Chaotic saddles in complex spatiotemporal systems paper

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Chaotic saddles in complex spatiotemporal
systems(paper 51)

• Abraham C.-L. Chian
• National Institute for Space Research (INPE),
  Brazil
• Rodrigo A. Miranda
• National Institute for Space Research (INPE),
  Brazil
• Centre for Quaternary Research (CEQUA), Chile
• Erico L. Rempel
• Institute of Aeronautical Technology (ITA),
  Brazil

www.cea.inpe.br/wiser
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Chaotic sets

• Chaotic Attractors
• - Set of unstable periodic orbits
• - Positive maximum Lyapunov exponent
• - Attract all initial conditions in a given
  neighbourhood
• - Basin of attraction (continuous stable
  manifolds, without gaps)
• - Responsible for asymptotic chaos
• Chaotic Saddles
• - Set of unstable periodic orbits
• - Positive maximum Lyapunov exponent
• - Repel most initial conditions from their
  neighbourhood, except those on stable manifolds
• - No basin of attraction (fractal stable
  manifolds, with gaps)
• - Responsible for transient chaos
• Ref Rempel Chian, PRL 98, 014101, 2007
• Rempel, Chian Miranda, PRE (submitted)

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Nonlinear regularized long-wave equation
?t? a?txx? c?x? f??x? - gf - ? sin(x - ?t)
a - 0.287, c 1, f - 6, ? - 0.1 and W
0.65 Periodic BC f(x,t) f(x2?, t) f(x,t)åk
bk(t)exp(ikx), k-N,...,N Pseudospectral
method with N32 Poincaré map Reb1(t)0 and
dReb1(t)/dt gt 0
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Spatiotemporal dynamics
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Energy time Series
E ò?2 a?x?2dx/4p
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Chaotic Sets chaotic saddles chaotic attractors
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Lyapunov exponents and correlation dimensions
Ref Rempel Chian, PRL 98, 014101, 2007
Rempel, Chian Miranda, PRE (submitted)