scholarly journals Examining the Chaotic Behavior in Dynamical Systems by Means of the 0-1 Test

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Loukas Zachilas ◽  
Iacovos N. Psarianos
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Efficient Method ◽  
Toda Lattice ◽  
Good Accuracy ◽  
Chaotic Behavior ◽  
Characteristic Number ◽  
The Stability ◽  
Lyapunov Characteristic Number

We perform the stability analysis and we study the chaotic behavior of dynamical systems, which depict the 3-particle Toda lattice truncations through the lens of the 0-1 test, proposed by Gottwald and Melbourne. We prove that the new test applies successfully and with good accuracy in most of the cases we investigated. We perform some comparisons of the well-known maximum Lyapunov characteristic number method with the 0-1 method, and we claim that 0-1 test can be subsidiary to the LCN method. The 0-1 test is a very efficient method for studying highly chaotic Hamiltonian systems of the kind we study in our paper and is particularly useful in characterizing the transition from regularity to chaos.

Stability of Dynamical Systems With Multiple Nonlinearities

1987 ◽  
Vol 109 (4) ◽  
pp. 410-413 ◽  
Author(s):  
Norio Miyagi ◽  
Hayao Miyagi
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Stability Region ◽  
Direct Method ◽  
Essential Feature ◽  
Point Of View ◽  
Stability Of Dynamical Systems ◽  
The Stability

This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Lure´ type Lyapunov function which surpasses the Lure´-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Lure´ type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.

Stability Analysis of Chaotic Wavy Stratified Fluid-Fluid Flow With the 1D Fixed-Flux Two-Fluid Model

2016 ◽  
Author(s):  
Avinash Vaidheeswaran ◽  
William D. Fullmer ◽  
Krishna Chetty ◽  
Raul G. Marino ◽  
Martin Lopez de Bertodano
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Two Phase Flow ◽  
Chaotic Behavior ◽  
Fluid Model ◽  
Phase Flow ◽  
Two Phase ◽  
Two Fluid Model ◽  
The Stability ◽  
Two Fluid

The one-dimensional fixed-flux two-fluid model (TFM) is used to analyze the stability of the wavy interface in a slightly inclined pipe geometry. The model is reduced from the complete 1-D TFM, assuming a constant total volumetric flux, which resembles the equations of shallow water theory (SWT). From the point of view of two-phase flow physics, the Kelvin-Helmholtz instability, resulting from the relative motion between the phases, is still preserved after the simplification. Hence, the numerical fixed-flux TFM proves to be an effective tool to analyze local features of two-phase flow, in particular the chaotic behavior of the interface. Experiments on smooth- and wavy-stratified flows with water and gasoline were performed to understand the interface dynamics. The mathematical behavior concerning the well-posedness and stability of the fixed-flux TFM is first addressed using linear stability theory. The findings from the linear stability analysis are also important in developing the eigenvalue based donoring flux-limiter scheme used in the numerical simulations. The stability analysis is extended past the linear theory using nonlinear simulations to estimate the Largest Lyapunov Exponent which confirms the non-linear boundedness of the fixed-flux TFM. Furthermore, the numerical model is shown to be convergent using the power spectra in Fourier space. The nonlinear results are validated with the experimental data. The chaotic behavior of the interface from the numerical predictions is similar to the results from the experiments.

STABILITY ANALYSIS OF A FLUID FLOW MODEL FOR TCP LIKE BEHAVIOR

2005 ◽  
Vol 15 (07) ◽  
pp. 2277-2282 ◽  
Author(s):  
WIM MICHIELS ◽  
SILVIU-IULIAN NICULESCU
Keyword(s):  
Fluid Flow ◽  
Stability Analysis ◽  
Chaotic Behavior ◽  
Quantitative Information ◽  
Computational Tools ◽  
Flow Models ◽  
Qualitative And Quantitative ◽  
Numerical Tools ◽  
The Stability ◽  
Delay Differential

This note focuses on the stability analysis of some classes of nonlinear time-delay models, encountered as fluid models for TCP/AQM network. By combining analytical and numerical tools, the attractors of these models, as well as the local and global behaviors of the solutions are studied. Among others, the presence of a chaotic attractor is shown, which supports the proposition that TCP itself as a deterministic process can cause or contribute to chaotic behavior in a network. The main goals of the paper are firstly to provide qualitative and quantitative information on the dynamics of the models under consideration, and secondly to illustrate the capabilities of computational tools for stability and bifurcation analysis of delay differential equations to analyze fluid flow models.

