A Singularity Analysis Approach to the Solutions of Duffing’s Equation

Abstract

The singularity structure of Duffing’s equation in the complex t-plane is investigated analytically and numerically. A series expansion for the general solution around each singularity t_* = t_R+it_I is given, and is subsequently used to approximate the locations of singularity “spirals” t _* ⁽ⁿ⁾ , n = 1,2,…, around every t_*. The main “chimney” patterns—on which singularities are observed to accumulate—are explained by deriving a simple expression for the distances between singularities on the “walls” of these chimneys |t_*−t _* ⁽¹⁾ |∼Q^−1/4exp(−t_I/2), Q being the amplitude of the (periodic) driving force. Thus, singularity patterns are seen to further “condense”, as Q increases and the motion becomes globally more chaotic in real t. These results suggest that series expansions near singularities in the complex t-plane can provide useful representations of the general solution of Duffing’s equation.