State variable mapping method for studying initial-dependent dynamics in memristive hyper-jerk system with line equilibrium

Memristive hyper-jerk system has been reported to exhibit the initial-dependent multistability phenomenon, whose dynamical mechanism and control strategy deserve further in-depth investigations. However, a more effective method is required for the existence of line equilibrium. Inspired by the dimensionality decreasing flux-charge analysis method, we handle these problems via state variable mapping (SVM) method of integral transformation. With this method, the hyper-jerk system characterized by four autonomous differential equations can be reconstructed and a dimensionality decreasing model is obtained. In the newly formulated system, the initial conditions of the original system are mapped as standalone system parameters. Thus, the initial-dependent multi-stability phenomenon is transformed into the easily controlled parameter-relied dynamics, which facilitates physical direction of multi-stable system's operation mode in hardware circuit. Specially, the line equilibrium point exhibited in the voltage-current model is mapped as two determined equilibria. Thus, the dynamics of coexisting infinitely many attractors and their forming mechanisms are readily investigated in the newly constructed system. Moreover, a hardware circuit is designed and fabricated, whose operation mode is rigorously controlled by applying additional DC signals which are proportional to the specified initial conditions. Through circuit simulations and experimental measurements, it is demonstrated that the SVM method offers a direction for interpretation and control of multistability in memristive systems without any determined equilibrium.