Period of the DE trajectory

[srpvx]

I have a second order nonlinear ODE. I know that a trajectory with specified initial conditions $$\left[ x(0) = x_0, \dot{x}(0) = \dot{x}_0 \right]$$ is periodic. How can I numerically calculate period of this trajectory without solve this DE?

 

[rcgldr]

Are you allowed to use a numerical method like Runge Kutta or would that be considered "solving" the ODE?

 

[srpvx]

Are you allowed to use a numerical method like Runge Kutta or would that be considered "solving" the ODE?

I mean that the numerical solution like the attachment paper.

 

[rcgldr]

That's a method that I'm not familiar with. Hopefully someone here may be able to help.

 

[LudusRex]

To calculate the period of a trajectory without solving the differential equation, you can use the Poincaré map method. This method involves plotting the trajectory in phase space and identifying the points where the trajectory crosses a certain plane or surface, known as the Poincaré section. The distance between two consecutive crossings of the Poincaré section is equal to the period of the trajectory. This method is particularly useful for nonlinear systems, where finding an analytical solution may be difficult. Additionally, numerical methods such as the shooting method or the Runge-Kutta method can also be used to approximate the period of the trajectory. These methods involve iterating through smaller time steps and comparing the results to the initial conditions until the trajectory returns to its starting point. However, it is important to note that these numerical methods may not be as accurate as solving the differential equation directly.