ALGORITHMS FOR OBTAINING A SADDLE TORUS BETWEEN TWO ATTRACTORS

2013 ◽  
Vol 23 (09) ◽  
pp. 1330032 ◽  
Author(s):  
KYOHEI KAMIYAMA ◽  
MOTOMASA KOMURO ◽  
TETSURO ENDO
Keyword(s):  
Lyapunov Exponent ◽  
Periodic Orbit ◽  
Pattern Matching ◽  
The Other ◽  
Lyapunov Spectrum ◽  
Positive Lyapunov Exponent ◽  
Template Method ◽  
Bisection Method ◽  
Basin Boundary ◽  
Wave Solutions

We develop two algorithms for obtaining an index 1 saddle torus between two attractors of which basin boundary is smooth by using the bisection method. In the first algorithm, which we name "overall template method", we make a template approximating a whole attractor and use it for pattern matching. The attractor can be a periodic orbit, a torus and even a chaos. In the second algorithm, which we name "partial template method", we make a template approximating only a part of an attractor. In this algorithm, the attractor is restricted to a periodic orbit and a two-torus, but calculation time is much smaller than that of the overall template method. We demonstrate these two algorithms for two solutions, observed in a ring of six-coupled bistable oscillators. One is a saddle torus in the basin boundary of the two switching solutions, and the other is of the two quasi-periodic propagating wave solutions. We calculate the Lyapunov spectrum of the obtained saddle torus, and confirm one positive Lyapunov exponent.

BASIN BIFURCATIONS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS: FRACTALIZATION OF BASINS

1994 ◽  
Vol 04 (02) ◽  
pp. 343-381 ◽  
Author(s):  
C. MIRA ◽  
D. FOURNIER-PRUNARET ◽  
L. GARDINI ◽  
H. KAWAKAMI ◽  
J.C. CATHALA
Keyword(s):  
Phase Plane ◽  
The Other ◽  
Two Dimensional ◽  
Basin Boundary ◽  
Critical Curves ◽  
Noninvertible Maps

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two preimages, the other with no preimage.

On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

2019 ◽  
Vol 7 (4) ◽  
pp. 1157-1172 ◽  
Author(s):  
Alejandro Silva-Juárez ◽  
Carlos Javier Morales-Pérez ◽  
Luis Gerardo de la Fraga ◽  
Esteban Tlelo-Cuautle ◽  
José de Jesús Rangel-Magdaleno
Keyword(s):  
Lyapunov Exponent ◽  
Positive Lyapunov Exponent ◽  
Chaotic Oscillators

REPOSITIONING BONE FRAGMENTS USING REGISTRATION OF PAIRED-POINTS AND ASSISTED-CONSTRAINTS IN VIRTUAL BONE REDUCTION SURGERY

2019 ◽  
Vol 31 (03) ◽  
pp. 1950021
Author(s):  
I. Irwansyah ◽  
Jiing-Yih Lai ◽  
Pei-Yuan Lee
Keyword(s):  
Orthopedic Surgery ◽  
Singular Value ◽  
The Other ◽  
Fracture Line ◽  
Template Method ◽  
Great Promise ◽  
Facial Aesthetics ◽  
Bone Fragments ◽  
Rms Error ◽  
Value Decomposition

When repositioning fractured bones in orthopedic surgery, correctness and accuracy are vital to allow the bone to regain the function and facial aesthetics of native uninjured bone. Various improvements to repositioning techniques have been proposed using points, curves, and surfaces to find correspondence between the fracture fragments. The aim of this study was to investigate the appropriate registration constraints for fractured bone reduction. One paired-point and three assisted-constraints registration methods were tested based on contralateral, landmark, and fracture line markers. The fractured proximal femur of a patient was used to compare the performance of these registration methods. Semi-automatic repositioning based on a singular value decomposition algorithm was performed to solve the problem of matching the data from two fragments. The repositioning results show that the proposed registration methods have great promise in assisting the user in defining the paired points, which is often difficult due to visibility limitations on images of fractured bone. Each of the proposed approaches was shown to yield different benefits. In terms of repositioning correctness, the use of contralateral constraints produced the smallest RMS error (1.853[Formula: see text]mm). The contralateral template method yielded the lowest fragment deviation error, but was not significantly superior to the other approaches. Fracture line-based constraints may potentially enable the relocation of fragments closest to pre-injured conditions.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

Three Conditions Lyapunov Exponents Should Satisfy

2003 ◽  
Author(s):  
Jingjun Lou ◽  
Shijian Zhu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
The Other ◽  
Cubic Nonlinearity ◽  
3 Dimensional ◽  
Duffing System ◽  
Dissipative Dynamical System

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

scholarly journals A relation between Lyapunov exponents, Hausdorff dimension and entropy

1981 ◽  
Vol 1 (4) ◽  
pp. 451-459 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Unstable Manifold ◽  
Positive Lyapunov Exponent ◽  
Axiom A ◽  
Generic Points

AbstractFor an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

CHARACTERIZATION OF CHAOTICITY IN THE TRANSIENT CURRENT THROUGH PMMA THIN FILMS

Fractals ◽  
2006 ◽  
Vol 14 (02) ◽  
pp. 125-131 ◽  
Author(s):  
A. HACINLIYAN ◽  
Y. SKARLATOS ◽  
H. A. YILDIRIM ◽  
G. SAHIN
Keyword(s):  
Thin Films ◽  
Time Series ◽  
Lyapunov Exponent ◽  
Power Law ◽  
Chaotic Behavior ◽  
Transient Current ◽  
Positive Lyapunov Exponent ◽  
Aluminum Films ◽  
Thin Aluminum

Chaotic behavior in the transient current through thin Aluminum-PMMA-Aluminum films has been analyzed for times ranging up to 30,000s, in the temperature range 293–363K for applied voltages in the range 10–80V. Time series analysis reveals a positive Lyapunov exponent consistently and reproducibly throughout this range. Power law relaxation as reflected by the autocorrelation function and the positive Lyapunov exponent show parallel behaviors as a function of applied electric field.

On fractional lyapunov exponent for solutions of linear fractional differential equations

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Nguyen Cong ◽  
Doan Son ◽  
Hoang Tuan
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Lyapunov Spectrum ◽  
Bounded Linear ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

AbstractOur aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.

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