# The Shilnikov type homoclinic orbits of perturbed cubic polynomial Moon-Rand systems

Volume 17, Issue 3, pp 408-419 Publication Date: July 23, 2017

### Authors

Dandan Xie - School of Mathematics and Statistics, Linyi University, Linyi, Shandong, 276005, P. R. China.
Yinlai Jin - School of Mathematics and Statistics, Linyi University, Linyi, Shandong, 276005, P. R. China.
Feng Li - School of Mathematics and Statistics, Linyi University, Linyi, Shandong, 276005, P. R. China.
Nana Zhang - School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250014, P. R. China.

### Abstract

In this paper, perturbed polynomial Moon-Rand systems are considered. The Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic orbits for three dimensional nonlinear dynamical systems. In order to get real bifurcation parameters, four undetermined coefficients are introduced including three initial values about position and the value of bifurcation parameter. By the eigenvectors of its all eigenvalues, the value of the bifurcation parameter and three initial values about position are obtained directly. And, the analytical expressions of the Shilnikov type homoclinic orbits are achieved and the deletion errors relative to the practical system are given. In the end, we roughly predict when the horseshoe chaos occurs.

### Keywords

• analytic solution
• Shilnikov theorem
• homoclinic orbit.

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