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.
Highlights
Duffing oscillator, named after Georg Duffing, is a famous damped and forced nonlinear dynamical system [1]
We mainly investigate the occurrence of the chaotic behaviors in double-well Duffing system forced by parametrical excitation and explore the dynamical phenomena between the occurrences of horseshoe chaos and permanent chaos
In order to find the origin of chaotic motions, Melnikov’s method is used
Summary
Duffing oscillator, named after Georg Duffing, is a famous damped and forced nonlinear dynamical system [1]. The Duffing-type nonlinear dynamical systems have been investigated uninterruptedly in so many fields by plenty of researches, such as physics, engineering, chemistry, economics, and biological and social sciences [2,3,4,5]. It is famous for the existence of chaos behavior in recent decades. In 1979, the chaotic phenomena in Duffing’s equation had been investigated by Ueda [6]. The chaotic behavior in Duffing’s equation under a harmonic excitation is known as a common dynamical phenomenon. The generation of chaos was not mentioned, and it is really important in theory and practice
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