Bifurcation and chaos in a smooth 3D dynamical system extended from Nosé-Hoover oscillator

Abstract

The chaotic system with complicated dynamical behaviors has high potential in practical applications. This paper reports a simple smooth 3D dynamical system that is derived from the Nosé-Hoover oscillator and has rich dynamical behaviors, such as fold-Hopf bifurcation, saddle-node bifurcation, transient chaos, and conservative chaos (hidden attractors), which respectively correspond to four cases of equilibrium: infinitely many equilibria, two equilibria, four equilibria and no equilibrium. We first concentrate on the theoretical investigation of degenerate fold-Hopf and saddle-node bifurcations. Then, the power-law distribution of average lifetime of transient chaos is calculated and the coexistence of steady state and periodic motion after transient chaos is partially exhibited by time series. For the case of no equilibrium, we further study the conservative and ergodic dynamics, and hidden attractors with extremely rare properties by time-domain waveforms, phase portraits, histograms and Poincaré sections. Finally, the FPGA-based experimental results are presented and they are consistent with the numerical simulation results.