Duffing equation

Example

Let $\delta, \alpha, \beta, \gamma, \omega$ be real constants. The Duffing equation is the second-order ordinary differential equation $$x'' + \delta x' + \alpha x + \beta x^3 = \gamma \cos(\omega t).$$ Here $\delta$ is a damping coefficient, $\alpha x + \beta x^3$ is a restoring force with a cubic term, and $\gamma \cos(\omega t)$ is a periodic forcing.

Take $\delta = \gamma = 0$, $\alpha = -1$ and $\beta = 1$, so $x'' - x + x^3 = 0$. This is a Hamiltonian system with $H(x, y) = \frac{1}{2} y^2 - \frac{1}{2} x^2 + \frac{1}{4} x^4$ and $y = x'$. Its equilibria are $(0, 0)$, a hyperbolic saddle, and $(\pm 1, 0)$, two centres. The function $x(t) = \sqrt{2}\, \operatorname{sech} t$ is a solution. Check: with $u = \operatorname{sech} t$ one has $u'' = u - 2u^3$, so $x'' - x + x^3 = \sqrt{2}(u - 2u^3) - \sqrt{2} u + 2\sqrt{2} u^3 = 0$. Since $x(t) \to 0$ and $x'(t) \to 0$ as $t \to \pm\infty$, this solution is a homoclinic orbit of the saddle. By the symmetry $x \mapsto -x$, there is a second one in the half-plane $x < 0$.