Multiplicative Ergodic Theorem Research Articles

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets’ multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman’s subadditive ergodic theorem.

Fractional stochastic vibration system under recycling noise

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems

Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are shown to equate with unit norm FTNMs at critical times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, are used to establish their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. The findings are deduced for systems with nondegenerate OLVs, and, as well, with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves. Efficient numerical methods for the calculation of leading CLVs are proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.

Analysis of a microfluidic chemostat model with random dilution ratios

In this paper, we first construct a microfluidic chemostat model for the growth of biofilms and planktonic populations with random dilution ratios and then investigate its dynamical behavior. Using the theory of monotone dynamical systems and the Multiplicative Ergodic Theorem, we show the existence of random attractors and stationary measures, and present Lyapunov exponents for the linearized cocycle with respect to the random model. Further on, if the top Lyapunov exponent is negative, we give the extinction of microbial populations, including the forward and pull-back trajectories.

The asymptotic behaviour of cocycles over flows

In 1968, V. I. Oseledets formulated the question of the convergence in Birkhoff’s theorem and in the multiplicative ergodic theorem for measurable cocycles over flows, under the condition of integrability at any fixed time. In 2016, A. M. Stepin and the author of this paper established convergence along subsets of density 1 on the time axis. Here we show that, moreover, convergence takes place modulo subsets of finite measure of the time axis.

A Dynamical Theory for Singular Stochastic Delay Differential Equations I: Linear Equations and a Multiplicative Ergodic Theorem on Fields of Banach Spaces

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

Studying finite-time (non)-domination in dynamical systems using Oseledec’s splitting. Application to the standard map

Oseledec’s theorem provides the necessary conditions for the existence of the decomposition of the tangent bundle called Oseledec’s splitting and the formal definition of the Lyapunov exponents. Using the concept of τ-domination of expansion or contraction rates of invariant subspaces after a time τ, together with the Oseledec theorem, we propose three quantifiers of τ-domination and τ-non-domination, which can be used to study the complex dynamics of physical systems. The interpretation of the Oseledec τ-domination in terms of the finite-time Lyapunov exponents is of great advantage for this purpose. Numerical results for the quantifiers are presented using the conservative standard map in a regime with mixed phase-space dynamics. Distinct typical regions in the phase-space are chosen, two containing a completely chaotic motion and two containing regularity islands. Results show that all quantifiers recognize that regions close to hyperbolic periodic points have a larger Oseledec τ-domination than regions around islands of regularity. Furthermore, Oseledec τ-non-dominated regions are also identified and are mostly close to regularity islands. We found a relation between the mean of expansion rates between the Oseledec subspaces and the mean of the angles between them.

Lagrangian chaos and scalar advection in stochastic fluid mechanics

We study the Lagrangian flow associated to velocity fields arising from various models of stochastic fluid mechanics. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive (almost surely, all particle trajectories are simultaneously exponentially sensitive with respect to initial conditions). Our main results are for the Navier–Stokes equations on $\mathbb{T}^2$ and the hyper-viscous regularized Navier–Stokes equations on $\mathbb{T}^3$ (at arbitrary fixed Reynolds number and hyper-viscosity parameters), subject to white-in-time, $H^s$-in-space stochastic forcing which is nondegenerate at high frequencies. Using these results, we further make a mathematically rigorous study of “passive scalar turbulence”. To this end, we study statistically stationary solutions to the passive scalar advection-diffusion advected by one of these velocity fields and subjected to a white-in-time random source. We show that the chaotic behavior of Lagrangian dynamics implies a type of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies Yaglom’s law of scalar turbulence – the universal scaling law analogous to the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems in infinite dimensions, namely the Multiplicative Ergodic Theorem and a version of Furstenberg’s Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.

Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay

The computational implementation of the algorithm for estimating the spectrum of Lyapunov exponents for systems of differential equations with delayed arguments is considered. Taking into account that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem, which makes it possible to efficiently calculate the required values, we only have to talk about estimates of characteristic exponents, in a sense close to Lyapunov ones. In this paper, we propose two methods for processing solutions of systems linearized on an attractor, one of which is based on a set of impulse functions; the other one is based on a set of trigonometric functions. The flexibility of application of the indicated algorithms is demonstrated in the case of quasi-stable structures, when several Lyapunov exponents are close to zero. The developed methods are tested on the logistic equation with delay. The results illustrate the “closeness” of the estimated characteristics and Lyapunov exponents.

