Divakar Viswanath 2003 Nonlinearity 16 1035 doi:10.1088/0951-7715/16/3/314
Symbolic dynamics and periodic orbits of the Lorenz attractor*
Divakar Viswanath
Show affiliationsRecommended by J Keating
The butterfly-like Lorenz attractor is one of the best known images of chaos. The computations in this paper exploit symbolic dynamics and other basic notions of hyperbolicity theory to take apart the Lorenz attractor using periodic orbits. We compute all 111011 periodic orbits corresponding to symbol sequences of length 20 or less, periodic orbits whose symbol sequences have hundreds of symbols, the Cantor leaves of the Lorenz attractor, and periodic orbits close to the saddle at the origin. We derive a method for computing periodic orbits as close as machine precision allows to a given point on the Lorenz attractor. This method gives an algorithmic realization of a basic hypothesis of hyperbolicity theory—namely, the density of periodic orbits in hyperbolic invariant sets. All periodic orbits are computed with 14 accurate digits.
- PACS
- MSC
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37D45 Strange attractors, chaotic dynamics
37B10 Symbolic dynamics (See also 37Cxx, 37Dxx)
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
- Subjects
- Dates
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Issue 3 (May 2003)
Received 4 November 2002, in final form 17 March 2003
Published 4 April 2003
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Symbolic dynamics and periodic orbits of the Lorenz attractor
Divakar Viswanath 2003 Nonlinearity 16 1035
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