Find initial condition such that ODE solution is periodic

[TomAlso]

I have the following ODE system

$\begin{cases} x' = v \\ v' = v - \frac{v^3}{3} - x \\ x(0) = x_0 \\ v(0) = 0 \end{cases}$

I am asked to find $x_0>0$ such that the solution $(x(t),v(t))$ is periodic. Also, I need to find the period $T$ of such solution.

I don't know how to solve the system in the first place (or if it is even possible), so is there a way to figure out what $x_0>0$ will give a periodic solution without solving the system? Thanks!

 

[jvicens]

Thank you for your question. This system of ODEs is known as the Van der Pol oscillator and it is a classic example of a non-linear system that exhibits periodic behavior. It is possible to find a value for x_0 that will result in a periodic solution without explicitly solving the system. Here are the steps you can follow to find such a value:

1. Rewrite the system in terms of the phase variable, defined as \theta = \arctan(v/x). This will result in a system of two first-order ODEs for \theta and r = \sqrt{x^2 + v^2}.

2. Use the fact that the system is periodic to write the solution in terms of a Fourier series. This will result in an infinite series with unknown coefficients.

3. Use the initial conditions x(0) = x_0 and v(0) = 0 to determine the values of the unknown coefficients in the Fourier series.

4. Use the fact that the system is periodic to determine the period T of the solution. This can be done by finding the smallest positive value of t for which the solution repeats itself.

5. Finally, use the values of x_0 and T to write down the periodic solution for the system.

I hope this helps in finding a periodic solution for your system. Good luck with your research!