2012/11/13 by Peter Giesl, Giesl, Peter, Sigurður Hafstein +1
Engineering · Physics and Astronomy · Chemistry · #Spacecraft Dynamics and Control #Quantum chaos and dynamical systems #Advanced NMR Techniques and Applications
paper · pdf · doi:10.48550/arxiv.1211.3022
A Riemannian metric with a local contraction property can be used to prove\nexistence and uniqueness of a periodic orbit and determine a subset of its\nbasin of attraction. While the existence of such a contraction metric is\nequivalent to the existence of an exponentially stable periodic orbit, the\nexplicit construction of the metric is a difficult problem.\n In this paper, the construction of such a contraction metric is achieved by\nformulating it as an equivalent problem, namely a feasibility problem in\nsemidefinite optimization. The contraction metric, a matrix-valued function, is\nconstructed as a continuous piecewise affine (CPA) function, which is affine on\neach simplex of a triangulation of the phase space. The contraction conditions\nare formulated as conditions on the values at the vertices.\n The paper states a semidefinite optimization problem. We prove on the one\nhand that a feasible solution of the optimization problem determines a CPA\ncontraction metric and on the other hand that the optimization problem is\nalways feasible if the system has an exponentially stable periodic orbit and\nthe triangulation is fine enough. An objective function can be used to obtain a\nbound on the largest Floquet exponent of the periodic orbit.\n