Exponential Stability of Invariant Manifold for a Nonlinear Impulsive Multifrequency System
- PDF / 135,670 Bytes
- 10 Pages / 594 x 792 pts Page_size
- 12 Downloads / 222 Views
EXPONENTIAL STABILITY OF INVARIANT MANIFOLD FOR A NONLINEAR IMPULSIVE MULTIFREQUENCY SYSTEM P. Feketa1;2 , T. Meurer,3 , M. M. Perestyuk4 , and Yu. M. Perestyuk5
UDC 517.956.3
We study the exponential stability of a trivial invariant manifold of nonlinear extension of the dynamical system on a torus with impulsive jumps at nonfixed moments of time. The deduced sufficient conditions for the exponential stability of the trivial torus take into account the information on the qualitative properties of the dynamics of the system on the invariant manifold and weaken sufficient conditions available in the literature for a wide class of dynamical systems. New theorems impose constraints on a nonwandering set of the dynamical system guaranteeing the exponential stability of trivial manifold and are especially beneficial for the stability analysis of the extensions of dynamical systems with simple structures of the limit sets and recurrent trajectories.
1. Introduction Invariant toroidal manifold is a central object of investigations in the qualitative theory of multifrequency oscillations. The existence of invariant tori is a necessary condition for the existence of multifrequency oscillations formed by quasiperiodic solutions to dynamical systems [1]. The fundamental results on the existence of invariant toroidal manifolds of linear systems in Tm ⇥ Rn ; perturbation theory of invariant manifolds for nonlinear systems, smoothness, and stability properties of invariant tori were obtained by A. M. Samoilenko and summarized in [1]. In [2], the stability properties of invariant tori for systems defined in the direct product of m-dimensional torus Tm and n-dimensional Euclidean space Rn were studied in terms of sign-definite quadratic forms. Sufficient conditions for the existence and stability of invariant tori of the linear extensions of dynamical systems on a torus subjected to impulsive perturbations [3–5] were established in [6, 7]. In the present paper, we obtain new sufficient conditions for the exponential stability of the trivial torus of nonlinear extension of a dynamical system on Tm subjected to impulsive perturbations when the trajectory on the torus intersects a predefined submanifold of Tm : We consider the invariant manifold not just as a set of points but rather as a set of trajectories of the dynamical system and take into account the dynamics of the system on the surface of the torus. This approach leads us to sufficient conditions for the exponential stability of the trivial torus in terms of quadratic forms, which are sign definite not on the entire surface of Tm but only in the nonwandering set of dynamical system. A similar technique for the stability analysis of invariant tori of the linear extensions of dynamical systems on Tm was used in [8, 9] for the impulse-free case and in [10–13] for systems with impulsive perturbations. Sufficient conditions for the exponential stability and instability of the trivial torus of nonlinear extensions of the dynamical systems on a torus without pulses were obtained
Data Loading...
