Complex Dynamics in Arrays of Memristor Oscillators via the Flux–Charge Method

The key intent of the work is to analyze complex dynamics and synchronization phenomena in a 1-D array of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> diffusively coupled memristor-based oscillatory/chaotic circuits, i.e., each uncoupled oscillator is a 3^rd–order memristor-based Chua’s circuit obtained by replacing the nonlinear resistor with an ideal flux-controlled memristor. It is shown that the state space <inline-formula> <tex-math notation="LaTeX">$ \mathbb {R}^{4N}$ </tex-math></inline-formula> in the voltage–current domain of the array can be decomposed in <inline-formula> <tex-math notation="LaTeX">$\infty ^{N}~3N$ </tex-math></inline-formula>-dimensional manifolds which are positively invariant for the nonlinear dynamics. Moreover, on each manifold the array obeys a different reduced-order dynamics in the flux-charge domain. These basic properties imply that two main types of bifurcations can occur, i.e., standard bifurcations on a fixed invariant manifold induced by changing the circuit parameters and bifurcations due to the variation of initial conditions and invariant manifold, but for fixed circuit parameters. The latter bifurcation phenomena are referred to as bifurcations without parameters. The reduced dynamics on invariant manifolds, and their analytic expressions, are the key tools for a comprehensive analysis of synchronization phenomena in the array of memristor-based Chua’s circuits. The main results are proved via a recently introduced technique for studying memristor-based circuits in the flux-charge domain.