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Mathematical Problems in Engineering
Volume 2016, Article ID 4246916, 7 pages
Research Article

Hopf Bifurcation of Compound Stochastic van der Pol System

School of Mathematics & Information Science, Beifang University of Nationalities, Yinchuan 750021, China

Received 9 December 2015; Revised 5 May 2016; Accepted 16 May 2016

Academic Editor: Haranath Kar

Copyright © 2016 Shaojuan Ma and Qianling Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strength and noise intensity on stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increased can relocate the critical value of bifurcation parameter forward while increased makes it backward and the influence of is more sensitive than . The results are verified by numerical simulations.