# American Institute of Mathematical Sciences

January  2011, 15(1): 231-254. doi: 10.3934/dcdsb.2011.15.231

## Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems

 1 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China 2 Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, China, China

Received  May 2009 Revised  March 2010 Published  October 2010

A semi-analytical procedure for studying stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous dynamical systems is developed. This procedure is based mainly on the incremental harmonic balance (IHB) method. It is composed of three key steps, namely, the determination of limit cycles by IHB method, the calculation of transition matrix by precise integration (PI) algorithm and the discrimination of limit cycle stability by Floquet theory. As an application, the procedure is used to investigate the dynamics of the limit cycle of a three-dimensional nonlinear autonomous system. The symmetry-breaking bifurcation, the first and the second period-doubling bifurcations of the limit cycle are identified. The critical parameter values corresponding to these bifurcations are calculated. The phase portraits and bifurcation points agree well with those of direct numerical integrations by using Runge-Kutta method.
Citation: Jianhe Shen, Shuhui Chen, Kechang Lin. Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 231-254. doi: 10.3934/dcdsb.2011.15.231
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