Lyapunov exponents for the random product of two shears

Abstract

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring devices. The bounds, obtained by considering invariant cones in tangent space, give excellent accuracy compared to standard and general bounds, and are increasingly accurate with increasing shear. Bounds on generalised exponents are useful for testing numerical methods, since these exponents are difficult to compute in practice.

arXiv comment18 pages, 12 figures, iopart LaTeX style

References in corpus (9)

• Frontiers of chaotic advection
• Topology, Braids, and Mixing in Fluids
• Estimating generalised Lyapunov exponents for products of random matrices
• Topological optimisation of rod-stirring devices
• Exact Lyapunov Exponent for Infinite Products of Random Matrices
• Exponential Decay of Correlations for Randomly Chosen Hyperbolic Toral Automorphisms
• Scalar Decay in Chaotic Mixing
• How do random Fibonacci sequences grow?
• The mathematics of taffy pullers

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• Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime
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