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86
SRB measures for partially hyperbolic systems whose central direction is mostly expanding
, 2000
"... We construct SinaiRuelleBowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms  the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting  under the assumption that the complementary subbundle is nonuniformly expanding. If the r ..."
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Cited by 107 (27 self)
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We construct SinaiRuelleBowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms  the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting  under the assumption that the complementary subbundle is nonuniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). 1 Introduction The following approach has been most effective in studying the dynamics of complicated systems: one tries to describe the average time spent by typical orbits in different regions of the phase space. According to the ergodic theorem of Birkhoff, such times are well defined for almost all point, with respect to any invariant probability measure. However, the...
Equations of motion from a data series
 Complex Systems
, 1987
"... Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reco ..."
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Cited by 41 (14 self)
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Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series. These equations of motion represent a vast reduction of a chaotic data set’s observed complexity to a compact, algorithmic specification. This approach employs an informational measure of model optimality to guide searching through the space of dynamical systems. As corollary results, we indicate how to estimate the minimum embedding dimension, extrinsic noise level, metric entropy, and Lyapunov spectrum. Numerical and experimental applications demonstrate the method’s feasibility and limitations. Extensions to estimating parametrized families of dynamical systems from bifurcation data and to spatial pattern evolution are presented. Applications to predicting chaotic data and the design of forecasting, learning, and control systems, are discussed. 1.
LowFrequency Variability in ShallowWater Models of the WindDriven Ocean Circulation. Part I: SteadyState Solutions
"... Successive bifurcations  from steady states through periodic to aperiodic solutions  are studied in a shallowwater, reducedgravity, 2.5layer model of the midlatitude ocean circulation subject to timeindependent wind stress. The bifurcation sequence is studied in detail for a rectangula ..."
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Cited by 26 (19 self)
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Successive bifurcations  from steady states through periodic to aperiodic solutions  are studied in a shallowwater, reducedgravity, 2.5layer model of the midlatitude ocean circulation subject to timeindependent wind stress. The bifurcation sequence is studied in detail for a rectangular basin with an idealized spatial pattern of wind stress. The aperiodic behavior is studied also in a NorthAtlanticshaped basin with realistic continental contours. The bifurcation sequence in the rectangular basin is studied in Part I, the present article. It follows essentially the one reported for singlelayer quasigeostrophic and 1.5layer shallowwater models. As the intensity of the NorthSouth symmetric wind stress is increased, the nearly symmetric doublegyre circulation is destabilized through a perturbed pitchfork bifurcation. The nearantisymmetry of the lowstress steady solution, with its nearly equal subtropical and subpolar gyres, is replaced by an approximately mirrorsymmetric pair of stable equilibria. On the upper branch, the subtropical gyre is stronger while on the lower one the subpolar gyre dominates. This perturbed pitchfork bifurcation is robust to changes in the interface friction between the two active layers and the thickness H 2 of the lower active layer. It persists in the presence of asymmetries in the wind stress and of changes in the model's spatial resolution and finitedi#erence scheme. Timedependent model behavior in the rectangular basin, as well as in the more realistic, NorthAtlanticshaped one, is studied in Part II. 2 1
Phase Transition in the Passive Scalar Advection
, 1998
"... The paper studies the behavior of the trajectories of fluid particles in a compressible generalization of the Kraichnan ensemble of turbulent velocities. We show that, depending on the degree of compressibility, the trajectories either explosively separate or implosively collapse. The two behaviors ..."
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Cited by 23 (0 self)
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The paper studies the behavior of the trajectories of fluid particles in a compressible generalization of the Kraichnan ensemble of turbulent velocities. We show that, depending on the degree of compressibility, the trajectories either explosively separate or implosively collapse. The two behaviors are shown to result in drastically different statistical properties of scalar quantities passively advected by the flow. At weak compressibility, the explosive separation of trajectories induces a familiar direct cascade of the energy of a scalar tracer with a shortdistance intermittency and dissipative anomaly. At strong compressibility, the implosive collapse of trajectories leads to an inverse cascade of the tracer energy with suppressed intermittency and with the energy evacuated by large scale friction. A scalar density whose advection preserves mass exhibits in the two regimes opposite cascades of the total mass squared. We expect that the explosive separation and collapse of Lagrangia...
