Volume 7, Issue 2, May 2000, Pages 198 - 212
Asymptotic Integration of Nonlinear Systems of Differential Equations whose Phase Portrait is Foliated on Invariant Tori
- Yuri A. IL'IN
- Corresponding Author
- Yuri A. IL'IN
Available Online 13 December 2006.
- https://doi.org/10.2991/jnmp.2000.7.2.9How to use a DOI?
- 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.
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Cite this article
TY - JOUR AU - Yuri A. IL'IN PY - 2006 DA - 2006/12 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 - https://doi.org/10.2991/jnmp.2000.7.2.9 ID - IL'IN2006 ER -