Poincare Analyticity and the Complete Variational Equations

TL;DR

This paper discusses how solutions to differential equations are analytic functions of initial conditions and parameters, and introduces a method to compute their Taylor expansions using complete variational equations, enabling accurate polynomial approximations.

Contribution

It presents a systematic way to obtain Taylor expansions of solutions to differential equations via integration of complete variational equations, demonstrating practical applications.

Findings

Taylor expansions can be computed to any order using the proposed method.

Polynomial approximations accurately reproduce complex dynamical behaviors.

The method effectively captures phenomena like bifurcations and strange attractors.

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.