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* = tR+itI 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(−tI/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.
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Bountis, T., Bier, M., Papageorgiou, V. (1990). A Singularity Analysis Approach to the Solutions of Duffing’s Equation. In: Lakshmanan, M., Daniel, M. (eds) Symmetries and Singularity Structures. Research Reports in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-76046-4_11
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DOI: https://doi.org/10.1007/978-3-642-76046-4_11
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