Memristive-Modified Hybrid System with a Line of Equilibria

This paper proposes and investigates a novel four-dimensional (4D) fractional-order chaotic system obtained by coupling a flux-controlled memristor with a classical Jerk oscillator. The resulting model is governed by a set of nonlinear fractional differential equations involving the Caputo derivative of order 0<α≤1, which introduces memory effects into the system dynamics. The memristive nonlinearity is incorporated through a quadratic memductance function W(w)=a+bw², while μ acts as a control parameter and β represents the damping coefficient.

The inclusion of both fractional-order operators and memristive coupling leads to richer dynamical behavior compared to the classical integer-order Jerk system. A rigorous qualitative analysis is performed to establish the existence and uniqueness of solutions. The equilibrium analysis shows that the system possesses a continuous line of equilibrium points E=(0,0,0,w∗), where w∗∈R, which is a typical feature of memristive systems.

Local stability is studied using eigenvalue analysis combined with Matignon’s fractional-order stability criterion. The results indicate that the interaction between the fractional order α and memristive parameters significantly affects stability regions and the onset of oscillatory behavior.

Extensive numerical simulations reveal a rich variety of nonlinear dynamics, including periodic oscillations, chaotic attractors, coexistence of attractors, and hidden attractors. Notably, hidden chaotic attractors are observed even when the equilibrium manifold is locally stable, highlighting the complexity induced by dual memory effects.

Overall, the proposed system combines fractional-order and memristive memory mechanisms, generating complex dynamics with potential applications in nonlinear circuits, secure communications, and neuromorphic engineering.