Numerical inclusion of exact periodic solutions for time delay Duffing equation

Numerical inclusion results for exact periodic solutions are presented for the time delay autonomous Duffing equations. Constructive implicit function theorem is used for including one dimensional solution manifolds consisting of exact periodic solutions. A conjecture of a lower bound for a number of periodic solutions is given as a function of the time delay. If the delay time is less than 30, we have proved this conjecture using verified numerical computations. Theory for proving the existence of periodic solutions of the forced delay Duffing equation is proposed based on the verified numerical computations. The forced term is sinusoidal waves. Stress is on a study of the bifurcation of periodic solutions synchronizing to the external forces. A rich bifurcation phenomena of periodic solutions are reported taking the delay time as parameters. Especially, a kind of fractal structure is observed concerning resonance peaks.