scholarly journals On the historical behavior of singular hyperbolic attractors

2019 ◽  
Vol 148 (4) ◽  
pp. 1641-1644
Author(s):  
Dawei Yang
Keyword(s):  
Hyperbolic Attractors

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.

scholarly journals Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers

2004 ◽  
Vol 160 (2) ◽  
pp. 375-432 ◽  
Author(s):  
Carlos Morales Rojas ◽  
Maria Pacifico ◽  
Enrique Pujals
Keyword(s):  
Singular Sets ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic
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