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Given the dynamical system $$ \begin{cases} \dot x = -(c - x^2 - y^2) y\\ \dot y = (x^2 + y^2) x \end{cases} $$ My aim is to study the stability at the origin. My feeling is that the origin is stable so, I should fine a Lyapunov in other to conclude.

Defintioon Given an autonomous system $$\dot x = f(x(t))$$ a Lyapunov function for this system at an equilibrium $x_0$ is any differentialbe function

$$ V: U_{x_0}\to \mathbb R$$ defines on a small neighborhood $U_{x_0}$ of $x_0$ such that,

$\bullet$ $V(x_0) =0$ and $V(x)>0$ for $x\neq 0$

$\bullet$ $\dot V(x(t)) \le 0$ for $x(t)\neq 0$.

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This system is invariant under $t \to -t$, $x \to -x$, $y \to -y$. Thus if there are solutions that approach the origin as $t \to +\infty$, there are also solutions that approach the origin as $t \to -\infty$. The only way you can have a Lyapunov function $V$ is if $\dot{V} = 0$. Then the function $V$ is an invariant, and the level curves of $V$ are trajectories.

In fact there is one:

$$ V = -2 x^2 + 2 y^2 - c \ln(1 - 2 x^2/c - 2 y^2/c) $$

If $c > 0$ this is positive for $x, y \ne (0,0)$ and sufficiently close to $(0,0)$: the origin is stable (but not asymptotically stable). However, for $c < 0$ it takes both positive and negative values, and the curves $V = 0$ are trajectories showing that the origin is not stable in this case.

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  • $\begingroup$ ROBERT, thanks for this function. May I ask who do you came up with this function? Precisely, is there any general rule which help to find the Lyapunov function ? I fact I tried to compute the Energy of the system I realized that the system i had to integrate was not integrable. Please any goog references for this type of issues? Thanks in advance. $\endgroup$
    – Guy Fsone
    Commented Jul 28, 2017 at 14:27
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    $\begingroup$ I asked Maple for an implicit solution of the DE $$ \dfrac{dy}{dx} = - \frac{(x^2+y^2)x}{(c-x^2-y^2)y}$$ $\endgroup$ Commented Jul 28, 2017 at 19:27

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