On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures II: the low-dimensional case

2017 ◽  
Vol 38 (8) ◽  
pp. 3042-3061
Author(s):  
MICHIHIRO HIRAYAMA ◽  
NAOYA SUMI
Keyword(s):  
Probability Measure ◽  
Closed Manifold ◽  
Dimensional Case ◽  
Anosov Diffeomorphism ◽  
Stable And Unstable Manifolds ◽  
Higher Dimensional ◽  
Unstable Manifolds ◽  
Low Dimensional

In this paper, we consider diffeomorphisms on a closed manifold $M$ preserving a hyperbolic Sinaĭ–Ruelle–Bowen probability measure $\unicode[STIX]{x1D707}$ having intersections for almost every pair of stable and unstable manifolds. In this context, we show the ergodicity of $\unicode[STIX]{x1D707}$ when the dimension of $M$ is at most three. If $\unicode[STIX]{x1D707}$ is smooth, then it is ergodic when the dimension of $M$ is at most four. As a byproduct of our arguments, we obtain sufficient (topological) conditions which guarantee that there exists at most one hyperbolic ergodic Sinaĭ–Ruelle–Bowen probability measure. Even in higher dimensional cases, we show that every transitive topological Anosov diffeomorphism admits at most one hyperbolic Sinaĭ–Ruelle–Bowen probability measure.

TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

scholarly journals Geometric and Differential Features of Scators as Induced by Fundamental Embedding

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1880
Author(s):  
Artur Kobus ◽  
Jan L. Cieśliński
Keyword(s):  
Building Block ◽  
Differential Calculus ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Main Building ◽  
Higher Dimensional ◽  
Algebraic Operations ◽  
Low Dimensional

The scator space, introduced by Fernández-Guasti and Zaldívar, is endowed with a product related to the Lorentz rule of addition of velocities. The scator structure abounds with definitions calculationally inconvenient for algebraic operations, like lack of the distributivity. It occurs that situation may be partially rectified introducing an embedding of the scator space into a higher-dimensonal space, that behaves in a much more tractable way. We use this opportunity to comment on the geometry of automorphisms of this higher dimensional space in generic setting. In parallel, we develop commutative-hypercomplex analogue of differential calculus in a certain, specific low-dimensional case, as also leaned upon the notion of fundamental embedding, therefore treating the map as the main building block in completing the theory of scators.

ALFVÉN INTERIOR CRISIS

2004 ◽  
Vol 14 (07) ◽  
pp. 2375-2380 ◽  
Author(s):  
F. A. BOROTTO ◽  
A. C.-L. CHIAN ◽  
E. L. REMPEL
Keyword(s):  
Periodic Orbits ◽  
Large Amplitude ◽  
Numerical Study ◽  
Alfven Wave ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Laboratory Plasmas ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Unstable Manifolds ◽  
Low Dimensional

A numerical study of an interior crisis of a large-amplitude Alfvén wave described by the driven-dissipative derivative nonlinear Schrödinger equation, in the low-dimensional limit, is reported. An example of Alfvén interior crisis is characterized using the unstable periodic orbits and their associated invariant stable and unstable manifolds in the Poincaré plane. We suggest that this type of chaotic transition can be observed in space and laboratory plasmas.

A NEW ALGORITHM FOR COMPUTING ONE-DIMENSIONAL STABLE AND UNSTABLE MANIFOLDS OF MAPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250018 ◽  
Author(s):  
HUIMIN LI ◽  
YANGYU FAN ◽  
JING ZHANG
Keyword(s):  
Bisection Method ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Numerical Examples ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Correction Scheme ◽  
Accuracy Criterion ◽  
Tangent Component ◽  
Unstable Manifolds

A new algorithm is presented to compute one-dimensional stable and unstable manifolds of fixed points for both two-dimensional and higher dimensional diffeomorphism maps. When computing the stable manifold, the algorithm does not require the explicit expression of the inverse map. The global manifold is grown from a local manifold and one point is added at each step. The new point is located with a "prediction and correction" scheme, which avoids searching the computed part of the manifold with a bisection method and accelerates the searching process. By using the fact that the Jacobian transports derivatives along the orbit of the manifold, the tangent component of the manifold is determined and a new accuracy criterion is proposed to check whether the new point that defines the manifold is acceptable. The performance of the algorithm is demonstrated with several numerical examples.

PSEUDO-DETERMINISTIC CHAOTIC SYSTEMS

2003 ◽  
Vol 13 (11) ◽  
pp. 3235-3253 ◽  
Author(s):  
R. L. VIANA ◽  
S. E. DE S. PINTO ◽  
J. R. R. BARBOSA ◽  
C. GREBOGI
Keyword(s):  
Chaotic Systems ◽  
Critical Value ◽  
Time Model ◽  
Stable And Unstable Manifolds ◽  
Chaotic Dynamical System ◽  
Discrete Time Model ◽  
Finite Time Lyapunov Exponents ◽  
Unstable Manifolds ◽  
Model Equations ◽  
Low Dimensional

We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

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