Existence of global attractor in duffing equation

Question

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.

Answer 1

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.

Answer 2

Hint.

This DE can be written as

$$ \dot x = y\\ \dot y = a x-x^3-\epsilon y $$

with equilibrium points

$$ \left[ \begin{array}{ccc} x & y & \text{type} \\ 0 & 0 & \text{saddle}\\ -\sqrt{a} & 0 & \text{sink}\\ \sqrt{a} & 0 & \text{sink}\\ \end{array} \right] $$

This qualification is done according to the jacobian eigenvalues

$$ J = \left( \begin{array}{cc} 0 & 1 \\ a-3 x^2 & -\epsilon \\ \end{array} \right) $$

Attached an orbit for $a = 1, \epsilon = 0.1, x_0 = 2, y_0 = 1$