Generation and Evolution of Chaos in Double-Well Duffing Oscillator under Parametrical Excitation

Abstract

The generation and evolution of chaotic motion in double-well Duffing oscillator under harmonic parametrical excitation are investigated. Firstly, the complex dynamical behaviors are studied by applying multibifurcation diagram and Poincaré sections. Secondly, by means of Melnikov’s approach, the threshold value of parameterμfor generation of chaotic behavior in Smale horseshoe sense is calculated. By the numerical simulation, it is obvious that asμexceeds this threshold value, the behavior of Duffing oscillator is still steady-state periodic but the transient motion is chaotic; until the top Lyapunov exponent turns to positive, the motion of system turns to permanent chaos. Therefore, in order to gain an insight into the evolution of chaotic behavior afterμpassing the threshold value, the transient motion, basin of attraction, and basin boundary are also investigated.