Abstract
In this paper, we study the Duffing equation with one degenerate saddle point and one external forcing and obtain the criteria of chaos of Duffing equation under periodic perturbation through Melnikov method. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit the more new complex dynamical behaviors, including homoclinic bifurcation, bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and Poincaré maps.
Highlights
Since in 1918, the German electrical engineer Georg Duffing introduced the Duffing equation, many scientists have been widely studied the equation in physics, economics, engineering, and found many other physical phenomena.The Duffing oscillator, is normally written as x + δ x + β x + α x3 = F cos(ωt). (1)Depending on the parameters chosen, the equation can take a number of special forms
We study the Duffing equation with one degenerate saddle point and one external forcing and obtain the criteria of chaos of Duffing equation under periodic perturbation through Melnikov method
Numerical simulations show the correctness of the theoretical analysis and exhibit the more new complex dynamical behaviors, including homoclinic bifurcation, bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and Poincaré maps
Summary
Since in 1918, the German electrical engineer Georg Duffing introduced the Duffing equation, many scientists have been widely studied the equation in physics, economics, engineering, and found many other physical phenomena. Huang and Jing [6] studied the three well Duffing equation with one external forcing x = y, y = −x(x2 −1)(x2 − a) − δ y + f cos(ωt),. Jing et al [7] [8] obtained complex dynamics of the three well Duffing equation with two external forcings, x = y, y = −x(x2 −1)(x2 − a) + δ y + f1 cos(ω1t) + f2 cos(ω2t). [10], studied bifurcation and chaos of the three well Duffing equation with parametric excitation and one external forcing x = y, y = −x(x2 −1)(x2 − a2 ) + f cos(ωt) + bx cos(Ωt). In this paper we studied the following Duffing equation x = x, y = −x3(x2 −1) − δ y + f cos(ωt),. In essence we use perturbation methods to study the system (7), we study how the dynamics of unperturbed system (8) are changed under the periodic perturbation in the following parts
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