Entropy of snakes and the restricted variational principle

1992 ◽  
Vol 12 (4) ◽  
pp. 791-802
Author(s):  
M. Misiurewicz ◽  
J. Tolosa
Keyword(s):  
Variational Principle ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Symmetric Case ◽  
Easy Method ◽  
Interval Maps ◽  
Interval Map ◽  
Ergodic Measures

AbstractFor interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

2004 ◽  
Vol 14 (07) ◽  
pp. 2161-2186 ◽  
Author(s):  
GOONG CHEN ◽  
TINGWEN HUANG ◽  
YU HUANG
Keyword(s):  
Nonlinear Systems ◽  
Initial Data ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Oscillatory Behavior ◽  
Exponential Rate ◽  
Interval Maps ◽  
Interval Map ◽  
Sensitive Dependence

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Representations of Artin’s braid groups and linking numbers of periodic orbits

1995 ◽  
Vol 04 (02) ◽  
pp. 197-212 ◽  
Author(s):  
John Guaschi
Keyword(s):  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Braid Groups ◽  
Matrix Representations ◽  
Linking Number ◽  
Linking Numbers ◽  
Small Period

Let P be a periodic orbit of period n≥3 of an orientation-preserving homeomorphism f of the 2-disc. Let q be the least integer greater than or equal to n/2−1. Then f admits a periodic orbit Q of period less than or equal to q such that the linking number of P about Q is non-zero. This answers a question of Franks in the affirmative in the case that P has small period. We also derive a result regarding matrix representations of Artin’s braid groups. Finally a lower bound for the topological entropy of a braid in terms of the trace of its Burau matrix is found.

Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria

2020 ◽  
Vol 30 (07) ◽  
pp. 2030016
Author(s):  
Marina Esteban ◽  
Enrique Ponce ◽  
Francisco Torres
Keyword(s):  
Periodic Orbit ◽  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Symmetric Case ◽  
Hysteretic Systems ◽  
Piecewise Linear Systems ◽  
Saddle Node

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some [Formula: see text]D piecewise linear systems.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

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.

UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 323-332 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Real Line ◽  
Positive Measure ◽  
Schwarzian Derivative ◽  
Large Range ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Central Piece ◽  
Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

scholarly journals Subspaces of interval maps related to the topological entropy

2019 ◽  
pp. 1 ◽  
Author(s):  
Xiaoxin Fan ◽  
Jian Li ◽  
Yini Yang ◽  
Zhongqiang Yang
Keyword(s):  
Topological Entropy ◽  
Interval Maps

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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