A Variational Approach to Dynamical Systems and Its Numerical Simulation

Abstract

We propose and describe an alternative perspective to the study and numerical approximation of dynamical systems. It is based on a variational approach that seeks to minimize the quadratic error understood as a deviation of paths from being a solution of the corresponding system. Although this philosophy has been examined recently from the point of view of the direct method, we exploit optimality conditions and steepest descent strategies to establish precise and easy-to-implement numerical schemes for the approximation. We show the practical performance in a number of selected examples and indicate how this strategy, with minor changes, may also be used to deal with boundary value problems. Our emphasis is placed more so on relevant results that justify the numerical implementation and less on abstract theoretical results under optimal sets of assumptions.

Keywords:

• Error functional
• Optimality conditions
• Variational methods

AMS Subject Classification:

• 34A34
• 49K05
• 49M05
• 65L05

ACKNOWLEDGMENTS

The author would like to thank several anonymous referees for their patience in positively criticizing former drafts of this article.

Research was supported in part by MTM2007-62945 of the MCyT (Spain) and PCI08-0084-0424 of the JCCM (Castilla-La Mancha).