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 21 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
On Irwin's Proof Of The Pseudostable Manifold Theorem
 Math. Z
, 1991
"... . We simplify and extend Irwin's proof of the pseudostable manifold theorem. 1 This preprint is available from the mathphysics electronic preprints archive. Send email to mp arc@math.utexas.edu for instructions 2 Supported in part by National Science Foundation Grants 3 email address: llave@ ..."
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Cited by 20 (6 self)
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. We simplify and extend Irwin's proof of the pseudostable manifold theorem. 1 This preprint is available from the mathphysics electronic preprints archive. Send email to mp arc@math.utexas.edu for instructions 2 Supported in part by National Science Foundation Grants 3 email address: llave@math.utexas.edu 4 email address: wayne@math.psu.edu  2  1. Introduction In [Ir1], Irwin introduced a very clever method to prove the stable manifold theorem near hyperbolic points. The proof was then, streamlined in [W]. Compared to previous proofs of the stable manifold theorem, the proof was technically quite simple since it only required the use of the implicit function theorem in Banach spaces. The Banach spaces considered had a very natural interpretation as spaces whose elements were orbits. This made the method very natural in the study of partially hyperbolic systems (Pesin theory) for which individual orbits are hyperbolic but there is little global hyperbolicity in the sys...
Measuring dynamical prediction utility using relative entropy
 J. Atmos. Sci
, 2002
"... A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. E ..."
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Cited by 20 (3 self)
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A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. Expressions for the new parameter in the case of Gaussian prediction ensembles are derived and a useful decomposition of utility into dispersion (roughly equivalent to ensemble spread) and signal components is introduced. Earlier measures of predictability have usually considered only the dispersion component of utility. A variety of simple dynamical systems with relevance to climate and weather prediction is introduced, and the behavior of their potential utility is analyzed in detail. For the climate systems examined here, the signal component is at least as important as the dispersion in determining the utility of a particular set of initial conditions. The simple ‘‘weather’ ’ system examined (the Lorenz system) exhibited different behavior with the dispersion being more important than the signal at short prediction lags. For longer lags there appeared no relation between utility and either signal or dispersion. On the other hand, there was a very strong relation at all lags between utility and the location of the initial conditions on the attractor. 1.
Simplest Normal Forms Of Hopf And Generalized Hopf Bifurcations
, 1999
"... The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, t ..."
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Cited by 19 (15 self)
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The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorems are presented to show that the conventional normal forms of Hopf and generalized Hopf bifurcations can be further simplified. The forms obtained in this paper for Hopf and generalized Hopf bifurcations are shown indeed to be the “simplest”, and at most only two terms remain in the amplitude equation of the “simplest normal form ” up to any order. An example is given to illustrate the applicability of the theory. A computer algebra system using Maple is used to derive all the formulas and verify the results presented in this paper.
Strange Attractors in SaddleNode Cycles: Prevalence and Globality
, 1996
"... We consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddlenode cycles. We show that they always exhibit Hénonlike strange attractors for a set of parameter values with positive Lebesgue density at the bifurcation value. In open classes of such families ..."
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Cited by 17 (4 self)
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We consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddlenode cycles. We show that they always exhibit Hénonlike strange attractors for a set of parameter values with positive Lebesgue density at the bifurcation value. In open classes of such families the strange attractors are of global type: their basins contain an a priori defined n eighbourhood of the cycle. Furthermore, the bifurcation parameter may also be a point of positive density of hyperbolic dynamics.
Symbolic dynamics of onedimensional maps: Entropies, finite precision, and noise
, 1982
"... In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical syste ..."
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Cited by 12 (6 self)
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In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetic and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.