# American Institute of Mathematical Sciences

January  2017, 37(1): 591-604. doi: 10.3934/dcds.2017024

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

 1 Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China 2 School of Computing and Engineering, Huddersfield University, HD 4, the United Kingdom

* Corresponding author: wangweifrancistju@yahoo.com

Received  July 2015 Revised  September 2016 Published  November 2016

Fund Project: 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).

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.

Citation: Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024
##### References:

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##### References:
(a) The phase portrait of homoclinic orbit; (b) The phase portrait of heteroclinic orbit
(a) The phase portrait of heteroclinic orbit; (b) The curve of $u$; (c) The curve of $\dot u$; 1-without perturbation, 2-with perturbation
The flow chart of the new approach
Heteroclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method
Heteroclinic orbit of the oscillator; 1-Second-order direct method, 2-Numerical simulation method
Critical bifurcation values; 1-Second-order direct method, 2-Numerical simulation method, 3-Melnikov integration method
Homoclinic orbit of the oscillator; 1-Direct method, 2-Numerical simulation method
Comparison of the variables obtained by different methods
 Group Parameter values $\omega_{10}$ $\theta$ $\mu$ $G_{i}$ $\omega_{0}$ $\varepsilon$ $\alpha$ $\gamma_{2,1}$ $M_{1}$ $M_{1}$ $M_{1}$ $M_{2}$ $M_{3}$ 1 1.5 2 2 1 1.154 -0.192 0.220 0.228 0.225 2 2 1.5 5 3 1.596 -0.230 0.481 0.488 0.480 3 2 1 5 4 1.556 -0.205 0.630 0.648 0.640 4 2 2 5 3 1.745 -0.298 0.506 0.493 0.480 Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method
 Group Parameter values $\omega_{10}$ $\theta$ $\mu$ $G_{i}$ $\omega_{0}$ $\varepsilon$ $\alpha$ $\gamma_{2,1}$ $M_{1}$ $M_{1}$ $M_{1}$ $M_{2}$ $M_{3}$ 1 1.5 2 2 1 1.154 -0.192 0.220 0.228 0.225 2 2 1.5 5 3 1.596 -0.230 0.481 0.488 0.480 3 2 1 5 4 1.556 -0.205 0.630 0.648 0.640 4 2 2 5 3 1.745 -0.298 0.506 0.493 0.480 Note: $M_{1}$-Direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Melnikov integration method
Comparison of the variables obtained by different methods
 Group Parameter values $\omega_{10}$ $\theta$ $\mu$ $G_{i}$ $\omega_{0}$ $\varepsilon$ $\alpha$ $\gamma_{2,1}$ $M_{1}$ $M_{1}$ $M_{1}$ $M_{2}$ $M_{3}$ 3 2 1 5 4 1.559 -0.211 0.630 0.648 0.645 4 2 2 5 3 1.737 -0.291 0.506 0.493 0.494 Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method
 Group Parameter values $\omega_{10}$ $\theta$ $\mu$ $G_{i}$ $\omega_{0}$ $\varepsilon$ $\alpha$ $\gamma_{2,1}$ $M_{1}$ $M_{1}$ $M_{1}$ $M_{2}$ $M_{3}$ 3 2 1 5 4 1.559 -0.211 0.630 0.648 0.645 4 2 2 5 3 1.737 -0.291 0.506 0.493 0.494 Note: $M_{1}$-First-order direct method, $M_{2}$-Numerical simulation method, $M_{3}$-Second-order direct method
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