Can Cosine Terms in Differential Equations Disrupt Limit Cycle Construction?

[Dustinsfl]

s = NDSolve[{x'[t] == y[t], 
   y'[t] == -x[t] - .2*2*y[t] - .2*x[t]^3 + 3*.2*Cos[t], x[0] == 1, 
   y[0] == 2}, {x, y}, {t, 100}]ParametricPlot[Evaluate[{x[t], y[t] /. s}], {t, 0, 100}]

So I have a plane autonomous system but there is a $\cos t$ in the ODEs. I would use this code to construct a limit cycle but I think having a $\cos t$ in the ODE is messing it up. How can I proceed?

 

[mmwave]

Hello there,

First of all, great job on using NDSolve to solve your system of differential equations! As you mentioned, having a cosine term in your equations can make it difficult to construct a limit cycle. However, there are a few approaches you can try to overcome this challenge.

One option is to use the method of averaging, which involves taking the average of the rapidly varying term (in this case, the cosine term) over one period and replacing it with a constant. This can simplify your equations and make it easier to analyze the system.

Another approach is to use numerical methods specifically designed for systems with periodic or oscillatory behavior, such as the Runge-Kutta-Fehlberg method or the Bulirsch-Stoer method. These methods can handle the oscillations caused by the cosine term and provide accurate solutions.

Lastly, you can also try to rewrite your equations in terms of polar coordinates, where the cosine term can be expressed as a sine term. This can also simplify your equations and make it easier to analyze the system.

I hope these suggestions are helpful in tackling the challenge of constructing a limit cycle in your system. Best of luck with your research!