We report the results of studies of nonlinear dynamics and dynamical chaos in Hamiltonian systems composed of many interacting particles. The importance of the Lyapunov exponents and the Kolmogorov-Sinai entropy is discussed in the context of ergodic theory and nonequilibrium statistical mechanics. Two types of systems are studied: hard-ball models for the motion of a tracer or Brownian particl...

Informational Entropies for Non-ergodic Brains Arturo Tozzi 1, James F. Peters 2,Marzieh Zare 3*, Mehmet Niyazi Çankaya 4 1 Center for Nonlinear Science, University of North Texas, Denton, Texas 76203, USA ; [email protected] 2 Department of Electrical and Computer Engineering, University of Manitoba 75A Chancellor’s Circle Winnipeg, MB R3T 5V6 Canada; [email protected] 3 Institute for ...

The Hille-Yosida and Lumer-Phillips theorems play an important role in the theory of linear operators and its applications to evolution equations, probability and ergodic theory. (See, for example, [17] and [9].) Different nonlinear generalizations and analogues of these theorems can be found, for instance, in [13] and [2]. We are interested in establishing analogues of these theorems for the c...

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 a semi-invertible Multiplicative Ergodic Theorem cocycles acting measurable fields of Banach spaces 20 ].

One of the fascinating results in the one dimensional nonlinear dynamical system is that a mapping which maps a compact connected subset of the real line into itself is regular if and only if it does not have any order periodic points except fixed points. However, in general, this result does not hold true in the high dimensional case. In this paper, we provide a counter example for such kind o...

Estimates of quantitative characteristics of nonlinear dynamics, e.g., correlation dimension or Lyapunov exponents, require long time series and are sensitive to noise. Other measures (e.g., phase space warping or sensitivity vector fields) are relatively difficult to implement and computationally intensive. In this paper, we propose a new class of features based on Birkhoff Ergodic Theorem, wh...

We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, HamiltonJacobi equations and fully nonlinear elliptic equations. Motivated by the application to ...

In this paper we study the problems of invariant and ergodic expectations under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion (G-SDEs) have the unique invariant and ergodic expectations. Moreover, the invariant and ergodic expectations of G-SDEs are also sublinear expectations. However, the invariant expectations may not coincide with ...

Let K be a nonempty subset of a Hilbert space , where K is not necessarily closed and convex. A family Γ= {T(t); t ≥ 0} of mappings T(t) is called a semigroup on K if (S1) T(t) is a mapping from K into itself for t ≥ 0, (S2) T(0)x = x and T(t+ s)x = T(t)T(s)x for x ∈ K and t,s≥ 0, (S3) for each x ∈ K , T(·)x is strongly measurable and bounded on every bounded subinterval of [0,∞). Let Γ be a se...

We describe an approximation scheme for a class of Hamilton-Jacobi equations associated to H1 control problems. We deene a sequence of discrete time nonlinear systems and we prove the equivalence between the discrete H1 control problem and an ergodic control problem. Then we show that the sequence of discrete H1 norms converges to the norm of the corresponding continuous problem both in the nit...