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

Dimensionality Reduction Reconstitution for Extreme Multistability in Memristor‐Based Colpitts System

“…To explore dynamical mechanism of the initials-dependent extreme multistability emerged in system (3), an equivalent dimensionality reduction model for system (3) needs to be built [11, 13, 14, 22, 23]. Pursuant to the SVM method [11], integrating the four equations of (3) from 0 to , one gets…”

“…Due to the natural nonlinearities [4], memristors can be introduced into some existing circuits or systems to easily achieve chaotic oscillations. In the past few years, various memristor-based nonlinear oscillating circuits and systems were proposed, such as memristive Hindmarsh-Rose neuron model [5], memristive cellular nonlinear/neural network [6], memristive band-pass filter circuit [7], memristive spiking and bursting neuron circuit [8], memristive jerk circuit [9], memristive hypogenetic jerk system [10], memristive hyper-jerk system [11], memristive Twin-T oscillator [12], memristive Chua's circuit [13], memristive canonical Chua's circuit [14], memristive multi-scroll Chua's circuit [15], and memristive Chua's hyperchaotic circuit [16]. However, relatively little attention has been received on the memristorbased Colpitts oscillator [17][18][19].…”

“…Latterly, to solve the abovementioned problem, fluxcharge analysis method [13,14,22,23] for the memristorbased dynamical circuits and state variable mapping (SVM) method [11,40] for the memristor-based dynamical systems were proposed to achieve an equivalent dimensionality reduction model, leading to the fact that the circuit goes from high-order to low-order or the system goes from highdimensional to low-dimensional. With these methods, the implicit initials in the original circuit or system can be changed into explicitly initials-related circuit/system parameters appearing in the dimensionality reduction model, and multiple stable states can be controlled by changing the initials-related circuit/system parameters [14], upon which the mechanism explanation for initials-dependent dynamics can be realized.…”

“…The aforementioned analytic strategies have been preliminarily verified in several memristor-based Chua's circuits [13,14,22] and memristor-based hyperjerk system [11]. However, for memristor-based Colpitts system, applicability and effectivity of the state variable mapping method still need comprehensive investigations and the concept of dimensionality reduction reconstitution is insistent to be clarified.…”

“…The proposed memristive Colpitts system exhibits the initials-dependent extreme multistability. To focus on the revelation and reconstitution of this special phenomenon, an equivalent 3-D dimensionality reduction model is obtained using the SVM method reported in [11] and several determined isolated equilibria are thereby yielded. Consequently, the implicit initials of the 4-D memristorbased Colpitts system are transformed into the explicitly initials-related system parameters of the 3-D dimensionality reduction model.…”