I am looking at systems of (nonlinear) differential equations in the spirit of Strogatz' Nonlinear Dynamics and Chaos.
Given a system of
- $\frac{dx}{dt} = y^3 - 4x$
- $\frac{dy}{dt} = y^3 - y- 3x$
show that $|x(t) - y(t)| \rightarrow 0$ for $t \rightarrow \infty$.
My basic idea was to consider (very informally) $$|x(t) - y(t)| = |x(t-1) + \frac{dx}{dt} - y(t-1) + \frac{dy}{dt}| \\= |x(t-1) - y(t-1) - x + y | "=" 0$$ i.e. looking at the change from one timepoint to another and then seeing that the whole thing cancels itself out.
However, I don't really know how to express this idea properly. Further, as of yet I am not considering "low" values of $t$ or using that $t \rightarrow \infty$.
I'd be grateful for any tips on how to go about this.
If you substract both equations, you get: $$x'-y'=y-x$$ $$\int \frac {d(x-y)}{x-y}=-\int dt$$ $$\int \frac {d(x-y)}{x-y}=-t+c$$ $$\implies \ln |x-y|=-t+c$$ $$|x-y|=Ke^{-t}$$ Then you have: $$|x-y|\rightarrow 0 \text { for } t \rightarrow \infty$$