Stability of Periodic Points of a Diffeomorphism of a Plane in a Homoclinic Orbit

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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Homoclinic orbits and Lie rotated vector fields

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171200010061 ◽

2000 ◽

Vol 24 (3) ◽

pp. 187-192

Author(s):

Jie Wang ◽

Chen Chen

Keyword(s):

Limit Cycles ◽

Homoclinic Orbit ◽

Vector Fields ◽

Homoclinic Orbits ◽

Singular Points ◽

Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

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Stabilization of the Furuta pendulum around its homoclinic orbit

International Journal of Control ◽

10.1080/0020717011011226 ◽

2002 ◽

Vol 75 (6) ◽

pp. 390-398 ◽

Cited By ~ 50

Author(s):

Isabelle Fantoni ◽

Rogelio Lozano

Keyword(s):

Homoclinic Orbit

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"CROSSROAD AREA–SPRING AREA" TRANSITION (II) FOLIATED PARAMETRIC REPRESENTATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127491000269 ◽

1991 ◽

Vol 01 (02) ◽

pp. 339-348 ◽

Cited By ~ 19

Author(s):

C. MIRA ◽

J. P. CARCASSÈS ◽

M. BOSCH ◽

C. SIMÓ ◽

J. C. TATJER

Keyword(s):

Complete Information ◽

Parametric Representation ◽

Periodic Points ◽

Parameter Plane ◽

Flip Bifurcation ◽

Cusp Point ◽

Dimensional Representation ◽

Bifurcation Structure ◽

Spring Area

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

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SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495001022 ◽

1995 ◽

Vol 05 (05) ◽

pp. 1351-1355

Author(s):

VLADIMIR FEDORENKO

Keyword(s):

Topological Entropy ◽

Complex Dynamics ◽

Periodic Points ◽

Dimensional Manifold ◽

One Dimensional ◽

Circle Maps ◽

Interval Maps ◽

Chain Recurrent Set ◽

Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

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Almost all $S$-integer dynamical systems have many periodic points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798113378 ◽

1998 ◽

Vol 18 (2) ◽

pp. 471-486 ◽

Cited By ~ 4

Author(s):

T. B. WARD

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Cellular Automata ◽

Zeta Function ◽

Periodic Points ◽

Radius Of Convergence ◽

Dynamical Zeta Function ◽

Hyperbolic Dynamical Systems ◽

Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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On orbits of endomorphisms of tori and the Schmidt game

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004673 ◽

1988 ◽

Vol 8 (4) ◽

pp. 523-529 ◽

Cited By ~ 19

Author(s):

S. G. Dani

Keyword(s):

Hausdorff Dimension ◽

Periodic Points ◽

Dimensional Torus

AbstractWe show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any x∈F and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.

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Generalizations of the Poincaré Birkhoff fixed point theorem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700010650 ◽

1977 ◽

Vol 17 (3) ◽

pp. 375-389 ◽

Cited By ~ 10

Author(s):

Walter D. Neumann

Keyword(s):

Fixed Point ◽

Lower Bound ◽

Fixed Point Theorem ◽

Point Theorem ◽

Periodic Points ◽

Open Problems ◽

Number Of Components ◽

Birkhoff Theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

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