Definition: A $\underline{\text{Limit Cycle}}$ in the phase-plane of a second-order system is an isolated closed trajectory.

A Trajectory needs to be closed to indicate the periodic nature of the motino and isolated to indicate the ????? nature of the system.

There are 3 kinds of Limit Cycles:

1. Stable: All trajectories in the vicinity of the Limit Cycle converge to the Limit Cycle as $t \rightarrow \infty$
2. Unstable: All trajectories in the vicinity of the Limit Cycle diverge from it as $t \rightarrow \infty$
3. Semi-Stable: Some of the trajectories in the vicinity of the Limit Cycle converge to it as $t \rightarrow \infty$

Example:

$\begin{cases} \dot{x}_1 = x_2 - x_1(x_1^2 + x_2^2 - 1) \\ \dot{x}_2 = -x_1 - x_2(x_1^2 + x_2^2 -1) \end{cases} \Longrightarrow \underline{x} = [0, 0]^T \text{ (equillibrium points)}$

The system shows a Limit Cycle $\Rightarrow$ best seen in polar coords:

$\displaystyle r = \sqrt{x_1^2 + x_2^2}; \quad \theta = \tan^-1 \left(\frac{x_2}{x_1}\right) \Rightarrow \text{ transformation}$

$\begin{cases} \dot{r} = r(r-1) \\ \dot{\theta} = -1 \end{cases} \Rightarrow \, \dot{r} = 0, \,\text{when } r=0,\, 1$

$r=0$ is an isolated point at the origin. $r=1$ is a unitary circle. Let's look at the overall solution:

$\displaystyle \begin{matrix} r(t) = && \frac{1}{\sqrt{1 + c_0 e^{-2 t}}} \\ \theta(t) = && \theta_0 - t \end{matrix} , \quad c_0 = \frac{1}{r_0^2} - 1$

For $r_0 = 1 \Rightarrow r(t) = 1$. $r(t)$ stays on the circle and rotates periodically with $\dot{\theta}(t) = -1$

For any other solution $(r_0 \neq 1) \Rightarrow c_0 e^{-2t} \rightarrow 0 \Rightarrow r(t) \rightarrow 1 \Rightarrow \underline{\text{ Stable Limit Cycle!}}$