scholarly journals Existence and smoothness of the stable foliation for sectional hyperbolic attractors

2017 ◽  
Vol 49 (2) ◽  
pp. 351-367 ◽  
Author(s):  
Vitor Araújo ◽  
Ian Melbourne
Keyword(s):  
Stable Foliation ◽  
Hyperbolic Attractors

scholarly journals Physical measures for certain partially hyperbolic attractors on 3-manifolds

2017 ◽  
Vol 39 (1) ◽  
pp. 74-104 ◽  
Author(s):  
RICARDO T. BORTOLOTTI
Keyword(s):  
Finite Number ◽  
Lebesgue Measure ◽  
Gibbs States ◽  
Stable Foliation ◽  
Ergodic Properties ◽  
Central Direction ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Full Lebesgue Measure

In this work, we analyze ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior; the main feature is a condition of transversality between the projections of unstable leaves, projecting through the stable foliation. We prove that partial hyperbolic attractors satisfying this condition of transversality, neutrality in the central direction and regularity of the stable foliation admit a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of robustly non-hyperbolic attractors satisfying these properties.

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

scholarly journals SRB measures for partially hyperbolic attractors of local diffeomorphisms

2018 ◽  
Vol 40 (6) ◽  
pp. 1545-1593
Author(s):  
ANDERSON CRUZ ◽  
PAULO VARANDAS
Keyword(s):  
Thermodynamic Formalism ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Local Diffeomorphism ◽  
Local Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Cone Field ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

scholarly journals Almost sure rates of mixing for partially hyperbolic attractors

2022 ◽  
Vol 311 ◽  
pp. 98-157
Author(s):  
José F. Alves ◽  
Wael Bahsoun ◽  
Marks Ruziboev
Keyword(s):  
Hyperbolic Attractors ◽  
Partially Hyperbolic

HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS

2005 ◽  
Vol 15 (11) ◽  
pp. 3567-3578 ◽  
Author(s):  
VLADIMIR BELYKH ◽  
IGOR BELYKH ◽  
ERIK MOSEKILDE
Keyword(s):  
Neuron Model ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Hyperbolic Attractor ◽  
Neuron Models ◽  
Neuron System ◽  
Physical Systems ◽  
System A ◽  
Bursting Neurons ◽  
Hyperbolic Attractors

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

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