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.

A continuous archetype of nonuniform chaos in area-preserving dynamical systems / Cerbelli, Stefano; Giona, Massimiliano. - In: JOURNAL OF NONLINEAR SCIENCE. - ISSN 0938-8974. - 15:6(2005), pp. 387-421. [10.1007/s00332-004-0673-2]

A continuous archetype of nonuniform chaos in area-preserving dynamical systems

CERBELLI, Stefano;GIONA, Massimiliano
2005

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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A continuous archetype of nonuniform chaos in area-preserving dynamical systems / Cerbelli, Stefano; Giona, Massimiliano. - In: JOURNAL OF NONLINEAR SCIENCE. - ISSN 0938-8974. - 15:6(2005), pp. 387-421. [10.1007/s00332-004-0673-2]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/236418
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