The chaotic mechanisms in some jerk systems

In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which $\epsilon$ is a small parameter, and $a$, $b$, $c$, $d$ are some other parameters. For $\epsilon =0$, the system is $Z_2$ symmetric. For $\epsilon \ne 0$, the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor's theorem and Hassard's formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter $a$, the mechanisms are identified by means of bifurcation diagrams. Cases A and B are $Z_2$ symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincaré sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system.