Lyapunov's stability

Asked Nov 20 '13 at 21:13

Active Nov 20 '13 at 23:06

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I'm working on the following differential equations: $$\frac{dN}{dt} = wN(1-\frac{N}{p})$$ $$\frac{dK}{dt} = sK - gKN$$

where $w, p, s, g$ are real numbers $\ge 0$ and $N, K$ are always positive (or equal $0$).

I'm trying to find equilibrium points and check if they are stable.

My attempt:

I know 3 types of equilibrium points (N,K): $ (0,0), (p,0), (p, K)$ (in the last one, $K$ is any real number $>0$ and $N = p = \frac{s}{g}$).

For the point (0,0) and point (p,0) (if $p \ne \frac{s}{g} $) it's easy to check from linearity. I was able to do that.

But I'm struggling with points $(p,K)$, where $K \ge 0$ and $N = p = \frac{s}{g}$. I tried to find maybe some Lyapunov function, or "antiLyapunov" function, but didn't succeed. I only know that it is not asymptotically stable (some real parts of eigenvalues from linearity are equal to $0$), but what about Lyapunov's stability? Can you help and show me how to check it? I've spent a lot of time on it, but can't get any results...

edited Nov 20 '13 at 23:06

asked Nov 20 '13 at 21:13

Anne

1,557

A related problem. – Mhenni Benghorbal Nov 20 '13 at 21:19
But there they use Jacobi matrix, which is useless in my case- the real part of at least one eigenvalue is zero, so I can't say anything about Lyapunov's stability. – Anne Nov 20 '13 at 21:23
Perhaps linearisation of the system around the equilibrium points may help. – daulomb Nov 20 '13 at 21:26
@Anne: You need to look in the literature. – Mhenni Benghorbal Nov 20 '13 at 21:27

Lyapunov's stability

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