Poincare Analyticity and the Complete Variational Equations

Abstract

According to a theorem of Poincare, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters providing the right sides of the differential equations are analytic in the variables, the time, and the parameters. We describe how these Taylor expansions may be obtained, to any desired order, by integration of what we call the complete variational equations. As illustrated in a Duffing equation stroboscopic map example, these Taylor expansions, truncated at an appropriate order thereby providing polynomial approximations, can well reproduce the behavior (including infinite period doubling cascades and strange attractors) of the solutions of the underlying differential equations.

86 pages including 22 figures

Cited by in corpus (4)

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• Violation of Cauchy-Schwarz inequalities by spontaneous Hawking radiation in resonant boson structures
• Regular and chaotic motion in general relativity: The case of an inclined black hole magnetosphere
• Convergence of Taylor Transfer Map for Duffing Equation