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.

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.

What Could Be Worse than the Butterfly Effect?

2008 ◽  
Vol 38 (4) ◽  
pp. 519-547 ◽  
Author(s):  
Robert C. Bishop
Keyword(s):  
Nonlinear Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Classical Mechanics ◽  
Physical Systems ◽  
Physics Community ◽  
Sensitive Dependence ◽  
Classical Models ◽  
In The Beginning

Our understanding of classical mechanics (CM) has undergone significant growth in the latter half of the twentieth century and in the beginning of the twenty-first. This growth has much to do with the explosion of interest in the study of nonlinear systems in contrast with the focus on linear systems that had colored much work in CM from its inception. For example, although Maxwell and Poincaré arguably were some of the first to think about chaotic behavior, the modern study of chaotic dynamics traces its beginning to the pioneering work of Edward Lorenz (1963). This work has yielded a rich variety of behavior in relatively simple classical models that was previously unsuspected by the vast majority of the physics community (see Hilborn 2001). Chaos is a property of nonlinear systems that is usually characterized by sensitive dependence on initial conditions (SDIC). In CM the behavior of simple physical systems is described using models (such as the harmonic oscillator) that capture the main features of the systems in question (Giere 1988).

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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.

Symmetry, bifurcations, and chaos in a distributed respiratory control system

1994 ◽  
Vol 77 (5) ◽  
pp. 2481-2495 ◽  
Author(s):  
M. Sammon
Keyword(s):  
Chaotic Behavior ◽  
Respiratory Control ◽  
Oscillatory Behavior ◽  
Phase Switching ◽  
Control Distribution ◽  
Complex Eigenvalues ◽  
Afferent Information ◽  
Switching Functions ◽  
The Matrix ◽  
Afferent Control

A multivariate model is outlined for a distributed respiratory central pattern generator (RCPG) and its afferent control. Oscillatory behavior of the system depends on structure and symmetry of a matrix of phase-switching functions (F omega, phi) that control distribution of central excitation (CE) and inhibition (CI) within the circuit. The matrix diagonal (F omega) controls activation of CI variables as excitatory inputs are altered (e.g., central and afferent contributions to inspiratory off switch); off-diagonal terms (F phi) distribute excitations within the CI system and produce complex eigenvalues at the switching points between inspiration and expiration. For null F phi, phase switchings of saddle equilibria located at end expiration and end inspiration are overdamped all-or-nothing events; graded control of CI is seen for phi > 0. When coupling is significant (phi >> 0), CI dynamics become underdamped, admitting a domain of inputs where chaotic behavior is predictably observed. For the homogeneous RCPG (symmetric F omega, phi), CE oscillations are one-dimensional limit cycles (D = 1) or weakly chaotic (D approximately equal to 1). When perturbations from symmetry are significant, the distributed RCPG becomes partitioned where strongly chaotic oscillations (D > or = 2) and central apnea (D = 0) are seen more frequently. The equations provide means for mapping Silnikov bifurcations that alter the geometry and dimension of the breathing pattern and formalisms for discussing RCPG processing of afferent information.

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

COMPUTER ASSISTED VERIFICATION OF CHAOS IN THREE-NEURON CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (12) ◽  
pp. 4381-4386 ◽  
Author(s):  
QUAN YUAN ◽  
XIAO-SONG YANG
Keyword(s):  
Neural Networks ◽  
Topological Entropy ◽  
Chaotic Behavior ◽  
Cellular Neural Networks ◽  
Computer Assisted ◽  
Topological Horseshoe ◽  
Poincaré Maps ◽  
New Class ◽  
Numerical Studies ◽  
Connection Matrices

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.

Strange Attractors and Chaos in Nonlinear Mechanics

1983 ◽  
Vol 50 (4b) ◽  
pp. 1021-1032 ◽  
Author(s):  
P. J. Holmes ◽  
F. C. Moon
Keyword(s):  
Mathematical Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Strange Attractors ◽  
Electrical Systems ◽  
Nonlinear Mechanical ◽  
Sensitive Dependence

We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

2012 ◽  
Vol 22 (10) ◽  
pp. 1250259 ◽  
Author(s):  
ZDENĚK KOČAN
Keyword(s):  
Topological Entropy ◽  
Metric Spaces ◽  
Chaotic Behavior ◽  
Homoclinic Trajectory ◽  
One Dimensional ◽  
Distributional Chaos ◽  
Compact Metric Spaces ◽  
Continuous Maps

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

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