Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

G. L. Alfimov ; M. E. Lebedev

Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 473-506

Résumé

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called “codes” of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a “great part” of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Détail

Keywords: Duffing-type equation, periodic coefficients, symbolic dynamics, Smale horseshoe