We prove new estimates for the error of the averaging method for oscillation systems with slowly varying frequencies subjected to pulse action at fixed times. The main assumption is imposed not on all harmonics of the right-hand side of the system but only on resonance ones.
UDC 517.9In the neighborhood of an asymptotically stable integral manifold of a multifrequency system with pulse influence at fixed times, we perform a decomposition of the equations for angular and position variables.A dynamical system with a constant vector of frequencies in the neighborhood of an invariant torus was studied in [1,2], and a decomposition of equations for systems with slowly varying phase was carried out in [3]. For oscillation systems with slowly varying frequencies, a decomposition of equations in the neighborhood of an asymptotically stable manifold was investigated in [4]. One should also mention the monograph [5], where the problem of decomposition of motions by the method of integral manifolds for singularly perturbed ordinary differential equations was studied in detail. In the present work, we consider analogous problems for multifrequency systems of differential equations with pulse influence at fixed times. Consider the following system of n + m ordinary differential equations with pulse influence at fixed times∆x τ τ ν = = ε ϕ τ ε ν p x ( , , , ), ∆ϕ τ τ ν = = ε ϕ τ ε ν q x ( , , , ). Here, τ = ε t ∈ R, ( 0, ε 0 ] ' ε is a small parameter, ε 0 << 1, x ∈ D ⊂ R n , ϕ ∈ R m , τ τ ν ν + − 1 = εθ for all ν ∈ Z, θ is a positive number, D is a bounded region, Z is the set of all integers, and the functions a, b, p, and q are 2π-periodic in each coordinate ϕ j , j = 1, m , of the vector ϕ. Assume that, for some l ≥ m, the functions d d s r s ω τ τ ( ) , s = 0, l , r = 1, m , are uniformly continuous on the entire axis and ( )where σ 1 is a positive constant, ω = ( ω 1 , … , ω m ) , and V l ( ) τ and V l T ( ) τ denote, respectively, the l × m matrix
517.928Using uniform estimates for oscillation integrals and sums, we prove theorems on asymptotic and conditional asymptotic stability of the integral manifold of a multifrequency system of ordinary differential equations with pulse action at fixed times.The method of integral manifolds in the theory of differential equations is an efficient mathematical apparatus because the problem of the determination of solutions of these equations becomes considerably simpler if they belong to a manifold of lower dimension than the dimension of the phase space. This method was generalized to various classes of differential equations [1][2][3][4][5], including quasilinear systems with pulse action [6]. Conditions for the existence of an integral manifold for a single-frequency pulse oscillation system were established in [7]. In [8], this result was generalized to the case of multifrequency resonance systems.In the present paper, using uniform estimates of oscillation integrals and sums, we study the problem of the stability of the integral manifold constructed in [8] for a pulse oscillation system with slowly varying frequencies.Consider the following nonlinear oscillation system of ordinary differential equations with slowly varying frequencies and pulse action at fixed times t ν = ε −1 τ ν : dx dτ = a(x, τ ) +ã(x, ϕ, τ ) + εA(x, ϕ, τ, ε),θ is a positive number, D is a bounded domain, and the right-hand sides of the equalities in (1) belong to certain classes of functions smooth and 2π-periodic in each coordinate ϕ s , s = 1, m, of the vector ϕ. Without loss of generality, we can assume that the averages with respect to ϕ in the cube of periods of the functionsã andp are identically equal to zero. Let
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.