From limit cycles to strange attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.Comment: 27 page

Generalized nonuniform dichotomies and local stable manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.Comment: 18 pages. New version with minor corrections and an additional theorem and an additional exampl

Perturbations of Noise: The origins of Isothermal Flows

We make a detailed analysis of both phenomenological and analytic background for the "Brownian recoil principle" hypothesis (Phys. Rev. A 46, (1992), 4634). A corresponding theory of the isothermal Brownian motion of particle ensembles (Smoluchowski diffusion process approximation), gives account of the environmental recoil effects due to locally induced tiny heat flows. By means of local expectation values we elevate the individually negligible phenomena to a non-negligible (accumulated) recoil effect on the ensemble average. The main technical input is a consequent exploitation of the Hamilton-Jacobi equation as a natural substitute for the local momentum conservation law. Together with the continuity equation (alternatively, Fokker-Planck), it forms a closed system of partial differential equations which uniquely determines an associated Markovian diffusion process. The third Newton law in the mean is utilised to generate diffusion-type processes which are either anomalous (enhanced), or generically non-dispersive.Comment: Latex fil

Bernstein Processes Associated with a Markov Process

Abstract. A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the rela-tions with statistical physics concepts (Gibbs measure, entropy,...) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (ac-tion functional, path integrals, Noether’s Theorem,...) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach. This is a review of various recent developments regarding the construction and properties of Bernstein processes, a class of diffusions originally introduced for the purpose of Euclidean Quantum Mechanics (EQM), a probabilistic analogue o

Minimizing orbits in the discrete Aubry-Mather model

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We consider a generalization of the Frenkel-Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one-dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual 'total ordering' of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study of the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton-Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as 'ground states' obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the noncrossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.242563611Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)ANR [BLANC07-3_187245]Hamilton-JacobiWeak KAM TheoryConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)ANR [BLANC07-3_187245

Description of Some Ground States by Puiseux Techniques

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Let (Sigma(+)(G), sigma) be a one-sided transitive subshift of finite type, where symbols are given by a finite spin set S, and admissible transitions are represented by an irreducible directed graph G subset of S x S. Let H : Sigma(+)(G) -> R be a locally constant function (that corresponds with a local observable which makes finite-range interactions). Given beta > 0, let mu(beta H) be the Gibbs-equilibrium probability measure associated with the observable -beta H. It is known, by using abstract considerations, that {mu(beta H)}(beta>0) converges as beta -> +infinity to a H-minimizing probability measure mu(H)(min) called zero-temperature Gibbs measure. For weighted graphs with a small number of vertices, we describe here an algorithm (similar to the Puiseux algorithm) that gives the explicit form of mu(H)(min) on the set of ground-state configurations.1461125180Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)ANR [BLANC07-3_18724]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)ANR [BLANC07-3_18724

On calibrated and separating sub-actions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)We consider a one-sided transitive subshift of finite type sigma : Sigma -> Sigma and a Holder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If (A) over bar denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A >= u o sigma - u + (A) over bar. We call contact locus of u with respect to A the subset of Sigma where A = u o sigma - u + (A) over bar. A calibrated sub-action u gives the possibility to construct, for any point x is an element of Sigma, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Omega (A), the set of non-wandering points with respect to A. We prove that separating sub-actions are generic among Holder sub-actions. We also prove that, under certain conditions on Omega(A), any calibrated sub- action is of the form u(x) = u(x(i)) + h(A)(x(i), x) for some x(i) is an element of Omega (A), where h(A)(x, y) denotes the Peierls barrier of A. We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type ((Sigma) over cap, (sigma) over cap).404577602Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)PRONEX - Sistemas DinamicosInstituto do MilenioCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)ANR [BLANC07-3_187245]Hamilton-JacobiWeak KAM TheoryConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)ANR [BLANC07-3_187245

LYAPUNOV EXPONENTS ON METRIC SPACES

DOI: 10.1017/S0004972717000703/a