Polynomial entropies and integrable Hamiltonian systems

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.     

The role of normally hyperbolic invariant manifolds (NHIMS) in the context of the phase space setting for chemical reaction dynamics

In this paper we give an introduction to the notion of a normally hyperbolic invariant manifold (NHIM) and its role in chemical reaction dynamics.We do this by considering simple examples for one-, two-, and three-degree-of-freedom systems where explicit calculations can be carried out for all of the relevant geometrical structures and their properties can be explicitly understood. We specifically emphasize the notion of a NHIM as a “phase space concept”. In particular, we make the observation that the (phase space) NHIM plays the role of “carrying” the (configuration space) properties of a saddle point of the potential energy surface into phase space.We also consider an explicit example of a 2-degree-of-freedom system where a “global” dividing surface can be constructed using two index one saddles and one index two saddle. Such a dividing surface has arisen in several recent applications and, therefore, such a construction may be of wider interest.     

Quadratic Hamiltonians on non-symmetric Poisson structures

Many dynamical systems may be represented in a set of non-canonical coordinates that generate an su(2) algebraic structure. The topology of the phase space is the one of the S² sphere, the Poisson structure is the one of the rigid body, and the Hamiltonian is a parametric quadratic form in these “spherical” coordinates. However, there are other problems in which the Poisson structure losses its symmetry. In this paper we analyze this case and, we show how the loss of the spherical symmetry affects the phase flow and parametric bifurcations for the bi-parametric cases.     

Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation

It had been found that some nonlinear wave equations have the so-called “W/M”-shape-peaks solitons. What is the dynamical behavior of these solutions? To answer this question, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given.     

Nonlinear processes in Hamiltonian reconnection

The generation of small spatial scales and their interplay with large scale coherent structures is one of the outstanding phenomena of plasma physics and fluid mechanics. In high temperature space and laboratory plasmas dissipative effects become important at length scales that are much smaller than those where microscopic dynamical effects, related e.g., to electron inertia, come into play. Here we discuss the role of this dissipationless small scale dynamics on the nonlinear evolution of collisionless magnetic reconnection within the framework of the so called “two-field” and “four-field models”.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Singularities in dynamics: A catastrophic viewpoint

The article is about singularities of dynamical systems, a notion which is far from being as clear as for smooth maps. To give an idea of it, the beginning of “catastrophe theory” for dynamical systems (including very recent results) is developed, first in parallel with the well-known analogous theory for potentials, showing that dynamical versions of the Morse lemma with parameters already lead to difficult open problems. The second part of the paper makes it clear that the short-sighted view that singularities in dynamics are rest points and periodic orbits cannot resist serious investigation since many other specifically dynamical “singularities” are born from statics. A homage to René Thom, the whole article is written in the language of stratifications of function spaces-even though it deals with semi-local phenomena.     

Cognitive processes of the brain: An ultrametric model of information dynamics in unconsciousness

We discuss differences in mathematical representations of the physical and mental worlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnected topological space. One of the basic models of such spaces is given by ultrametric spaces and more specially by m-adic trees. We use dynamical systems in such spaces to model flows of unconscious information at different level of mental representation hierarchy, for “mental points”, categories, and ideas. Our model can be interpreted as an unconventional computational model: non-algorithmic hierarchic “computations” (identified with the process of thinking at the unconscious level).     

The Reduction of Equivariant Dynamics to the Orbit Space for Compact Group Actions

Symmetry introduces degeneracies in dynamical systems, as well as in bifurcation problems. An “obvious” idea in order to remove these degeneracies is to project the dynamics onto the quotient space obtained by identifying points in phase space which lie in the same group orbits (the so-called orbit space). Unfortunately, several difficulties arise when one tries to implement this idea. First, the orbit space is not, in general a manifold. Second, how does one explicitely realize the orbit space, and how does one compute and analyze the projected dynamics? In this paper I will describe the methods which have been developped in order to answer these questions, and I will show on three examples how they apply. We shall see that, although not always suitable to treat equivariant dynamics, these methods sometimes lead to insightful reductions.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.     

Construction of the viability kernel for a generalized dynamical system

The problem of the approximate construction of the viability kernel for a generalized dynamical system, the evolution of which is specified directly by an attainability set, is investigated under phase constraints. A backward grid method, based on the substitution of the phase space by pixels and a consideration of “inverse” attainability sets, is proposed. The convergence of the method is proved.     

Fractional standard map: Riemann–Liouville vs. Caputo

Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann–Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems demonstrate are different from properties of the regular and chaotic attractors of systems without memory: they exist in the asymptotic sense, different types of trajectories may lead to the same attracting points, trajectories may intersect, and chaotic attractors may overlap. Two maps have significant differences in the types of attractors they demonstrate and convergence of trajectories to the attracting points and trajectories. Still existence of the most remarkable new type of attractors, “cascade of bifurcation type trajectories”, is a common feature of both maps.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Sur la classification des systèmes dynamiques non commutatifs

The purpose of this paper is to extend certain results on commutative dynamical systems and on von Neumann algebras (provided with their inner automorphism groups) to general dynamical systems: “decompositions” into finite, semifinite, properly infinite, purely infinite, discrete and continuous systems; induced systems, system extensions, properties of invariant weights, etc.     

STRUCTURE-PRESERVING ALGORITHMS FOR DYNAMICAL SYSTEMS

We study structure-preserving algorithms to phase space volume for linear dynamical systems y = Ly for which arbitrarily high order explicit symmetric structure-preserving schemes,i.e. the numerical solutions generated by the schemes satisfy det( ) =ehtrL, where trL is the trace of matrix L, can be constructed. For nonlinear dynamical systems y = f(y) Feng-Shang first-order volume-preserving scheme can be also constructed starting from modified θ- methods and is shown that the scheme is structure-preserving to phase space volume.     

Focus on the Clausius Inequalities as a consequence of modeling thermodynamic systems as a series of open Carnot cycles

Various results in thermodynamics developed recently are brought into focus by further refinement to set in less ambiguous form topics connected to irreversibility and the so called “Clausius Inequality”. This singular “Clausius Inequality” for both closed and open systems was traditionally deduced from the Riemann “integration” of closed Carnot cycle loops for irreversible transitions. Evidently topological problems might be expected to arise concerning boundary conditions when “open” and “closed” systems exists simultaneously in such a scheme. It has hitherto been assumed that in this scheme, only one central Clausius Inequality can exist coupling all processes. Based on a new recent development of open system Carnot cycles, it is shown that other analogous inequalities can be derived, due to the presence of another fundamental entropy state function derived in the recent development, implying non-singularity. Their properties are such as to indicate that no new non-equilibrium entropy can arise from the inequalities as has been proposed over the decades. It is shown that a sequence of points along a non-equilibrium state space must have excess variables augmenting those for the equilibrium situation, which demonstrates that the often used Principle of Local Equilibrium (PLE) is only an approximation, implying that far-from-equilibrium theories should be developed ab initio from irreversible dynamical laws rather than from PLE. Examples presented from actual computations for both systems in equilibrium and non-equilibrium appears to support this deduction. Large scale and extensive thermodynamical theories have been created based on the assumption of a single Clausius-like inequality, such as those stemming from the very influential and extensive Truesdale school, and so such pervasive developments are also open to question.