A Comparative Study of Liapunov Stability Analyses of Flexible Damped Satellites

1978 ◽  
Vol 45 (3) ◽  
pp. 657-663 ◽  
Author(s):  
H. B. Hablani ◽  
S. K. Shrivastava
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Comparative Study ◽  
Density Function ◽  
Orbital Plane ◽  
Hybrid Dynamical Systems ◽  
Stability Criteria ◽  
Liapunov Stability ◽  
Dynamic Potential ◽  
The Stability

A literal Liapunov stability analysis of a spacecraft with flexible appendages often requires a division of the associated dynamic potential into as many dependent parts as the number of appendages. First part of this paper exposes the stringency in the stability criteria introduced by such a division and shows it to be removable by a “reunion policy.” The policy enjoins the analyst to piece together the sets of criteria for each part. Employing reunion the paper then compares four methods of the Liapunov stability analysis of hybrid dynamical systems illustrated by an inertially coupled, damped, gravity stabilized, elastic spacecraft with four gravity booms having tip masses and a damper rod, all skewed to the orbital plane. The four methods are the method of test density function, assumed modes, and two and one-integral coordinates. Superiority of one-integral coordinate approach is established here. The design plots demonstrate how elastic effects delimit the satellite boom length.

Time Domain Approaches to the Stability Analysis of Flexible Dynamical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Jielong Wang ◽  
Xiaowen Shan ◽  
Bin Wu ◽  
Olivier A. Bauchau
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Time Domain ◽  
Floquet Theory ◽  
Characteristic Exponent ◽  
Periodic Coefficient ◽  
Theoretical Background ◽  
System Stability ◽  
Lyapunov Characteristic Exponent ◽  
The Stability

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

scholarly journals Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems

2019 ◽  
Vol 24 (1) ◽  
pp. 13 ◽  
Author(s):  
Francisco Solis
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Dynamical Behavior ◽  
Discrete Dynamical Systems ◽  
Exponential Polynomial ◽  
Linear Dynamical System ◽  
The Stability ◽  
Polynomial Family ◽  
Complex Dynamical Behavior

In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, from which we show the intricate and complex dynamical behavior of their orbits.

A REVIEW STUDY OF THE 3-PARTICLE TODA LATTICE AND HIGHER-ORDER TRUNCATIONS: THE ODD-ORDER CASES — PART I

2010 ◽  
Vol 20 (10) ◽  
pp. 3007-3064 ◽  
Author(s):  
LOUKAS ZACHILAS
Keyword(s):  
Periodic Orbits ◽  
Toda Lattice ◽  
Characteristic Number ◽  
Higher Order ◽  
Surface Of Section ◽  
Odd Order ◽  
Poincaré Surface Of Section ◽  
Families Of Periodic Orbits ◽  
Review Study ◽  
Lyapunov Characteristic Number

The numerical behavior of the truncated 3-particle Toda lattice (3pTL) is reviewed and studied in more detail (than in previous papers) and at higher energies (at odd-orders n ≤ 9). We further extended our study to higher truncations at odd-orders, n = 2k + 1, k = 1, …, 9. We have located the majority of the families of periodic orbits along with their main bifurcations. By using: (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits, we studied the topology of the nine odd-order Hamiltonians with respect to their order of truncation.

SOLITARY WAVES AND CHAOTIC BEHAVIOR FOR A CLASS OF COUPLED FIELD EQUATIONS

2003 ◽  
Vol 13 (03) ◽  
pp. 643-651
Author(s):  
KENG-HUAT KWEK ◽  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Hamiltonian Systems ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Solitary Wave Solutions ◽  
Field Equations ◽  
Wave Solutions ◽  
Coupled Field ◽  
Coupled Field Equations

Using the theory of Hamiltonian systems and dynamical systems to a class of coupled field equations, the existence of uncountably infinite many solitary wave solutions, arbitrarily many distinct periodic solutions and chaotic behavior is obtained. Some sufficient conditions to guarantee the existence of the above solutions are given.

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