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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.

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 x 3 = F c o s ( ω t ) . (1)

Depending on the parameters chosen, the equation can take a number of special forms. For example, Bender and Orszag [

x ¨ + β x + α x 3 = 0. (2)

Wiggins [

x ¨ + δ x ˙ − x + x 3 = F c o s ( ω t ) . (3)

Ravichandran et al. [

x ˙ = y , y ˙ = − x ( x 2 − 1 ) ( x 2 − a ) − δ y + f c o s ( ω t ) , (4)

and obtained the conditions of existence and bifurcations for harmonics, subharmonics and superharmonics under small perturbations and the threshold values of chaotic motion under periodic perturbation. And Jing et al. [

x ˙ = y , y ˙ = − x ( x 2 − 1 ) ( x 2 − a ) + δ y + f 1 c o s ( ω 1 t ) + f 2 c o s ( ω 2 t ) . (5)

Wang [

x ˙ = y , y ˙ = − x ( x 2 − 1 ) ( x 2 − a 2 ) + f c o s ( ω t ) + b x c o s ( Ω t ) . (6)

But less attention was focused on the two well Duffing equation with one degenerated saddle. In this paper we studied the following Duffing equation

x ˙ = x , y ˙ = − x 3 ( x 2 − 1 ) − δ y + f c o s ( ω t ) , (7)

where δ , f , ω are real parameters. Physically, δ can be regarded as dissipation or damping; f and ω is the amplitude and frequency of the external force.

The structure of the paper is as follows. In Section 2, the fixed points and phase portraits are obtained for the unperturbed system of (7). In Section 3, the conditions of existence of chaos under periodic perturbation resulting from the homoclinic bifurcations are performed by Melnikov method. Finally, we make some numerical computations which give support to the theoretical analysis and some complex dynamics in Section 4.

In this section, we obtain the stability of fixed points and phase portrait of unperturbed system of (7).

For we take δ = f = 0 and obtain the unperturbed system of (7) as follows

x ˙ = y , y ˙ = − x 3 ( x 2 − 1 ) . (8)

The unperturbed system (7) can be easily obtained three fixed points: a degenerate saddle S ( 0,0 ) and two centers C 1 ( − 1,0 ) and C 2 ( 1,0 ) . The phase portrait of the unperturbed system (7) is plotted in

Γ 1 ( x 0 ( t ) , y 0 ( t ) ) = ( − 6 4 + 3 t 2 , 3 6 t ( 4 + 3 t 2 ) 3 ) and

Γ 2 ( x 0 ( t ) , y 0 ( t ) ) = ( 6 4 + 3 t 2 , − 3 6 t ( 4 + 3 t 2 ) 3 ) , respectively.

In essence we use perturbation methods to study the system (7), we therefore study how the dynamics of unperturbed system (8) are changed under the periodic perturbation in the following parts.

In this section, we consider the chaotic behaviors of system (7) in which δ , f are assumed to be small parameters with order ε . The Duffing system can be written an as follows:

x ˙ = y , y ˙ = − x 3 ( x 2 − 1 ) + ε ( − δ 1 y + f 1 c o s ( ω t ) ) , (9)

where ε δ 1 = δ , ε f 1 = f .

The closed homoclinic orbits break when the perturbation is added, and system (7) may have transverse homoclinic orbits. By the Smale-Birkhoff Theorem [

For the homoclinic orbit Γ 1 , we have the Melnikov function,

M 1 ( t 0 ; f 1 , δ 1 , ω ) = ∫ − ∞ + ∞ y 0 ( t ) ( − δ 1 y 0 ( t ) + f 1 c o s ( ω ( t + t 0 ) ) ) d t = − 3 32 3 π δ 1 + 2 2 f 1 ω s i n ( ω t 0 ) B e s s e l K ( 0, 2 ω 3 ) , (10)

where B e s s e l K ( 0, 2 ω 3 ) is Bessel functions of the second. If we define

R 0 ( ω ) = 3 3 π 64 2 ω B e s s e l K ( 0, 2 ω 3 ) , (11)

then it follows from Theorem 4.5.3 in [

For the homoclinic orbit Γ 2 , the computation is identical and the similar result is obtained.

In this section we give numerical simulations to look for other new dynamics. In the process of numerical simulation, we vary one parameter and fix the other parameters of system (7) as follows:

1) Varying f in the range 0 ≤ f ≤ 5 and fixing δ = 0.2 and for rational and irrational values of ω .

2) Varying δ in the range 0 ≤ δ ≤ 2 and fixing f = 1 and for rational and irrational values of ω .

For case 1). The bifurcation diagram of system (7) in ( f , x ) plane and the corresponding Lyapunov exponents for δ = 0.2, ω = 2 are given in

The bifurcation diagram of system (7) in ( f , x ) plane and the corresponding Lyapunov exponents for δ = 0.2, ω = 2 / 2 are given in

For case 2). The bifurcation diagram of system (7) in ( f , x ) plane and the corresponding Lyapunov exponents for f = 1, ω = 2 are given in

interior crisis occurs. Poincaré maps of chaotic attractors for δ = 0.621 and δ = 0.622 are shown in

The bifurcation diagram of system (7) in ( f , x ) plane and the corresponding Lyapunov exponents for f = 1, ω = 2 / 2 are given in

bifurcations for 0.315 < δ < 0.325 . At δ = 0.3541 an intermittence of chaos occurs and chaotic motion becomes inverse period-doubling bifurcation. There is a bubble for 1.432 < δ < 1.531 in the local amplification

This work was supported by the National Science Foundations of China (10671063, 10801135 and 61571052).

Yang, Z.Y. and Jiang, T. (2017) Bifurcations and Chaos in the Duffing Equation with One Degenerate Saddle Point and Single External Forcing. Journal of Applied Mathematics and Physics, 5, 1908-1916. https://doi.org/10.4236/jamp.2017.59161