Sinai-Ruelle-Bowen Research Articles

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

Conditional mixing in deterministic chaos

AbstractWhile on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the Sinai–Ruelle–Bowen (SRB) measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized baker’s maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.

On bifurcation of statistical properties of partially hyperbolic endomorphisms

AbstractWe give an example of a path-wise connected open set of $C^{\infty }$ partially hyperbolic endomorphisms on the $2$ -torus, on which the (unique) Sinai–Ruelle–Bowen (SRB) measure exists for each system and varies smoothly depending on the system, while the sign of its central Lyapunov exponent changes.

Reclassification of Mood Disorders with Comorbid Medical Diseases based on Sinai-Ruelle-Bowen/ SRB Entropy Measures

Background: Current classification systems ignore the family histories of patients and psychiatric and medical comorbidity. Methods: We study a new approach of applying spectral clustering to determine distinct bipolar disorder subtypes, which is data whose clusters are of various sizes and densities. We discovered clusters by processing a SRB (Sinai-Ruelle-Bowen) similarity matrix that reflects the proximity of Von Bertalanffy’s functions fitted to phase growth dynamics of EEG (electroencephalography) within a new pipeline architecture. For this purpose, 109 patients diagnosed with bipolar disorder according to DSM-V (Diagnostic and Statistical Manual of Mental Disorders, fifth edition) were evaluated in remission period cross-sectionally. Results: We found three distinct bipolar disorder subtypes with the p-values < 0.001. We exhibit mixing sub-shifts of EEG phase gradients such that there are chaotic phase transitions but higher order phase gradients in a cone basin is always strictly convex. More surprisingly, we show that the SRB entropy measures on some time interval although there exist several equilibrium states each corresponds to equilibrium state. Conclusion: It seems subtypes of the bipolar spectrum were shaped according to seasonality, comorbidity for anxiety disorder and presence of psychotic symptom.

Entropy formula for systems with inducing schemes

We obtain entropy formulas for Sinai–Ruelle–Bowen (SRB) measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures are given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold.

Approximating the linear response of physical chaos

Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

SRB measures of singular hyperbolic attractors

<p style='text-indent:20px;'>It is known that hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.</p>

Rank-one strange attractors <i>versus</i> heteroclinic tangles

<p style='text-indent:20px;'>We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.</p><p style='text-indent:20px;'>Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a <i>prevalent</i> phenomenon.</p>

SRB Measures and Young Towers for Surface Diffeomorphisms

We give geometric conditions that are necessary and sufficient for the existence of Sinai–Ruelle–Bowen (SRB) measures for \(C^{1+\alpha }\) surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in non-uniformly hyperbolic sets which is of independent interest.

SRB measures for pointwise hyperbolic systems on open regions

A diffeomorphism f : M → M is pointwise partially hyperbolic on an open invariant subset N ⊂ M if there is an invariant decomposition TNM = Eu ⊕ Ec ⊕ Es such that Dxf is strictly expanding on E and contracting on E at each x ∈ N. We show that under certain conditions f has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure μ. If f is pointwise hyperbolic on N, then μ is a Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.

Some statistical properties of almost Anosov diffeomorphisms

Abstract For a kind of almost Anosov diffeomorphisms, we study the relationship among the existence of Sinai-Ruelle-Bowen (SRB) measures, the local differentiability near the indifferent fixed points, and space dimension, where the almost Anosov diffeomorphisms are hyperbolic everywhere except for the indifferent fixed points. As a consequence, there are C2 almost Anosov diffeomorphisms that admit σ-finite (infinite) SRB measures in spaces with dimensions bigger than one; there exist C2 almost Anosov diffeomorphisms with finite SRB measures in spaces with dimensions bigger than three. Further, we obtain the lower and upper polynomial bounds for the decay rates of the correlation functions of the Holder observables for the maps admitting finite SRB measures.

SRB measures in chaotic dynamical systems

In this paper, we present the construction of natural invariant measures so called SRB(Sinai-Ruelle-Bowen) measures by the properties of geometric t-potential and Bowen's equation for the hyperbolic attractors.

Polynomial decay of correlations for almost Anosov diffeomorphisms

We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai–Ruelle–Bowen (SRB) measures. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and are expressed by using coefficients of the third-order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed point.

Watermarking based on discrete wavelet transform and q-deformed chaotic map

Hierarchy of one-dimensional ergodic chaotic maps with Tsallis type of q-deformation are studied. We find that in the chaotic region, these maps with q-deformation are ergodic as the Birkhoff ergodic theorem predicts. q-deformed maps are defined as ratios of polynomials of degree N. Hence, by using the Stieltjes transform approach (STA), invariant measure is proposed. In addition, considering Sinai-Ruelle-Bowen (SRB) measure, Kolmogorov-Sinai (KS) entropy for q-deformed maps is calculated analytically. The new q-deformed scheme have ability to keep previous significant properties such as ergodicity, sensitivity to initial condition. By adding q-parameter to the hierarchy in order increase the randomness and one-way computation, we present a new scheme for watermarking. The introduced algorithm tries to improve the problem of failure of encryption such as small key space, encryption speed and level of security. To illustrate the effectiveness of the proposed scheme, some security analyses are presented. By considering the obtained results, it can be concluded that, this scheme have a high potential to be adopted for watermarking. It can be concluded that, the proposed novel watermarking scheme for image authentication can be applied for practical applications.

The Geometric Approach for Constructing Sinai–Ruelle–Bowen Measures

An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.

Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics

We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in ${R}^{N}$ with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we establish the emergence of strange attractors and Sinai--Ruelle--Bowen (SRB) measures with strong statistical properties (central limit theorem, exponential decay of correlations, etc.). We identify and study the mechanism responsible for the nonuniform hyperbolicity: saddle point shear. Our results apply to concrete systems of interest in the biological and physical sciences, such as May--Leonard models of Lotka--Volterra dynamics.

Convergence of rare event point processes to the Poisson process for planar billiards

We show that for planar dispersing billiards the distribution of return times is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB (Sinai–Ruelle–Bowen) measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied, as they require the maps to have bounded second derivatives everywhere. We first prove the Poisson limiting distribution assuming exponentially decaying correlations. For the case where the correlations decay polynomially, we induce on a subset on which the induced map has exponentially decaying correlations. We then prove a general theorem according to which the limiting return times statistics of the original map and the induced map are the same.

Statistical properties of generalized Viana maps

We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular, we construct a class of multi-dimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB (Sinai–Ruelle–Bowen) measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of SRB measures for skew-products associated with hyperbolic parameters by the study of fibred hyperbolic maps.

STRANGE ATTRACTORS IN A PERIODICALLY PERTURBED LORENZ-LIKE EQUATION

This paper studies a periodically perturbed Lorenz-like equation.We obtain three types of attractors: (i) periodic sinks, (ii) H¶enon-like attrac-tors, and (iii) rank one attractors. Among the three, (i) represent the stabledynamics of equation, and (ii) and (iii) represent chaotic behaviors character-ized by an Sinai-Ruelle-Bowen(SRB) measure. Each attractor admits an basinof positive Lebesgue measure, hence we observe it in numerical simulations.

Equilibrium states and SRB-like measures of $C^1$-expanding maps of the circle

For any C^1 -expanding map f of the circle we study the equilibrium states for the potential \psi=-\log |f'| . We formulate a C^1 generalization of Pesin’s Entropy Formula that holds for all the Sinai–Ruelle–Bowen (SRB) measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C^1 -generic case Pesin’s Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non-generic case for which no SRB measure exists.