The figure below shows a solution to the system of ODEs $$\dot{q} = p$$ $$\dot{p} = -\lambda p - V'(q)$$ where $\lambda= 0.1$ and $V'(q) = q - q^3$. The figure suggests that the system converges to the fixed-point $q=1,p=0$. I want to prove the convergence of the system, by finding a Lyapunov function or finding contraction in some other way.
( solution curves for an initial condition $p=0.1, q=0$. note that the $p$ and $q$ axis were accidently swapped in the 2d figure. From http://www.asc.tuwien.ac.at/~juengel/sde/Lord.pdf)
I tried to look at the Eigenvalues of the Jacobian for the system, but I don't think that the criteria works for this system.
How can I prove that $(q(t), p(t) ) \rightarrow (1,0)$ as $t \to \infty$?