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 ẇ=aΦ1(w)+bΦ2(t,w), w∈ℝ3, where aΦ1 is the spacial part and bΦ2 is the time-varying part, and a and b 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 |b|≫max{1,|a|}, and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and |b|≫max{1,|a|}. For |b| sufficiently small, the parameter regions for (a,b) are roughly divided into three parts: the spacial region (|a|≫|b|), the balance region (|a|≈|b|), and the time-varying region (|a|≪|b|); (i) for |a|≫|b|, the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for |a|≈|b|, 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 |a|≪|b|, 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 a and b 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.
