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The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.

It is well known that many kinds of nonlinearities exist in engineering systems, such as parametric excitation, nonsmoothness, time delay, discontinuity, and large deformation [

The heteroclinic bifurcation and chaos behaviors are two of the most important characteristics in nonlinear dynamical systems. Many scholars paid attention to the heteroclinic bifurcation and chaos phenomena of nonlinear Duffing oscillator [

All the above analyses mainly focused on qualitative, numerical, or simplified analytical solutions about the necessary condition for generating chaos. In this paper, the first-order exact analytical solution of the necessary condition for generating chaos in sense of Smale horseshoes in a Duffing oscillator with both delayed displacement feedback and delayed velocity feedback is obtained based on Melnikov theory. Besides, the numerical heteroclinic bifurcation results were presented and some new phenomena were found in the Duffing oscillator with time-delay feedback. The basic structure of this paper is arranged as follows. The Melnikov function is obtained based on Melnikov method, and the analytical necessary condition for generating chaos is also obtained in Section

Duffing oscillator is one of the most familiar systems in nonlinear dynamics. Under the function of both delayed displacement feedback and delayed velocity feedback, a Duffing oscillator with forcing excitation would be investigated in this section. The dynamic equation is

Introducing the transformation

If the delayed displacement feedback coefficient

If there is only the delayed displacement feedback in (

By the analysis of (

In order to verify the validity of the analytical necessary condition for generating chaos, a set of illustrative system parameters is chosen as

The system responses for different excitation amplitude

Bifurcation diagram

The largest Lyapunov exponents

Bifurcation diagram for Type 1

The system response at

The system response at

The system response at

Bifurcation diagram for Type 2

The system response at

The system response at

The system response at

The transitions observed in Type 1 occur at

When

When

When

The transitions observed in Type 2 occur at

When

When

When

At last,

The system response at

If the same system parameters are substituted into (

The relation curves of

From the analysis of (

The relation curves of

If there is only the delayed velocity feedback in (

Here system typical parameters are chosen as

The relation curves of

The system parameters are chosen as

The relation curves of

Through all the above analysis, it could be found that tendencies of the analytical solutions for the influences of all the delayed feedback parameters are consistent with the numerical iterative simulations. That is to say, a qualitative agreement between the numerical and analytical solutions is obtained. It is generally well known that the analytical necessary condition for the chaos in sense of Smale horseshoes by Melnikov method is the first-order approximate result, so that the quantitative differences between the numerical results and analytical solutions are acceptable. Although there are some differences of analytical solutions with numerical results, the conclusions of the analytical necessary condition for generating chaos are helpful to design this kind of delayed system or control the chaos by choosing appropriate system parameters.

In this paper, the heteroclinic bifurcation behavior of a Duffing oscillator under forcing excitation with both delayed displacement feedback and delayed velocity feedback is investigated. The Melnikov function is established and the analytical necessary condition for generating chaos in (

The authors declare that they have no conflicts of interest.

The authors are grateful to the support by National Natural Science Foundation of China (no. 11602152 and no. 11272219) and the Education Department Project of Hebei Province (QN2016258).