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.

Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters $(\mu ,a,c)$ vary. Here, $\mu$ serves as the dissipation parameter in the $y$ direction. Bifurcation analysis for four scenarios with $\mu =0$ was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying $\mu$. The parameter $\mu$ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters $(a,c)$ also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.