Controlling effect of geometrically defined local structural changes on chaotic Hamiltonian systems

Yossi Ben Zion, Lawrence Horwitz

Research output: Contribution to journalArticlepeer-review

Abstract

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy-dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a minimal method for achieving control of a chaotic system.

Original languageEnglish
Article number046217
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume81
Issue number4
DOIs
StatePublished - 29 Apr 2010

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