On the Dynamics of a Time-periodic Equation

In this paper we use the second order equation dq dt + (λ − γq2) dt − q + q = μq sinωt as a demonstrative example to illustrate how to apply the analysis of [WO] and [WOk] to the studies of concrete equations. We prove, among many other things, that there are positive measure sets of parameters (λ, γ, μ, ω) corresponding to the case of intersected (See Fig. 1(a)) and the case of separated (See Fig. 1(b)) stable and unstable manifold of the solution q(t) = 0, t ∈ R respectively, so that the corresponding equations admit strange attractors with SRB measures. In the history of the theory of dynamical systems, ordinary differential equations have served as a source of inspirations and a test ground. There are mainly two ways in relating the studies of differential equations to the studies of maps, both originated from the work of H. Poincaré. The first is to use return maps locally defined on Poincaré sections for the studies of autonomous equations. The second is to use the globally defined time-T maps for the studies of time-periodic equations. The studies of periodically forced second order equations, such as van de Pol’s equation, Duffing’s equation and the equation for non-linear pendulums, have played substantial roles in shaping the chaos theory in modern times ([Ar], [D], [V], [Lev1], [Le], [GH]). When a homoclinic solution is periodically perturbed, the stable and the unstable manifold of the perturbed saddle intersect each other within a certain range of forcing parameters, generating homoclinic tangles and chaotic dynamics ([P], [S], [M]). See Fig. 1(a). There are also forcing parameters for which the stable and the unstable manifold of the perturbed saddle are pulled apart. See Fig. 1(b). (a) (b) Fig. 1 The stable and the unstable manifold of a perturbed saddle. Date: January, 2008.