Recurrence time statistics in low-dimensional dynamical systems

2008 ◽  
pp. 42-54 ◽  
Author(s):  
C. Nicolis ◽  
G. Nicolis ◽  
V. Balakrishnan ◽  
M. Theunissen
Keyword(s):  
Dynamical Systems ◽  
Recurrence Time ◽  
Low Dimensional

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

scholarly journals Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

2016 ◽  
Vol 27 (6) ◽  
pp. 904-922 ◽  
Author(s):  
STEPHEN COOMBES ◽  
RÜDIGER THUL
Keyword(s):  
Dynamical Systems ◽  
Variational Equation ◽  
Period Doubling ◽  
Star Graph ◽  
Stability Function ◽  
Bifurcation Of Limit Cycles ◽  
Master Stability Function ◽  
Coupled Networks ◽  
Low Dimensional ◽  
Insight Into

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

scholarly journals ON ONSET OF CHAOTIC CONVECTION IN COUPLE-STRESS FLUIDS

2014 ◽  
Vol 19 (3) ◽  
pp. 359-370 ◽  
Author(s):  
Jadallah M. Jawdat ◽  
Ishak Hashim ◽  
Beer S. Bhadauria ◽  
Shaher Momani
Keyword(s):  
Dynamical Systems ◽  
Nonlinear System ◽  
Couple Stress ◽  
Fluid Layer ◽  
Kutta Method ◽  
Fluid Field ◽  
Runge Kutta Method ◽  
Chaotic Convection ◽  
Low Dimensional ◽  
Couple Stress Fluids

The effect of couple-stress fluid field on chaotic convection in a fluid layer heated from below was studied in this paper based on the theory of dynamical systems. A low-dimensional, Lorenz-like model was obtained using Galerkin truncated approximations. The fourth-order Runge–Kutta method was employed to solve the nonlinear system. The results show that inhibition of chaotic convection depends strongly on the couple-stress parameter.

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.

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