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Coherent structures and isolated spectrum for PerronFrobenius cocycles
, 2008
"... We present an analysis of onedimensional models of dynamical systems that possess “coherent structures”; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation u ..."
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Cited by 20 (8 self)
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We present an analysis of onedimensional models of dynamical systems that possess “coherent structures”; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for noninvertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalise the notions of almostinvariant and almostcyclic sets to nonautonomous dynamical systems and provide a new ensemblebased formalism for coherent structures in onedimensional nonautonomous dynamics.
On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps,
 DYN. SYST. SER. A
, 2007
"... PerronFrobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almo ..."
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Cited by 12 (5 self)
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PerronFrobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a PerronFrobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam’s method well approximates the isolated spectrum and eigenfunctions of this map.
Probability Density Functions of Decaying Passive Scalars in Periodic Domains: An Application of SinaiYakhot Theory
, 2008
"... Employing the formalism introduced by Sinai and Yakhot [PRL, 63(18), p. 1962, 1989], we study the probability density functions (pdf’s) of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields. The particular regime we focus on is one where the normal ..."
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Cited by 1 (0 self)
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Employing the formalism introduced by Sinai and Yakhot [PRL, 63(18), p. 1962, 1989], we study the probability density functions (pdf’s) of decaying passive scalars in periodic domains under the influence of smooth large scale velocity fields. The particular regime we focus on is one where the normalized scalar pdf’s attain a selfsimilar profile in finite time, i.e., the so called strange or statistical eigenmode regime. In accordance with the work of Sinai and Yakhot, the central regions of the pdf’s are power laws. But the details of the pdf profiles are dependent on the physical parameters in the problem. Interestingly, for small Peclet numbers the pdf’s resemble stretched or pure exponential functions, whereas in the limit of large Peclet numbers, there emerges a universal Gaussian form for the pdf. Numerical simulations are used to verify these predictions. 1.
Scalar decay in chaotic mixing
 Transport and Mixing in Geophysical Flows, volume 744 of Lecture Notes in Physics
, 2008
"... Summary. I review the local theory of mixing, which focuses on infinitesimal blobs of scalar being advected and stretched by a random velocity field. An advantage of this theory is that it provides elegant analytical results. A disadvantage is that it is highly idealised. Nevertheless, it provides i ..."
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Cited by 1 (1 self)
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Summary. I review the local theory of mixing, which focuses on infinitesimal blobs of scalar being advected and stretched by a random velocity field. An advantage of this theory is that it provides elegant analytical results. A disadvantage is that it is highly idealised. Nevertheless, it provides insight into the mechanism of chaotic mixing and the effect of random fluctuations on the rate of decay of the concentration field of a passive scalar. 1
Alternate rotating walls for thermal chaotic mixing
, 2009
"... In this study, we numerically investigate the evolution of twodimensional mixing and heat transfer enhancement within a tworod stirring device. The fluid is heated by the walls, which are maintained at a constant temperature. We show by analysis of different stirring protocols that the use of disc ..."
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In this study, we numerically investigate the evolution of twodimensional mixing and heat transfer enhancement within a tworod stirring device. The fluid is heated by the walls, which are maintained at a constant temperature. We show by analysis of different stirring protocols that the use of discontinuous wall rotations is necessary to promote heat transfer by chaotic mixing. This condition is also required to avoid hot spots in the vicinity of the walls. The statistics of temperature scalars (mean and standard deviation of dimensionless temperature fields) allow us to determine the influence of geometrical and physical parameters on mixing and heating performance. Thermal strange eigenmodes are revealed during the mixing process by the development of complex recurrent patterns, and the selfsimilar character of temperature evolutions is confirmed by the probability distribution functions of the rescaled nondimensional temperature.
Pattern Formation Induced by TimeDependent Advection
"... Abstract. We study patternforming instabilities in reactionadvectiondiffusion systems. We develop an approach based on LyapunovBloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space ..."
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Abstract. We study patternforming instabilities in reactionadvectiondiffusion systems. We develop an approach based on LyapunovBloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a patternforming instability in a twocomponent system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants. Key words: pattern formation, reactionadvectiondiffusion equation AMS subject classification: 35B36, 35K57, 92E20 1.
Advected fields in maps: III. Passive scalar decay in baker’s maps
"... The decay of passive scalars is studied in baker’s maps with uneven stretching, in the limit of weak diffusion. The map is alternated with diffusion, and three different boundary conditions are employed, zero boundaries, noflux boundaries, and periodic boundaries. Numerical results are given for sc ..."
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The decay of passive scalars is studied in baker’s maps with uneven stretching, in the limit of weak diffusion. The map is alternated with diffusion, and three different boundary conditions are employed, zero boundaries, noflux boundaries, and periodic boundaries. Numerical results are given for scalar decay modes. A set of eigenmode branches and eigenfunctions is also set up for case of zero diffusion, using a complex variable formulation. The effects of diffusion may then be included by means of a boundary layer theory. Depending on the boundary conditions, the effect of diffusion is to either simply perturb or entirely destroy each zerodiffusion branch. The paper considers analytically the decay of passive scalar fluctuations for each boundary condition, and elucidates scaling laws that govern the behaviour of eigenvalues in the limit of weak diffusion.
Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study
, 2014
"... Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems. These vectors identify spatially varying directions of ..."
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Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems. These vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles. In recent years new computational methods for approximating Oseledets vectors have been developed, motivated by increasing model complexity and greater demands for accuracy. In this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of Ginelli et al. [17] and Wolfe and Samelson [34]. We compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry, spectral separation, and dimension. We also investigate which methods perform well with limited data. 1