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 Investigations of Nonlinear Triopoly Models with Different Mechanisms

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
S. S. Askar ◽  
A. Al-khedhairi
Keyword(s):  
Dynamical System ◽  
Global Analysis ◽  
Discrete Dynamical System ◽  
The Other ◽  
3 Dimensional ◽  
Heterogeneous Players ◽  
State Point ◽  
The Stability ◽  
The Impact ◽  
Nash Point

This paper studies the dynamic characteristics of triopoly models that are constructed based on a 3-dimensional Cobb–Douglas utility function. The paper presents two parts. The first part introduces a competition among three rational firms on which their prices are isoelastic functions. The competition is described by a 3-dimensional discrete dynamical system. We examine the impact of rationality on the system’s steady state point. Studying the stability/instability of this point, which is Nash equilibrium and is unique in those models, is illustrated. Numerically, we give some global analysis of Nash point and its stability. The second part deals with heterogeneous scenarios. It consists of two different models. In the first model, we assume that one competitor adopts the local monopolistic approximation mechanism (LMA) while the other opponents are rational. The second model assumes two heterogeneous players with LMA mechanism against one rational firm. Studies show that the stability of NE point of those models is not guaranteed. Furthermore, simulation shows that when firms behave rational with symmetric costs, the stability of NE point is achievable.

scholarly journals Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tai-Ping Chang
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear System ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Duffing System ◽  
Linear And Nonlinear

This paper investigates the chaotic motion in forced Duffing oscillator due to linear and nonlinear damping by using Melnikov technique. In particular, the critical value of the forcing amplitude of the nonlinear system is calculated by Melnikov technique. Further, the top Lyapunov exponent of the nonlinear system is evaluated by Wolf’s algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs. It is concluded that the chaotic motion of the nonlinear system occurs when the forcing amplitude exceeds the critical value, and the linear and nonlinear damping can generate pronounced effects on the chaotic behavior of the forced Duffing oscillator.

scholarly journals Active beating modes of two clamped filaments driven by molecular motors

2022 ◽  
Vol 19 (186) ◽  
Author(s):  
Laura Collesano ◽  
Isabella Guido ◽  
Ramin Golestanian ◽  
Andrej Vilfan
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Limit Cycle ◽  
Molecular Motors ◽  
The Other ◽  
Stable Fixed Point ◽  
Euler Buckling ◽  
Planar Shape ◽  
Secondary Bifurcation ◽  
Surrounding Fluid

Biological cilia pump the surrounding fluid by asymmetric beating that is driven by dynein motors between sliding microtubule doublets. The complexity of biological cilia raises the question about minimal systems that can re-create similar patterns of motion. One such system consists of a pair of microtubules that are clamped at the proximal end. They interact through dynein motors that cover one of the filaments and pull against the other one. Here, we study theoretically the static shapes and the active dynamics of such a system. Using the theory of elastica, we analyse the shapes of two filaments of different lengths with clamped ends. Starting from equal lengths, we observe a transition similar to Euler buckling leading to a planar shape. When further increasing the length ratio, the system assumes a non-planar shape with spontaneously broken chiral symmetry after a secondary bifurcation and then transitions to planar again. The predicted curves agree with experimentally observed shapes of microtubule pairs. The dynamical system can have a stable fixed point, with either bent or straight filaments, or limit cycle oscillations. The latter match many properties of ciliary motility, demonstrating that a two-filament system can serve as a minimal actively beating model.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

A SIMPLE OBSERVER OF THE HYPERCHAOTIC RÖSSLER SYSTEM

2010 ◽  
Vol 24 (22) ◽  
pp. 4325-4331
Author(s):  
XING-YUAN WANG ◽  
JUN-MEI SONG
Keyword(s):  
Dynamical System ◽  
Time Domain ◽  
Global Exponential Stability ◽  
Observer Design ◽  
The Other ◽  
State Observation ◽  
Rossler System ◽  
Error System ◽  
Rössler System ◽  
The Time Domain

This paper studies the hyperchaotic Rössler system and the state observation problem of such a system being investigated. Based on the time-domain approach, a simple observer for the hyperchaotic Rössler system is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design that it does not need to transmit all signals of the dynamical system. It is proved theoretically, and numerical simulations show the effectiveness of the scheme finally.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

2002 ◽  
Vol 69 (3) ◽  
pp. 346-357 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Stochastic Fluctuation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

scholarly journals Well-definedness and observational equivalence for inductive–coinductive programs

2019 ◽  
Vol 29 (4) ◽  
pp. 419-468
Author(s):  
Henning Basold ◽  
Helle Hvid Hansen
Keyword(s):  
Fixed Point ◽  
Modal Logic ◽  
The Other ◽  
Observational Equivalence ◽  
Other Hand ◽  
Inductive Types ◽  
Point Types ◽  
Usual Topology

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

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