properties of this model have been investigated, in particular, the equilibrium points and the dissipation property of the model explored. Based on usual diagnostic tools such as phase portraits, bifurcation diagrams, frequency spectrum, Lyapunov spectrum and basins of attraction, the nonlinear behaviours of the model have been highlighted. We demonstrated that the three-dimensional autonomous system experiences the especial feature of subcritical Hopf bifurcation (when the control parameter exceeds a critical value); equally the period-doubling bifurcation, saddle-node bifurcation, offset boosting, reverse period doubling and antimonotonicity have been found. With the use of the normal form theory (Hassard algorithm), a formula for finding the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented. Furthermore, by using the Marsden–MacCracken index, we show that this Hopf bifurcation at equilibria is non-degenerate. At the end, the experimental results are given to further support the theoretical analysis.