Some Jerk Systems with Hidden Chaotic Dynamics

Hidden chaotic attractors is a fascinating subject of study in the field of nonlinear dynamics. Jerk systems with a stable equilibrium may produce hidden chaotic attractors. This paper seeks to enhance our understanding of hidden chaotic dynamics in jerk systems of three variables (x,y,z) with nonlinear terms from a predefined set: {-xy + dz(2),-xy + dztanh(z)}, where d is a real parameter. The behavior of the systems is analyzed using rigorous Hopf bifurcation analysis and numerical simulations, including phase portraits, bifurcation diagrams, Lyapunov spectra, and basins of attraction. For certain jerk systems with a subcritical Hopf bifurcation, adjusting the coefficient of a linear term can lead to hidden chaotic behavior. The adjustment modifies the subcritical Hopf equilibrium, transforming it from an unstable state to a stable one. One such jerk system, while maintaining its equilibrium stability, experiences a sudden transition from a point attractor to a stable limit cycle. The latter undergoes a period-doubling route to chaos, which may be followed by a reverse route. Therefore, by perturbing certain jerk systems with a subcritical Hopf equilibrium, we can gain insights into the formation of hidden chaotic attractors. Furthermore, adjusting the coefficient of the nonlinear term ztanh(z) in certain systems with a stable equilibrium can also lead to period-doubling routes or reverse period-doubling routes to hidden chaotic dynamics. Both findings are significant for our understanding of the hidden chaotic dynamics that can emerge from nonlinear systems with a stable equilibrium.

Automated summary

This paper investigates the hidden chaotic attractors in jerk systems with three variables. The behavior of the systems is analyzed using Hopf bifurcation analysis and numerical simulations. By adjusting the coefficient of a linear term, the systems can undergo a sudden transition from a point attractor to a stable limit cycle. These findings contribute to our understanding of hidden chaotic dynamics in nonlinear systems.