The solution to the above system looks like,
the frequencies ,, and are functions of a,b and c My goal is to determine that values of a,b and c such that all w are Real. However, my original system of differential equations has nonlinear terms. I managed to derive the solution but in power expansion form or,
I need to determine the frequencies so that I can find the values of a,b and c such that all w are Real. Is it possible to derive w from limited power expansion (say 8th term) of x and y?
My first attempt:
I did a test of my method by taking Fourier Sine Transform of analytical solution which give me the answer in form of where w is the frequency. Then solved the g(w) for w which gives me four answers for w which are the frequencies ,, and . However when I attempted with Fourier Sine Transform of series solutions, the answer is different. This is due to the limited power expansion. Is here a way to improve this?
My second attempt:
I linearize the nonlinear terms of differential equations and used the matrix to calculate Determinant of (A-wI) where A is the matrix of the system of differential equation. I managed to calculate the values of a,b and c but they were incorrect because of linear terms I made.
A last method that I am considering is Monte Carlo. Is it possible to get a frequency equation from limited power expansion of differential equation solution? Any ideas for other methods that I missed?