Search results for "Systems Theory"

On the collision property of chaotic iterations based post-treatments over cryptographic pseudorandom number generators

2018

International audience; There is not a proper mathematical definition of chaos, we have instead a quite big amount of definitions, each of one describes chaos in a more or less general context. Taking in account this, it is clear why it is hard to design an algorithm that produce random numbers, a kind of algorithm that could have plenty of concrete appliceautifat (anul)d bions. However we must use a finite state machine (e.g. a laptop) to produce such a sequence of random numbers, thus it is convenient, for obvious reasons, to redefine those aimed sequences as pseudorandom; also problems arise with floating point arithmetic if one wants to recover some real chaotic property (i.e. propertie…

Dynamics, Operator Theory, and Infinite Holomorphy

2014

Stability of Hamiltonian Systems of Two Degrees of Freedom and of Formally Conservative Mappings Near a Singular Point

1985

We restrict ourselves to the stability problems considered in our lecture because the length of this paper is limited. In contrast to the lecture, however, we consider here not only area preserving mappings but a more general class of mappings.

Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems

2005

In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class…

SURVEY Towards a global view of dynamical systems, for the C1-topology

2011

AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

Finiteness properties of pseudo-hyperbolic varieties

2019

Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi-Ochiai's finiteness result for…

Unifying vectors and matrices of different dimensions through nonlinear embeddings

2020

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, derivi…

Generic Properties of Dynamical Systems

2006

The state of a concrete system (from physics, chemistry, ecology, or other sciences) is described using (finitely many, say n) observable quantities (e.g., positions and velocities for mechanical systems, population densities for echological systems, etc.). Hence, the state of a system may be represented as a point $x$ in a geometrical space $\mathbb R^n$. In many cases, the quantities describing the state are related, so that the phase space (space of all possible states) is a submanifold $M\subset \mathbb R^n$. The time evolution of the system is represented by a curve $x_t$, $t \in\mathbb R$ drawn on the phase space $M$, or by a sequence $x_n \in M$, $n \in\mathbb Z$, if we consider disc…

A description of pseudo-bosons in terms of nilpotent Lie algebras

2017

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

Generalized Geometric Quantum Speed Limits

2016

The attempt to gain a theoretical understanding of the concept of time in quantum mechanics has triggered significant progress towards the search for faster and more efficient quantum technologies. One of such advances consists in the interpretation of the time-energy uncertainty relations as lower bounds for the minimal evolution time between two distinguishable states of a quantum system, also known as quantum speed limits. We investigate how the non uniqueness of a bona fide measure of distinguishability defined on the quantum state space affects the quantum speed limits and can be exploited in order to derive improved bounds. Specifically, we establish an infinite family of quantum spee…