A Continuous Archetype of Nonuniform Chaos in Area-Preserving Dynamical Systems |
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Authors: | S Cerbelli M Giona |
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Institution: | (1) Centro Interuniversitario sui Sistemi Disordinati e sui Frattali, nell’Ingegneria Chimica, Dipartimento di Ingegneria Chimica Universita di Roma "La Sapienza," Via Eudossiana, 18, 00184 Rome, Italy |
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Abstract: | We propose a piecewise linear, area-preserving, continuous map of the two-torus as a prototype of nonlinear two-dimensional
mixing transformations that preserve a smooth measure (e.g., the Lebesgue measure). The model lends itself to a closed-form
analysis of both statistical and geometric properties. We show that the proposed model shares typical features that characterize
chaotic dynamics associated with area-preserving nonlinear maps, namely, strict inequality between the line-stretching exponent
and the Lyapunov exponent, the dispersive behavior of stretch-factor statistics, the singular spatial distribution of expanding
and contracting fibers, and the sign-alternating property of cocycle dynamics. The closed-form knowledge of statistical and
geometric properties (in particular of the invariant contracting and dilating directions) makes the proposed model a useful
tool for investigating the relationship between stretching and folding in bounded chaotic systems, with potential applications
in the fields of chaotic advection, fast dynamo, and quantum chaos theory. |
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