A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one-dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

Automated summary

We propose a modification of a Fourier analysis based method to obtain the Floquet characteristic exponents for periodic homogeneous linear systems with high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once the Floquet exponents are obtained, explicit approximated solutions are determined. The accuracy of our method is verified by testing on systems with known exact solutions, including one-dimensional periodic potentials of interest in quantum physics. We also study approximate solutions for homogeneous linear equations with periodic coefficients using the equivalent linear system.