Bifurcation mechanism of periodic bursting in a simple three-element-based memristive circuit with fast-slow effect

Abstract

Abstract The design and analysis of simple autonomous memristive circuits are of great significance in theoretical research and practical application of complex dynamics. This paper designs a simple autonomous three-element-based memristive circuit with fast-slow effect, which is constructed by parallel connecting an active second-order memristive diode bridge with a parallel LC filter. The corresponding mathematical model is established, upon which stability analysis of the equilibrium point is performed by Routh–Hurwitz criterion. Theoretical derivation results indicate that the presented memristive circuit is always unstable, and then complex dynamical behaviors of period, chaos, quasi-period, and periodic bursting oscillation are numerically revealed through phase plane orbit, time-domain sequence, Lyapunov exponent spectrum and bifurcation diagram. In addition, bifurcation mechanisms of the periodic bursting behavior are further explored by fast-slow dynamics analysis, from which it can be found that the periodic bursting oscillation of the presented memristive circuit is a type of symmetric fold/Hopf cycle-cycle bursting oscillation. Especially, two types of dynamical behaviors of quasi-periodic oscillation and periodic bursting oscillation with a symmetric fold/Hopf cycle-cycle burster simultaneously appear in such a simple autonomous memristive circuit is amusing, which has never been reported in the previous literatures.