| Abstract: |
| Motivated by some recent studies on the Allen--Cahn phase
transition model with a periodic non-autonomous term, we prove the
existence of complex dynamics for the second order equation
$$-\ddot{x} + (1 + \varepsilon^{-1} A(t)) G`(x) = 0,$$
where $A(t)$ is a non-negative $T$-periodic function and
$\varepsilon > 0$ is sufficiently small. In particular, we find a
full symbolic dynamics made by solutions which oscillate between
any two different strict local minima $x_0$ and $x_1$ of $G(x).$
Our approach is based on a variant of the theory of topological
horseshoes. |
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