Convex Optimization Methods for Bounding Lyapunov Exponents




Oeri, Hans

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In dynamical systems, the stability of orbits is quantified by Lyapunov exponents (LEs), which are computed from the average rate of divergence of trajectories. We develop techniques for computing sharp upper bounds on the largest LE over trajec- tories using methods from convex optimization, which have previously been used to compute sharp bounds on the time averages of scalar quantities on bounded orbits of dynamical systems. For discrete-time dynamics we develop an optimization-based approach for computing sharp bounds on the geometric mean of scalar quantities. We therefore express LEs as infinite-time averages and as geometric means in continuous- time systems and discrete-time systems, respectively, and then derive optimization problems whose solutions give sharp bounds on LEs. When the system’s dynamics is governed by a polynomial vector field, the problems can be relaxed to computa- tionally tractable sum-of-squares (SOS) whose solutions also give sharp bounds on LEs. An approach for the practical implementation of a sequence of SOS feasibility problems whose solutions converge to the maximal LE of discrete systems is provided. We explain how symmetries can be used to simplify and generalize the optimization problems in both continuous-time and discrete-time systems. We conclude by dis- cussing the extension of the techniques developed here to the problem of bounding the sum of the leading LEs. Tractable SOS programs are derived for some special cases of this problem. The applicability of all the techniques developed here is shown by applying them to various explicit examples. For some systems we numerically compute sharp bounds that agree with the the maximal LEs, and for some we prove analytic bounds on maximal LEs by solving the optimization problems by hand.



Lyapunov exponents, Convex optimization, sum-of-squares programming, Dynamical systems, Chaos