A geometric approach to regular perturbation theory with an application to hydrodynamics

Author:
Carmen Chicone

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4559-4598

MSC:
Primary 58F22; Secondary 34C25, 58F30, 76E99

MathSciNet review:
1311905

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Abstract: The Lyapunov-Schmidt reduction technique is used to prove a persistence theorem for fixed points of a parameterized family of maps. This theorem is specialized to give a method for detecting the existence of persistent periodic solutions of perturbed systems of differential equations. In turn, this specialization is applied to prove the existence of many hyperbolic periodic solutions of a steady state solution of Euler's hydrodynamic partial differential equations. Incidentally, using recent results of S. Friedlander and M. M. Vishik, the existence of hyperbolic periodic orbits implies the steady state solutions of the Eulerian partial differential equation are hydrodynamically unstable. In addition, a class of the steady state solutions of Euler's equations are shown to exhibit chaos.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1311905-4

Keywords:
Lyapunov-Schmidt reduction,
resonance,
normal nondegeneracy,
Euler equations,
chaos

Article copyright:
© Copyright 1995
American Mathematical Society