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Thursday 12 Mar 2020Dynamics Seminar: Bounds on Lyapunov Exponents for Random and Deterministic Products of Shears

Rob Sturman - University of Leeds

Harrison 106 13:30-14:30

Lyapunov exponents are generally difficult to calculate rigorously. A well-known exception is the system of alternating shears on the 2-torus, or Arnold Cat Map. When such shear matrices are applied in a random order, the calculation is famously challenging. We give rigorous lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. More difficult still is the case where the randomness is replaced by deterministic dynamics. We study Lyapunov exponents for linked twist maps (canonical examples of non-uniformly hyperbolic systems), and again give rigorous lower and upper bounds. The motivation is two-fold: first, a wide class of fluid mixing and stirring devices can be modelled by composition of shears; second, although numerical computation of Lyapunov exponents may seem a simple task, we note that for linked twist maps, convergence of Lyapunov exponents is anomalously slow.

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