Asymptotic Integration of Nonlinear Systems of Differential Equations whose Phase Portrait is Foliated on Invariant Tori

DOI

10.2991/jnmp.2000.7.2.9How to use a DOI?

Abstract

We consider the class of autonomous systems x = f(x), where x R2n , f C1 (R2n ) whose phase portrait is a Cartesian product of n two-dimensional centres. We also consider perturbations of this system, namely x = f(x) + g(t, x), where g C1 (R × R2n ) and g is asymptotically small, that is g 0 as t + uniformly with respect to x. The rate of decrease of g is assumed to be t-p where p > 1. We prove under this conditions the existence of bounded solutions of the perturbed system and discuss their convergence to solutions of the unperturbed system. This convergence depends on p. Moreover, we show that the original unperturbed system may be reduced to the form r = 0, = A(r), and taking r Rm + , Tn , where Tn denotes the n-dimensional torus, we investigate the more general case of systems whose phase portrait is foliated on invariant tori. We notice that integrable Hamiltonian systems are of the same nature. We give also several examples, showing that the conditions of our theorems cannot be improved.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Yuri A. Il'in
PY  - 2000
DA  - 2000/05/01
TI  - Asymptotic Integration of Nonlinear Systems of Differential Equations whose Phase Portrait is Foliated on Invariant Tori
JO  - Journal of Nonlinear Mathematical Physics
SP  - 198
EP  - 212
VL  - 7
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2000.7.2.9
DO  - 10.2991/jnmp.2000.7.2.9
ID  - Il'in2000
ER  -