I have this exercise extracted from an exercise sheet the teacher gave us about qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov)
I don't know how to solve it: For 1. it's ok but from 2. and 3. I don't have any idea.
Let $k> 0$. Note $(x(t),y(t)),t\in I$ the maximal solution of
$\left\{ \begin{array}{lcc} \frac{dx}{dt}\, =\, y & ,x(0)=x_0 \\ \\ \frac{dy}{dt}\, =\, x(1-x^2)-ky & ,y(0)=y_0 \end{array} \right.$
Let $A(x,y)=\frac{1}{2}(x^2-1)^2+y^2$ and for $k>0$ the function $A$ is a function of Lyapunov
- Show that the solution exists for all positive times
- Show that $\lim_{t\to+\infty}(x(t),y(t))$ exists.
- What happens for $k <0$ and $k = 0$
For 1. I used the following result
Theorem: If $\dot{x}=f(x)$ has a Lyapunov function and if for all $R>0$, the set $\{x\in \mathbb{R}^n: V(x)\leq R\}$ is bounded then the maximal solution of Cauchy's problem is global in positive time.
I don't know how to solve 2. and 3.
Can someone help me?
Thank you
For $k>0$, can the solution stabilize on any positive level of $A$?
$k=0$ should be easy, as $A$ is also a first integral.
For $k<0$, can you bound the growth of $A$ in terms of $A$?