On a convergent nonlinear perturbation theory without small denominators or secular terms

Abstract

We demonstrate a method for solving the dynamics of systems of extremely nonlinear coupled anharmonic oscillators with polynomial force laws. Several convergent perturbative and iterative techniques are described and applied, yielding the Fourier frequencies and coefficients of the time dependent solution. The usual stumbling block in a series analysis of coupled nonlinear systems is the appearance of terms with arbitrarily small denominators resulting in a rapidly divergent series. We avoid this classic ’’small denominators problem’’ by concentrating on the periodic solutions of a system. We employ the ’’Poincaré recurrence time’’ T_r of a system as the period of the solution, i.e., each Fourier frequency can be expressed as n (2π/T_r), with some integer n. These periodic solutions are dense among all possible solutions and suffice for our practical calculations (just as the rational numbers suffice for most practical computations). Hence we construct the solutions with rationally dependent Fourier frequencies whereas the Kolmogorov–Arnold–Moser theorems demonstrate the existence of certain solutions whose frequencies are rationally independent. As an example we treat a two oscillator system equivalent to the ’’Duffing equation,’’ a driven anharmonic oscillator appearing in models of the laser, the driven pendulum, etc. In our analysis no small denominators arise. We extend the region of bounded solutions beyond the (numerical) estimates in the literature and add to the list of peculiar phenomena exhibited by the Duffing equation.