Non-asymptotic Results for Singular Values of Gaussian Matrix Products

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N, the number of terms in the product, is large and n, the size of the matrices, may be large or small and may depend on N. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate for convergence of the empirical measure of the squared singular values to the uniform distribution on [0, 1], and results on the joint normality of Lyapunov exponents when N is sufficiently large as a function of n. Our technique consists of non-asymptotic versions of the ergodic theory approach at \(N=\infty \) due originally to Furstenberg and Kesten (Ann Math Stat 31(2):457–469, 1960) in the 1960s, which were then further developed by Newman (Commun Math Phys 103(1):121–126, 1986) and Isopi and Newman (Commun Math Phys 143(3):591–598, 1992) as well as by a number of other authors in the 1980s. Our key technical idea is that small ball probabilities for volumes of random projections gives a way to quantify convergence in the multiplicative ergodic theorem for random matrices.

A Multiplicative Ergodic Theorem for von Neumann Algebra Valued Cocycles

The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution.

Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.

Computing Covariant Lyapunov Vectors in Hilbert spaces

<p style='text-indent:20px;'>Covariant Lyapunov Vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions.</p><p style='text-indent:20px;'>We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a convergence theorem in the setting of [<xref ref-type="bibr" rid="b19">19</xref>]. The theorem relates the speed of convergence to the spectral gap between Lyapunov exponents. While the theorem is restricted to the above setting, our proof requires only basic properties that are given in many other versions of the multiplicative ergodic theorem.</p>

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Building on results obtained in [ 21 ], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [ 20 ].

Ergodic Theorems for Laminations and Foliations: Recent Results and Perspectives

This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, and Lyapunov exponents. We deal with various ergodic theorems for such laminations: random and operator ergodic theorems, (geometric) Birkhoff ergodic theorems, Oseledec multiplicative ergodic theorem, and unique ergodicity theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Section 5.2) are new ones.

Subadditive and multiplicative ergodic theorems

A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.

Variational principles for spectral radius of weighted endomorphisms of 𝐶(𝑋,𝐷)

We give formulas for the spectral radius of weighted endomorphisms a α : C ( X , D ) → C ( X , D ) a\alpha : C(X,D)\to C(X,D) , a ∈ C ( X , D ) a\in C(X,D) , where X X is a compact Hausdorff space and D D is a unital Banach algebra. Under the assumption that α \alpha generates a partial dynamical system ( X , φ ) (X,\varphi ) , we establish two kinds of variational principles for r ( a α ) r(a\alpha ) : using linear extensions of ( X , φ ) (X,\varphi ) and using Lyapunov exponents associated with ergodic measures for ( X , φ ) (X,\varphi ) . This requires considering (twisted) cocycles over ( X , φ ) (X,\varphi ) with values in an arbitrary Banach algebra D D , and thus our analysis cannot be reduced to any of the multiplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with α : C ( X , D ) → C ( X , D ) \alpha : C(X,D)\to C(X,D) . In particular, they are far-reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin, and others. They are most efficient when D = B ( F ) D=\mathcal {B}(F) , for a Banach space F F , and endomorphisms of B ( F ) \mathcal {B}(F) induced by α \alpha are inner isometric. As a by-product we obtain a dynamical variational principle for an arbitrary operator b ∈ B ( F ) b\in \mathcal {B}(F) and that its spectral radius is always a Lyapunov exponent in some direction v ∈ F v\in F when F F is reflexive.

Three problems solved by Sébastien Gouëzel

We present three results of Sebastien Gouezel's: the local limit theorem for random walks on hyperbolic groups, a multiplicative ergodic theorem for non-expansive mappings (joint work with Anders Karlsson), and the description of the essential spectrum of the Laplacian on \begin{document}$ SL(2,{\mathbb R}) $\end{document} orbits in the moduli space (joint work with Artur Avila).

Entropies of commuting transformations on Hilbert spaces

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $C^{2}$ $\mathbb{N}^2$-actions.

Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay

We consider the computational implementation of the algorithm for Lyapunov exponents spectrum numerical estimation for delay differential equations. It is known that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem which allows us to calculate the required parameters very efficiently. Therefore, we can only talk about the estimates of the characteristics in some sense close to the Lyapunov exponents. In this paper, we propose two methods of linearized systems solutions processing. One of them is based on a set of impulse functions, and the other is based on a set of trigonometric functions. We show the usage flexibility of these algorithms in the case of quasi-stable structures when several Lyapunov exponents are close to zero. The developed methods are tested on a logistic equation with a delay, and these tests illustrate the “proximity” of the obtained numerical characteristics and Lyapunov exponents.