I'm preparing for my exam and I stumbled upon a question and I am a bit lost on how to write the correct solution.
The question goes as follows:
Prove that the equation $\ddot{x} + \mu(x^{4}-1)\dot{x} + x = 0$ has a unique stable limit cycle when $\mu > 0$. Determine whether it has a limit cycle when $\mu < 0$ and if it has, determine its stability.
I don't really understand how to go about this proof in a logical manner and provide a solution for all of these.
One way of establishing the existence of periodic orbits finding a quantity that is conserved along the trajectory of the dynamical system.
Let $\dot{x}=f(x), x\in \mathbb{R}^2$, be your, in this case, autonomous dynamical system.
Let $V(x) = \lVert x \rVert^2 = x_1^2 + x_2^2, x=(x_1,x_2)\in \mathbb{R}^2$, be a candidate conserved quantity.
Then if you consider the sign of $$\dot{V}=(\nabla_x V)^T\cdot f(x)$$
you will gain great insight on the character of the solution.
Consider also using the Matlab GUI pplane (http://math.rice.edu/~dfield/index.html) which allows you to easily simulate systems like this one.