scholarly journals "Large" strange attractors in the unfolding of a heteroclinic attractor

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexandre Rodrigues
Keyword(s):  
Strange Attractor ◽  
Invariant Manifolds ◽  
Strange Attractors ◽  
Positive Lebesgue Measure ◽  
Dimensional Sphere ◽  
3 Dimensional ◽  
Srb Measures ◽  
The Family ◽  
Parameter Values ◽  
Heteroclinic Network

<p style='text-indent:20px;'>We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.</p>

scholarly journals Complete Bredon cohomology and its applications to hierarchically defined groups

2016 ◽  
Vol 161 (1) ◽  
pp. 143-156
Author(s):  
BRITA E. A. NUCINKIS ◽  
NANSEN PETROSYAN
Keyword(s):  
Integral Characteristic ◽  
Linear Groups ◽  
Dimensional Model ◽  
Classifying Space ◽  
Soluble Groups ◽  
3 Dimensional ◽  
Cyclic Groups ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
The Family

AbstractBy considering the Bredon analogue of complete cohomology of a group, we show that every group in the class$\cll\clh^{\mathfrak F}{\mathfrak F}$of type Bredon-FP∞admits a finite dimensional model for$E_{\frak F}G$.We also show that abelian-by-infinite cyclic groups admit a 3-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for$\mathfrak {F}$, the class of virtually cyclic groups, the class of$\cll\clh^{\mathfrak F}{\mathfrak F}$-groups contains all locally virtually soluble groups and all linear groups over${\mathbb{C}}$of integral characteristic.

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.

THE BASIN OF ATTRACTION OF LOZI MAPPINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 1043-1049 ◽  
Author(s):  
DIOGO BAPTISTA ◽  
RICARDO SEVERINO ◽  
SANDRA VINAGRE
Keyword(s):  
Strange Attractor ◽  
Basin Of Attraction ◽  
Parameter Values ◽  
The Basin Of Attraction

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

WEAKLY SYMMETRIC FINSLER SPACES

2010 ◽  
Vol 12 (02) ◽  
pp. 309-323 ◽  
Author(s):  
SHAOQIANG DENG ◽  
ZIXIN HOU
Keyword(s):  
Open Problem ◽  
Finsler Space ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Finsler Spaces ◽  
Geometrical Properties ◽  
3 Dimensional ◽  
Weakly Symmetric

In this paper, we introduce the notion of weakly symmetric Finsler spaces and study some geometrical properties of such spaces. In particular, we prove that each maximal geodesic in a weakly symmetric Finsler space is the orbit of a one-parameter subgroup of the full isometric group. This implies that each weakly symmetric Finsler space has vanishing S-curvature. As an application of these results, we prove that there exist reversible non-Berwaldian Finsler metrics on the 3-dimensional sphere with vanishing S-curvature. This solves an open problem raised by Z. Shen.

Ground States and Singular Ground States for Quasilinear Partial Differential Equations with Critical Exponent in the Perturbative Case

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
Matteo Franca ◽  
Russell Johnson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Dichotomy ◽  
Invariant Manifolds ◽  
Ground States ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Quasilinear Partial Differential Equations ◽  
Melnikov Functions ◽  
The Family

AbstractWe study the structure of the family of radially symmetric ground states and singular ground states for certain elliptic partial differential equations with p- Laplacian. We use methods of Dynamical systems such as Melnikov functions, invariant manifolds, and exponential dichotomy.

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

scholarly journals MinimalCR-submanifolds of a six-dimentional sphere

1995 ◽  
Vol 18 (1) ◽  
pp. 63-66
Author(s):  
M. Hasan Shahid ◽  
S. I. Husain
Keyword(s):  
Dimensional Sphere ◽  
3 Dimensional

We establish several formulas for a 3-dimensionalCR-submanifold of a six-dimensional sphere and state some results obtained by making use of them.

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