Related papers: Accurate bounds on Lyapunov exponents for expandin…

In this paper, we consider a monic, centred, hyperbolic polynomial of degree $d \ge 2$, restricted on its Julia set and compute its Lyapunov exponents with respect to certain weighted Lyubich's measures. In particular, we show a certain…

In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is point-wise smaller (in the sense…

We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of…

The predictability problem for systems with different characteristic time scales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability…

Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the…

We apply the maximum entropy principle to construct the natural invariant density and Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique that is based on the solution of Hausdorff moment…

Let (M,d) be a compact metric space and f:M --> M an expansive homeomorphism. We define Lyapunov exponents L(f,m)_{max} and l(f,mu)_{min} for an f-invariant measure m. When L(f,m)_{max} > 0 and l(f,mu)_{min} < 0 can be interpreted as a weak…

First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to find and…

For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(\omega_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists,…

We present a novel way of generating Lyapunov functions for proving linear convergence rates of first-order optimization methods. Our approach provably obtains the fastest linear convergence rate that can be verified by a quadratic Lyapunov…

The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos this property is usually quantified by…

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov…

In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $\varphi$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for…

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques…

Extending recent work of Corrado, we derive an algorithm that computes rigorous upper and lower bounds for rectangle scan probabilities for Markov increments. We experimentally examine the closeness of the bounds computed by the algorithm…

For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_\sigma$ $(0<\sigma<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_\sigma$ has an observable (i.e. Lebesgue positive) Lyapunov…

We prove existence of positive solutions to a nonlinear fractional boundary value problem. Then, under some mild assumptions on the nonlinear term, we obtain a smart generalization of Lyapunov's inequality. The new results are illustrated…

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical…

In this paper we study the Lyapunov spectrum rigidity for random walks of expanding maps on unit circle $\mathbb{S}^1$ and Anosov diffeomorphisms on $d$-torus $\mathbb{T}^d$. Let $\nu$ be a probability supported on the set of expanding maps…

Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…