I am searching for a proper Lyapunov function, supporting the following system of differential equations.
$$ \begin{array} {} \dot{x_1}=a\;x_1-b\;x_1^3 \\ \dot{x_2}=a\;x_2-b\;x_2^3 \end{array} $$
$a,b\in \mathbb{R}$ are constant parameters of the system.
Would appreciate, if anyone could help with such a function.
Well, since the system is decoupled, let's take $\dot{x} = - a x - b x^{3}$. Begin by choosing a Lyapunov function candidate such that $V(0) = 0$, and $V(x) > 0$ for all $x$ with $x \neq 0$ $$ V(x) = \frac{1}{2} x^{2}. $$ Taking the time derivative of $V$ along the trajectory of the system yields $$ \begin{aligned} \dot{V}(x) &= x \dot{x} \\ \dot{V}(x) &= x \left(- a x - b x^{3}\right) \\ \dot{V}(x) &= - a x^{2} - b x^{4}. \\ \end{aligned} $$ Case #1: If $a > 0$ and $b > 0$, then $\dot{V}(x) < 0$ for all $x$ with $x \neq 0$, which implies that the $x = 0$ is asymptotically stable.
Case #2: If $a > 0$ and $b < 0$, then $\dot{V}(x) < 0$ for $|x| < \sqrt{|\frac{a}{b}|}$ with $x \neq 0$, which implies that the origin is asymptotically stable but surrounded by an unstable limit cycle, that is outside of $x = \pm \sqrt{|\frac{a}{b}|}$. Its region of attraction is the interior of the limit cycle, and the boundary of the region of attraction is the limit cycle itself. The region of attraction can be enlarged by increasing $\sqrt{|\frac{a}{b}|}$.
Case #3: If $a < 0$ and $b > 0$, then $\dot{V}(x) < 0$ for $|x| > \sqrt{|\frac{a}{b}|}$ with $x \neq 0$, which implies that the origin is unstable but surrounded by a stable limit cycle, that is outside of $x = \pm \sqrt{|\frac{a}{b}|}$. Practical stabilization can be achieved by decreasing $\sqrt{|\frac{a}{b}|}$.
Case #4: If $a < 0$ and $b < 0$, then $\dot{V}(x) > 0$ for all $x$ with $x \neq 0$, which implies that the origin is unstable.