Existence of global attractor in duffing equation How to prove the existence and identify global attractor in Duffing equation $$\ddot{x}+\epsilon \dot{x}+x^3-ax=0$$ where $\epsilon >0$ and $a>0$? I found a definition: A bounded closed set $A_1 \subset X$ is called a global attractor for a dynamical system $(X, S_t)$, if * $A_1$ is an invariant set * the set $A_1$ uniformly attracts all trajectories starting in bounded sets, i.e. for any bounded set $B$ from $X$ $$\lim_{t\to \infty} \sup \lbrace \operatorname{dist}(S_t y, A_1): y\in B \rbrace=0$$ where $\operatorname{dist}(z,A)=\inf\lbrace\operatorname{d}(z,y): y\in A\rbrace$ where $\operatorname{d}(z,y)$ is the distance between the elements $z$ and $y$ in $X$. I finished only ODE course and I don't know a lot about dynamical systems.

The equation describes a mechanical system with friction/energy dissipation. Or in formulas $$ \frac{d}{dt}\left[\frac12\dot x^2+\frac14(x^2-a)^2\right]=-ϵ\dot x^2. $$ So as long as the particle the system describes is in motion, it will lose energy and move down to one of the minima $x=\pm\sqrt{a}$, which both are stable equilibrium points of the equation.