Birkhoff normal forms, KAM theory and continua of periodic points for certain planar system
Date of Original Version
By using the KAM theory and time reversal symmetries we investigate the stability of the equilibrium solutions of the system: (Formula Presented) where the parameter a > 0; and initial conditions x0 and y0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We also use the time reversal symmetry method to find effectively some feasible periods and the corresponding periodic orbits. Finally, we give computational procedure for finding an infinite number of periodic solutions with the given period. The second order difference equation obtained by eliminating xn from this system is an equation of the type yn+1 = f(yn; yn-1), where f is decreasing in both variables. Such equation can be embedded into fifth order difference equation which is increasing in all its arguments and it exhibits chaotic behavior.
Journal of Computational Analysis and Applications
Kulenović, M. R., E. Pilav, and N. Mujić. "Birkhoff normal forms, KAM theory and continua of periodic points for certain planar system." Journal of Computational Analysis and Applications 27, 3 (2019): 470-480. https://digitalcommons.uri.edu/math_facpubs/149