Diffeomorphisms with infinitely many irrational invariant curves

2010 ◽  
Vol 31 (5) ◽  
pp. 1517-1535
Author(s):  
LEONARDO MORA ◽  
BLADISMIR RUIZ
Keyword(s):  
Saddle Points ◽  
Irrational Rotation ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Invariant Curves ◽  
Open Set ◽  
Rotation Numbers ◽  
Surface Diffeomorphism ◽  
Dense Set

AbstractFor a surface diffeomorphism f∈Diff l(M), with l≥8, we prove that if f exhibits a non-transversal heteroclinic cycle composed of two fixed saddle points Q1 and Q2, one dissipative and the other expansive, then there exists an open set 𝒱⊂Diff l(M) such that $ f \in \overline {\mathcal {V}}$ and there exists a dense set 𝒟⊂𝒱 such that for all g∈𝒟, g exhibits infinitely many invariant periodic curves with irrational rotation numbers. Moreover, these curves are C1 conjugated to an irrational rotation on 𝕊1.

Structurally stable heteroclinic cycles

1988 ◽  
Vol 103 (1) ◽  
pp. 189-192 ◽  
Author(s):  
John Guckenheimer ◽  
Philip Holmes
Keyword(s):  
Dynamical Systems ◽  
Symmetry Group ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Finite Subgroup ◽  
Heteroclinic Cycle ◽  
Heteroclinic Cycles ◽  
Open Set ◽  
Structurally Stable

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

scholarly journals Recognition of Synthetic Speech by Hearing-Impaired Elderly Listeners

1991 ◽  
Vol 34 (5) ◽  
pp. 1180-1184 ◽  
Author(s):  
Larry E. Humes ◽  
Kathleen J. Nelson ◽  
David B. Pisoni
Keyword(s):  
Hearing Loss ◽  
Speech Recognition ◽  
Recognition Performance ◽  
The Elderly ◽  
Hearing Impaired ◽  
The Other ◽  
Natural Speech ◽  
Synthetic Speech ◽  
Closed Set ◽  
Open Set

The Modified Rhyme Test (MRT), recorded using natural speech and two forms of synthetic speech, DECtalk and Votrax, was used to measure both open-set and closed-set speech-recognition performance. Performance of hearing-impaired elderly listeners was compared to two groups of young normal-hearing adults, one listening in quiet, and the other listening in a background of spectrally shaped noise designed to simulate the peripheral hearing loss of the elderly. Votrax synthetic speech yielded significant decrements in speech recognition compared to either natural or DECtalk synthetic speech for all three subject groups. There were no differences in performance between natural speech and DECtalk speech for the elderly hearing-impaired listeners or the young listeners with simulated hearing loss. The normal-hearing young adults listening in quiet out-performed both of the other groups, but there were no differences in performance between the young listeners with simulated hearing loss and the elderly hearing-impaired listeners. When the closed-set identification of synthetic speech was compared to its open-set recognition, the hearing-impaired elderly gained as much from the reduction in stimulus/response uncertainty as the two younger groups. Finally, among the elderly hearing-impaired listeners, speech-recognition performance was correlated negatively with hearing sensitivity, but scores were correlated positively among the different talker conditions. Those listeners with the greatest hearing loss had the most difficulty understanding speech and those having the most trouble understanding natural speech also had the greatest difficulty with synthetic speech.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

scholarly journals Periodic perturbations of quadratic planar polynomial vector fields

2002 ◽  
Vol 74 (2) ◽  
pp. 193-198 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Vector Fields ◽  
Dynamical Behavior ◽  
Saddle Points ◽  
Heteroclinic Cycle ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Poincaré Compactification ◽  
Stable And Unstable Manifolds ◽  
Perturbed System ◽  
Periodic Perturbations

In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

scholarly journals Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

1988 ◽  
Vol 8 (4) ◽  
pp. 555-584 ◽  
Author(s):  
Raphaël Douady
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Positive Measure ◽  
Invariant Curves ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractWe prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

ON THE SLIDING BIFURCATION OF A CLASS OF PLANAR FILIPPOV SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350040 ◽  
Author(s):  
DINGHENG PI ◽  
JIANG YU ◽  
XIANG ZHANG
Keyword(s):  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Qualitative Theory ◽  
Piecewise Smooth ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Differential Systems ◽  
Sliding Bifurcation ◽  
Bifurcation Phenomena ◽  
Limit Cycle Bifurcation

In this paper, we study the sliding bifurcation phenomena of a class of planar piecewise smooth differential systems consisting of linear and quadratic subsystems. Using the differential inclusion and the qualitative theory of ordinary differential equations, we find some new interesting phenomena appearing in the piecewise smooth differential systems. In brief, we prove that the system may have sliding homoclinic bifurcation, sliding cycle bifurcation, semistable limit cycle bifurcation and heteroclinic cycle bifurcation. In addition, the mentioned systems can have at most two limit cycles, and the maximal number of limit cycles can be realized and central nested with one bifurcated from the sliding–crossing bifurcation of a sliding cycle and the other from the saddle homoclinic bifurcation. These two limit cycles collide and then both disappear. This novel scenario is verified by our systems.

Boundary-layer flow between nodal and saddle points of attachment

1970 ◽  
Vol 41 (4) ◽  
pp. 823-835 ◽  
Author(s):  
J. C. Cooke ◽  
A. J. Robins
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Saddle Point ◽  
Difference Method ◽  
Series Solution ◽  
Nodal Point ◽  
Saddle Points ◽  
The Other ◽  
The Finite Difference Method ◽  
Layer Flow

A simplified example of this type of flow was examined in detail by developing two series, eventually matched, one about the nodal point and the other about the saddle point, and also by finite differences, marching from the nodal point to the saddle point. It was found that the results of marching the two series were in agreement with the finite difference method. The series solution near the saddle point is not unique, but numerical evidence indicates that the correct solution is that which has ‘exponential decay’ at infinity, and that this type of solution, if such exists, automatically emerges when the finite difference method is used.

REMARKS ON SKEW PRODUCTS OVER IRRATIONAL ROTATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3675-3689 ◽  
Author(s):  
L. M. LERMAN
Keyword(s):  
Sufficient Conditions ◽  
Irrational Rotation ◽  
Invariant Sets ◽  
Asymptotically Stable ◽  
Skew Product ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Skew Products ◽  
Irrational Rotation Number ◽  
Periodic Equation

We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without an almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones.Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincaré–Pontryagin theorem which sets conditions when a perturbation of a one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable).In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.

scholarly journals Simultaneous dense and non-dense orbits for toral diffeomorphisms

2016 ◽  
Vol 37 (4) ◽  
pp. 1308-1322 ◽  
Author(s):  
JIMMY TSENG
Keyword(s):  
Baire Category ◽  
Geometric Characterization ◽  
The Other ◽  
Baire Category Theorem ◽  
Open Set ◽  
Category Theorem ◽  
Dense Orbits ◽  
Set Of Points ◽  
Toral Automorphisms

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

Sign in / Sign up

Export Citation Format

Share Document