Dynamics of Nonautonomous Ordinary Differential Equations with Quasi-Periodic Coefficients

Abstract

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as [Formula: see text], [Formula: see text], where [Formula: see text] is the spacial part and [Formula: see text] is the time-varying part, and [Formula: see text] and [Formula: see text] are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and [Formula: see text], and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and [Formula: see text]. For [Formula: see text] sufficiently small, the parameter regions for [Formula: see text] are roughly divided into three parts: the spacial region ([Formula: see text]), the balance region ([Formula: see text]), and the time-varying region ([Formula: see text]); (i) for [Formula: see text], the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for [Formula: see text], there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for [Formula: see text], the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters [Formula: see text] and [Formula: see text] for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.