Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients

In this paper, we consider the equation $u_{xx}+Q(x)u+P(x)u^3=0$ where $Q(x)$ and $P(x)$ are periodic functions. It is known that, if $P(x)$ changes sign, a ``great part'' of the solutions for this equation are singular, i.e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i.e., not singular) on $\mathbb{R}$. For this purpose we consider the Poincaré map $\mathcal{P}$ (i.e., the map-over-period) for this equation and analyse the areas of the plane $(u,u_x)$ where $\mathcal{P}$ and $\mathcal{P}^{-1}$ are defined. We give sufficient conditions for hyperbolic dynamics generated by $\mathcal{P}$ in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of ``numerical evidence''. This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given.