This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.

In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which $ \varepsilon $ is a small parameter, and $ a $, $ b $, $ c $, $ d $ are some other parameters. For $ \varepsilon = 0 $, the system is $ Z_{2} $ symmetric. For $ \varepsilon \neq {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.

This paper proposes a novel three-dimensional chameleon system derived from the classical Duffing oscillator. The innovation lies in transforming an externally driven Duffing system into an internally modulated autonomous system through a coupling between the driving force and system state. This coupling mechanism leads to a chameleon system with rich multistable dynamics. Through rigorous mathematical analysis, we establish the system’s basic properties, including dissipativity, local stability and a supercritical Hopf bifurcation. Using a bifurcation diagram, for small values of parameter c, we detect that c = 0 serves as a critical boundary between hidden and self-excited chaotic regimes. Using a continuation diagram, we find that a supercritical Hopf bifurcation occurs at c = 0 followed by a period-doubling route to self-excited chaos as c increases. When c = 0, by varying two independent parameters, we conduct comprehensive bifurcation analysis, revealing intricate dynamical behaviors. A remarkable discovery is the system’s ability to exhibit four nested hidden chaotic attractors, with their chaotic intensity (characterized by maximum Lyapunov exponents) monotonically decreasing from 0.1044 to 0.0226 as the distance from the equilibrium point increases. Additionally, we introduce an offset boosting mechanism that enables controlled shifting of attractors - a feature with potential applications in chaos-based engineering systems. The system is validated through numerical simulations and physical implementation using an electronic circuit that employs MC33078 and OPA2134 operational amplifiers with an 8th-order polynomial approximation of the cosine function.