scholarly journals Centre manifolds of forced dynamical systems

1991 ◽  
Vol 32 (4) ◽  
pp. 401-436 ◽  
Author(s):  
S. M. Cox ◽  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Rational Approach ◽  
Centre Manifold ◽  
Formal Scheme ◽  
Dimensional Models ◽  
Centre Manifolds ◽  
Low Dimensional ◽  
Sivashinsky Equation

AbstractCentre manifolds arise in a rational approach to the problem of forming low-dimensional models of dynamical systems with many degrees of freedom. When a dynamical system with a centre manifold is subject to a small forcing, F, there are two effects: to displace the centre manifold; and to alter the evolution thereon. We propose a formal scheme for calculating the centre manifold of such a forced dynamical system. Our formalism permits the calculation of these effects, with errors of order |F|2. We find that the displacement of the manifold allows a reparameterisation of its description, and we describe two “natural” ways in which this can be carried out. We give three examples: an introductory example; a five-mode model of the atmosphere to display the quasi-geostrophic approximation; and the forced Kuramoto-Sivashinsky equation.

DYNAMICAL SYSTEMS AND TESSELATIONS: DETECTING DETERMINISM IN DATA

1991 ◽  
Vol 01 (04) ◽  
pp. 777-794 ◽  
Author(s):  
ALISTAIR I. MEES
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
State Space ◽  
Scientific Theory ◽  
Invariant Manifolds ◽  
Geometric Information ◽  
Topological Information ◽  
Low Dimensional ◽  
Dynamical Information

Data measurements from a dynamical system may be used to build triangulations and tesselations which can — at least when the system has relatively low-dimensional attractors or invariant manifolds — give topological, geometric and dynamical information about the system. The data may consist of a time series, possibly reconstructed by embedding, or of several such series; transients can be put to good use. The topological information which can be found includes dimension and genus of a manifold containing the state space. Geometric information includes information about folds, branches and other chaos generators. Dynamical information is obtained by using the tesselation to construct a map with stated smoothness properties and having the same dynamics as the data; the resulting dynamical model may be tested in the way that any scientific theory may be tested, by making falsifiable predictions.

scholarly journals On the Number of Isolating Integrals in Systems with Three Degrees of Freedom

1971 ◽  
Vol 10 ◽  
pp. 110-117
Author(s):  
Claude Froeschle
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Weak Coupling ◽  
Degrees Of Freedom ◽  
Model Problem ◽  
Energy Integral ◽  
Dimensional Mapping ◽  
Three Degrees Of Freedom

AbstractDynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations: We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).

Hierarchical Organization in Smooth Dynamical Systems

2005 ◽  
Vol 11 (4) ◽  
pp. 493-512 ◽  
Author(s):  
Martin N. Jacobi
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Hierarchical Structures ◽  
Hierarchical Organization ◽  
Necessary And Sufficient ◽  
Deterministic Dynamical System ◽  
Functional Components ◽  
Different Levels

This article is concerned with defining and characterizing hierarchical structures in smooth dynamical systems. We define transitions between levels in a dynamical hierarchy by smooth projective maps from a phase space on a lower level, with high dimensionality, to a phase space on a higher level, with lower dimensionality. It is required that each level describe a self-contained deterministic dynamical system. We show that a necessary and sufficient condition for a projective map to be a transition between levels in the hierarchy is that the kernel of the differential of the map is tangent to an invariant manifold with respect to the flow. The implications of this condition are discussed in detail. We demonstrate two different causal dependences between degrees of freedom, and how these relations are revealed when the dynamical system is transformed into global Jordan form. Finally these results are used to define functional components on different levels, interaction networks, and dynamical hierarchies.

scholarly journals Integrals of dynamical systems linear in the velocities

1971 ◽  
Vol 17 (3) ◽  
pp. 241-244
Author(s):  
C. D. Collinson
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Integral Conditions ◽  
Two Degrees Of Freedom ◽  
Generalized Coordinates ◽  
Image Position ◽  
Linear Integrals

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangianrepeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).

scholarly journals Synthetic Birdsongs as a Tool to Induce, and Iisten to, Replay Activity in Sleeping Birds

