How should I show that the index of a limit cycle is ## 1 ##?

[Math100]

Homework Statement

Show that the index of a limit cycle is $1$.

Relevant Equations

If $\Gamma$ surrounds $n$ equilibrium points $P_{1}, P_{2}, ..., P_{n}$, then $I_{\Gamma}=\sum_{i=1}^{n}I_{i}$, where $I_{i}$ is the index of the point $P_{i}$ for $i=1, 2, ..., n$.

Proof:

Consider the index of a limit cycle.
By definition, a limit cycle is an isolated periodic solution of an autonomous system represented in the phase plane by an isolated closed path.
The theorem states: If $\Gamma$ surrounds $n$ equilibrium points $P_{1}, P_{2}, ..., P_{n}$, then $I_{\Gamma}=\sum_{i=1}^{n}I_{i}$, where $I_{i}$ is the index of the point $P_{i}$ for $i=1, 2, ..., n$.
Let $I_{\Gamma}=\frac{1}{2\pi}\triangle\theta_{\Gamma}$ where $I_{\Gamma}$ is the index of the curve $\Gamma$ and $\triangle\theta_{\Gamma}$ is the total change in the angle of the vector field along $\Gamma$.
Note that the vector field along a limit cycle $\Gamma$ behaves such that the direction of the vector field is always tangent to $\Gamma$.
Since the limit cycle traverses once, it follows that the vector field rotates once, and the total angular change of the vector field along the limit cycle is $2\pi$.
Thus, $I_{\Gamma}=\frac{1}{2\pi}\triangle\theta_{\Gamma}=\frac{1}{2\pi}(2\pi)=1$.
Therefore, the index of a limit cycle is $1$.

Above is the proof for this problem. May anyone please take a look and verify/confirm if it's accurate/correct?

 

[pasmith]

The result follows essentially by inspection, but you should at least define the index of a closed curve $\Gamma$ for the system $(\dot x, \dot y) = (f_1,f_2)$ as $$\frac{1}{2\pi}\int_{\Gamma} d\arctan\left( \frac{f_2}{f_1} \right) = \frac{1}{2\pi}\int_{\Gamma} \frac{f_1df_2 - f_2df_1}{f_1^2 + f_2^2}.$$