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Complicated Bifurcations of Periodic Solutions in Some System of Ode

Published online by Cambridge University Press:  20 November 2018

A. Soleev
A. Soleev*
Affiliation:
Faculty of Mathematics and Mechanics, Samarkand State University, University Boulevard 15, Samarkand 703004, Uzbekistan, e-mail:soleev@samarkand.silk.glas.apc.org
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Abstract

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In a vicinity of a stationary solution we consider a real analytic system of ODE of order four, depending on a small parameter. We look for families of periodic solutions which contract to the stationary solution, when the parameter tends to zero. We apply the general methods developed in [2] for the study of complex bifurcations and in [4] for local resolutions of singularities.

Keywords

34C2034C25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 3 , 01 September 1996 , pp. 360 - 366
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Bierstone, E. and Milman, P. D., Uniformization of analytic spaces, J. Amer. Math. Soc. 2( 1989), 801–836.Google Scholar
2. Bruno, A. D., Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin, 1989.Google Scholar
3. Bruno, A. D., Bifurcation of the periodic solutions in the symmetric case of a multiple pair of imaginary eigenvalues, Selecta Math. Soviet. 12(1993), 1—12.Google Scholar
4. Bruno, A. D. and Soleev, A., Local uniformization of branches of a space curve and Newton polyhedra, St. Petersburg Math. J. 3( 1992), 53–841.Google Scholar
5. Bruno, A. D. and Soleev, A., First approximations of algebraic equations, Dokl. Akad. Nauk 335( 1994) 211—21%, (Russian), Russian Ac. Sci. Dokl. Math. 48(1994), 291–293, (English).Google Scholar
6. Drazin, P. G., Nonlinear Systems, Cambridge Univ. Press, 1992.Google Scholar
7. Holmes, P. J., Center manifolds, normal forms, and bifurcation of vector fields, Physica 2D( 1981 ), 449–481.Google Scholar
8. Khovanskii, A. G., Newton polyhedra (resolution of singularities), J. Soviet Math. 27(1984), 207–239.Google Scholar
9. Soleev, A., Algebraic aspects of the arrangement of periodic solutions of a Hamiltonian system, Qualitative and Analytic Methods in Dynamical Systems, Samarkand University, 1987, 55—67, (Russian).Google Scholar