Transitions to chaos, fractal basin boundaries, bursting and mixed modes oscillations in a cubic-quintic-septic duffing oscillator with four wells of equal depth

Abstract

This work analyses the various dynamical states that can be delivered by a sinusoidally excited cubic-quintic-septic Duffing oscillator with a potential having four wells of equal depth and three bumps of the same level using mathematical methods and numerical simulation based on the fourth order Runge-Kutta method. This special potential can be obtained in mechanics using magnets with appropriate magnetic inductions placed appropriately close to a line on which a magnetic body is moving. The frequency response curves are plotted for asymmetric oscillations around the stable equilibria. The transition routes to chaos are obtained through the bifurcation diagrams. Chaos appears through several routes. The signature of the four well potential on the phase portraits is clearly visible by the display of periodic or chaotic rounds near the equilibrium points. The horseshoes chaos is observed by plotting the attraction basins. The model shows four basins of attraction corresponding to each of the four wells which are regular for dynamics or fractal for chaos in the Melnikov sense. Bursting and mixed modes oscillations are obtained, some of which are chaotic. The justification of the appearance of bursting oscillations is conducted using the analysis of the equilibrium points in case of a slow variation of the excitation.