Affine-Periodic Solutions for Dissipative Systems

Abstract

As generalizations of Yoshizawa’s theorem, it is proved that a dissipative affine-periodic system admits affine-periodic solutions. This result reveals some oscillation mechanism in nonlinear systems.

The existence of affine-periodic solutions for nonlinear impulsive differential equations

“…Recently, these conceptions and existence results of the solutions have been introduced and proved by Li and his coauthors; see [5] for Levinson's problem, [6] for Lyapunov function type theorems, [7] for averaging methods of affine-periodic solutions, and [8] for some dissipative dynamical systems. The aim of this paper is to touch such a topic for affine-periodic solutions of nonlinear impulsive differential equations.…”

Existence of rotating-periodic solutions for nonlinear second order vector differential equations

“…However, some differential equations often exhibit certain symmetries rather than periodicity, such as anti-periodicity, harmonic-periodicity, and quasi-periodicity. Recently, the existence of affine-periodic solutions and rotating-periodic solutions for nonlinear differential equations, which was firstly introduced in [24], has become a very interesting topic. Especially, Chang and Li [4,5] studied the existence of rotating-periodic solutions for second order dynamical systems by using the coincidence degree theory.…”

“…x is the Hessian matrix of f , κ > 0 is a constant. Then (Q, T)-rotating-periodic system (23)-(24) has at least one (Q, T)-rotating-periodic solution.Proof Let φ(t) = C for some C > 0. For any t ∈ [0, T], |x| = C, we havex, f (t, x) = x, f (t, 0) + f x t, ξ (t) x ≤ x, f (t, 0) -κ|x| 2 ≤ 0 for C ≥ M 0 /κ, where M 0 = max t∈[0,T] |f (t, 0)|.…”

Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions

“…When dealing with a system instead of a scalar equation, the (ω, c)-periodic solutions can be regarded as a particular case of the so-called affine-periodic functions; namely, continuous vector functions X ∈ C(R, R n ) such that X(t + ω) = QX(t) for some invertible matrix Q. These functions have been introduced in [24] in order to represent solutions of nonlinear problems with geometric meaning, e.g. rotation, simmetry, etc [8].…”