LettersNo Access

CLOSED-FORM CONDITIONS OF BIFURCATION POINTS FOR GENERAL DIFFERENTIAL EQUATIONS

P. YU

Department of Applied Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7, Canada

https://doi.org/10.1142/S0218127405012582Cited by:70 (Source: Crossref)

Abstract

This note presents closed-form formulas for determining the critical points of general n-dimensional differential equations. The formulas do not require commutating the eigenvalues of the Jacobian of a system. Based on the Hurwitz criterion, explicit necessary and sufficient conditions are obtained. Particular attention is focused on Hopf and double Hopf bifurcations. A model of induction machine is presented to show the application of the results.

Keywords:

References

G. Chen, J. L. Moiola and H. O. Wang, Int. J. Bifurcation and Chaos 10, 515 (2000). Google Scholar
L. O. Chua , C. A. Desoer and E. S. Kuh , Linear and Nonlinear Circuits ( McGraw-Hill , NY , 1987 ) . Google Scholar
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 4th edn. (Springer-Verlag, NY, 1993). Google Scholar
P. C. Krause , O. Wasynczuk and S. D. Sudhoff , Analysis of Electrical Machinery and Drive Systems ( IEEE Press and Wiley Interscience , NJ , 2002 ) . Crossref, Google Scholar
J. E. Marsden and M. McCracken , The Hopf Bifurcation and its Applications ( Springer-Verlag , NY , 1976 ) . Crossref, Google Scholar
L. Orlando, Math. Ann. 71, 233 (1911). Crossref, Google Scholar
B. Porter , Stability Criteria for Linear Dynamical Systems ( Oliver & Boyd , London , 1967 ) . Google Scholar
F. Takens, Publ. Math. IHES 43, 47 (1974). Crossref, Google Scholar
P. Yu, J. Sound Vibr. 211, 19 (1998). Crossref, Web of Science, Google Scholar
P. Yu, A method for computing center manifold and normal forms, Proc. Int. Conf. Diff. Eqns.2 (World Scientific, Singapore, 2000) pp. 832–837. Google Scholar
P. Yu, J. Sound Vibr. 247, 615 (2001). Crossref, Web of Science, Google Scholar
P. Yu, Nonlin. Dyn. 27, 19 (2002). Crossref, Web of Science, Google Scholar
P. Yu and A. Y. T. Leung, Nonlinearity 16, 277 (2003). Crossref, Web of Science, Google Scholar