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Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

This work is supported by the National Natural Science Foundation of China (Grand No. 11102127,11372210) and Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 12JCYBJC12500)

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  • In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

    Mathematics Subject Classification: 37L10, 74H60.


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  • Figure 1.  (a) The phase portrait of homoclinic orbit; (b) The phase portrait of heteroclinic orbit

    Figure 2.  (a) The phase portrait of heteroclinic orbit; (b) The curve of $u$; (c) The curve of $\dot u$; 1-without perturbation, 2-with perturbation

    Figure 3.  The flow chart of the new approach

    Figure 4.  Heteroclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method

    Figure 5.  Heteroclinic orbit of the oscillator; 1-Second-order direct method, 2-Numerical simulation method

    Figure 6.  Critical bifurcation values; 1-Second-order direct method, 2-Numerical simulation method, 3-Melnikov integration method

    Figure 7.  Homoclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method

    Table 1.  Comparison of the variables obtained by different methods

    GroupParameter values$\omega_{10} $$\theta $ $\mu $
    $G_{i}$$\omega_{0} $$\varepsilon $$\alpha $ $\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
    Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method
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    Table 2.  Comparison of the variables obtained by different methods

    GroupParameter values$\omega_{10} $$\theta $$\mu $
    $G_{i}$ $\omega_{0} $ $\varepsilon $$\alpha $$\gamma_{2,1} $$M_{1}$$M_{1}$$M_{1}$$M_{2}$$M_{3}$
    Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method
     | Show Table
    DownLoad: CSV
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