TY - JOUR

T1 - Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori

AU - Il’in, Yuri A.

N1 - Yu. A. Il'in. Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori, Journal of Nonlinear Mathematical Physics, 2000. Vol. 7, No 2, pp. 198-212

PY - 2000/1/1

Y1 - 2000/1/1

N2 - We consider the class of autonomous systems (Figure presented.) where x ∈ R 2x , ƒ ∈ C 1 (R 2n ) whose phase portrait is a Cartesian product of n two-dimensional centres. We also consider perturbations of this system, namely (Figure presented.) where g ∈ C 1 (R × R 2n ) 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 (Figure presented.) and taking (Figure presented.) where T n 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.

AB - We consider the class of autonomous systems (Figure presented.) where x ∈ R 2x , ƒ ∈ C 1 (R 2n ) whose phase portrait is a Cartesian product of n two-dimensional centres. We also consider perturbations of this system, namely (Figure presented.) where g ∈ C 1 (R × R 2n ) 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 (Figure presented.) and taking (Figure presented.) where T n 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.

UR - http://www.scopus.com/inward/record.url?scp=0347015051&partnerID=8YFLogxK

U2 - 10.2991/jnmp.2000.7.2.6

DO - 10.2991/jnmp.2000.7.2.6

M3 - Article

AN - SCOPUS:0347015051

VL - 7

SP - 198

EP - 212

JO - Journal of Nonlinear Mathematical Physics

JF - Journal of Nonlinear Mathematical Physics

SN - 1402-9251

IS - 2

ER -