2021 ◽  
Vol 15 ◽  
Author(s):  
Ana Amador ◽  
Gabriel B. Mindlin
Keyword(s):  
Nervous System ◽  
Dynamical Systems ◽  
Sensorimotor Integration ◽  
Computational Models ◽  
Acoustic Signals ◽  
Vocal Behavior ◽  
Song System ◽  
Highly Nonlinear ◽  
Dimensional Models ◽  
Low Dimensional

Birdsong is a complex vocal behavior, which emerges out of the interaction between a nervous system and a highly nonlinear vocal device, the syrinx. In this work we discuss how low dimensional dynamical systems, interpretable in terms of the biomechanics involved, are capable of synthesizing realistic songs. We review the experimental and conceptual steps that lead to the formulation of low dimensional dynamical systems for the song system and describe the tests that quantify their success. In particular, we show how to evaluate computational models by comparing the responses of highly selective neurons to the bird’s own song and to synthetic copies generated mathematically. Beyond testing the hypothesis behind the model’s construction, these low dimensional models allow designing precise stimuli in order to explore the sensorimotor integration of acoustic signals.

scholarly journals Planform evolution in convection–an embedded centre manifold

1992 ◽  
Vol 34 (2) ◽  
pp. 174-198 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Dynamical Systems ◽  
Wide Class ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Centre Manifold ◽  
Higher Dimensional ◽  
Dimensional Approximation ◽  
Low Dimensional ◽  
Geometric Picture ◽  
Better Than

AbstractThe new motion of embedding a centre manifold in some higher-dimensional manifold leads to a practical approach to the rational low-dimensional approximation of a wide class of dynamical systems; it also provides a simple geometric picture for these approximations. In particular, I consider the problem of finding an approximate, but accurate, description of the evolution of a two-dimensional planform of convection. Inspired by a simple example, the straightforward adiabatic iteration is proposed to estimate an embedding manifold and arguments are presented for its effectiveness. Upon applying the procedure to a model convective planform problem I find that the resulting approximations perform remarkably well–much better than the traditional Swift-Hohenberg approximation for planform evolution.

Dynamical system approaches to climate variability

2020 ◽  
pp. 96-182
Author(s):  
Henk A. Dijkstra
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Climate Variability ◽  
Systems Theory ◽  
Ergodic Theory ◽  
Climate Dynamics ◽  
Dynamical Systems Analysis ◽  
Deterministic Dynamical Systems ◽  
Low Dimensional ◽  
System Approaches

A tutorial is provided on the application of dynamical systems theory to problems in climate dynamics. We start with the analysis of low-dimensional deterministic dynamical systems using bifurcation theory and provide examples in conceptual climate models.We then proceed to stochastic low-dimensional systems and eventually end with operator-based techniques within ergodic theory. In these notes, we start each section from a climate dynamics problem, motivate the choice of the model to study it, and use dynamical systems analysis to understand the behavior of the model solutions. In each of the chapters, a different phenomenon, a different type of model, and/or a different dynamical system tool will be presented.

scholarly journals Low-dimensional models of stellar and galactic dynamos

2008 ◽  
Vol 4 (S259) ◽  
pp. 419-420
Author(s):  
Dimitry Sokoloff ◽  
S. Nefyodov
Keyword(s):  
Dynamical System ◽  
Celestial Body ◽  
Dynamo Model ◽  
Galactic Disc ◽  
Regular Method ◽  
Dimensional Models ◽  
Low Dimensional ◽  
Galactic Dynamos

AbstractA regular method how to simplify dynamo model for a particular celestial body up to a dynamical system is suggested. Dynamical system obtained occurs specific for a thin galactic disc, a fully convective star and a thin convective shell.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

scholarly journals Entropy and irreversibility in dynamical systems

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120349 ◽  
Author(s):  
Oliver Penrose
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic System ◽  
Degrees Of Freedom ◽  
Usual Method ◽  
Chaotic Dynamical System ◽  
Two Degrees Of Freedom ◽  
Arnold’S Cat Map ◽  
Non Equilibrium

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann’s principle , does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold’s ‘cat’ map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

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