Nonlinear Systems & Linearization · Superchallenge problem

The Duffing system $x' = y,\; y' = x - x^3$ has a homoclinic orbit in the half-plane $x > 0$: a solution $x(t)$ with $x(t) \to 0$ as $t \to \pm\infty$. Compute $\displaystyle\int_{-\infty}^{\infty} x(t)\,dt$ along it.