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

Chicone, Carmen

doi:10.1090/S0002-9947-1995-1311905-4

A geometric approach to regular perturbation theory with an application to hydrodynamics
HTML articles powered by AMS MathViewer

by Carmen Chicone PDF

Trans. Amer. Math. Soc. 347 (1995), 4559-4598 Request permission

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.

References

V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391

Geometric methods in the theory of ordinary differential equations

Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
Carmen Chicone, The topology of stationary curl parallel solutions of Euler’s equations, Israel J. Math. 39 (1981), no. 1-2, 161–166. MR 617299, DOI 10.1007/BF02762862
Carmen Chicone, Bifurcations of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal. 23 (1992), no. 6, 1577–1608. MR 1185642, DOI 10.1137/0523087
Carmen Chicone, Lyapunov-Schmidt reduction and Mel′nikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations 112 (1994), no. 2, 407–447. MR 1293477, DOI 10.1006/jdeq.1994.1110
Carmen Chicone, Periodic solutions of a system of coupled oscillators near resonance, SIAM J. Math. Anal. 26 (1995), no. 5, 1257–1283. MR 1347420, DOI 10.1137/S0036141093243538
Stephen P. Diliberto, On systems of ordinary differential equations, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, no. 20, Princeton University Press, Princeton, N.J., 1950, pp. 1–38. MR 0034931
Susan Friedlander, Andrew D. Gilbert, and Misha Vishik, Hydrodynamic instability for certain ABC flows, Geophys. Astrophys. Fluid Dynam. 73 (1993), no. 1-4, 97–107. Magnetohydrodynamic stability and dynamos (Chicago, IL, 1992). MR 1289022, DOI 10.1080/03091929308203622
Susan Friedlander and Misha M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett. 66 (1991), no. 17, 2204–2206. MR 1102381, DOI 10.1103/PhysRevLett.66.2204
Susan Friedlander and Misha M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos 2 (1992), no. 3, 455–460. MR 1184488, DOI 10.1063/1.165888
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515
Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
V. K. Mel′nikov, On the stability of a center for time-periodic perturbations, Trudy Moskov. Mat. Obšč. 12 (1963), 3–52 (Russian). MR 0156048
Stephen Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics, vol. 2, Springer-Verlag, New York, 1990. MR 1056699, DOI 10.1007/978-1-4757-4067-7

A course of modern analysis