Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

2020 ◽  
Author(s):  
Murat Adıvar ◽  
Youssef N. Raffoul
Keyword(s):  
Dynamical Systems ◽  
Time Scales

SYMBOLIC DYNAMICS, COARSE GRAINING AND THE MONITORING OF COMPLEX SYSTEMS

2011 ◽  
Vol 21 (12) ◽  
pp. 3465-3475 ◽  
Author(s):  
VASILEIOS BASIOS ◽  
DÓNAL MAC KERNAN
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Systems ◽  
Error Estimates ◽  
Time Scales ◽  
Symbolic Dynamics ◽  
Prediction Error ◽  
Probabilistic Approach ◽  
Coarse Graining ◽  
Complex Nonlinear Systems

Coarse graining techniques and their associated symbolic dynamics are reviewed with a focus on probabilistic aspects of complex dynamical systems. The probabilistic approach initiated by Nicolis and coworkers has been elaborated. One of the major issues when dealing with the dynamics of complex nonlinear systems, the fact that the inherent time-scales of the unfolding phenomena are not well separated, is brought into focus. Recent results related to this interdependence, which is one of the most characteristic aspects of complexity and a major challenge in prediction, error estimates and monitoring of nonlinear complex systems, are discussed.

Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

Computer Simulation of Multibody Dynamical Systems in a TeraGrid Environment

2010 ◽  
Author(s):  
Shanzhong Shawn Duan ◽  
Abdul Muqtadir Mohammed
Keyword(s):  
Computer Simulation ◽  
Dynamical Systems ◽  
Time Scales ◽  
Multibody Dynamics ◽  
Virtual Prototyping ◽  
System Size ◽  
Computer Hardware ◽  
Foreseeable Future ◽  
Computing Systems ◽  
Great Growth

Despite the great growth in capability of computer hardware, the system size, complexity of structures, and time scales present in virtual prototyping of multibody dynamical systems will continue to challenge the field of computational multibody dynamics for the foreseeable future. In this paper, the scientific problems in virtual prototyping of multibody dynamical systems are articulated. Implementation of an efficient parallelizable algorithm on TeraGrid computing systems is further discussed. Various simulation cases and computing results are presented to demonstrate impact of the TeraGrid to the performance of the algorithm.

scholarly journals Generalized theorem of Hartman-Grobman on measure chains

1996 ◽  
Vol 60 (2) ◽  
pp. 157-191 ◽  
Author(s):  
Stefan Hilger
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Difference Equation ◽  
Time Scales ◽  
Imaginary Axis ◽  
Topological Equivalence ◽  
Critical Component ◽  
One Dimensional ◽  
Closed Intervals ◽  
Critical Spectrum

AbstractWe will prove the Theorem of Hartman-Grobman in a very general form. It states the topological equivalence of the flow of a nonlinear non-autonomous differential or difference equation with critical component to the flow of a partially linearized equation. The critical spectrum has not necessearily to be contained in the imaginary axis or the unit circle respectively. Further on we will employ the socalled calculus on measure chains within dynamical systems theory. Within this calculus the usual one dimensional time scales can be replaced by measure chains which are essentially closed subsets of R. The paper can be understood without knowledge of this calculus.So our main theorem will be valid even for equations defined on very strange time scales such as sequences of closed intervals. This is especially interesting for applications within the theory of differential-difference equations or within numerical analysis of qualitative phenomena of dynamical systems.

QUANTIFYING LOCAL INSTABILITY AND PREDICTABILITY OF CHAOTIC DYNAMICAL SYSTEMS BY MEANS OF LOCAL METRIC ENTROPY

2000 ◽  
Vol 10 (01) ◽  
pp. 135-154 ◽  
Author(s):  
MOZHENG WEI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Time Scales ◽  
Growth Rates ◽  
Metric Entropy ◽  
Random Errors ◽  
Local Instability ◽  
Instability Index ◽  
Local Growth ◽  
Chaotic Dynamical Systems

A local metric entropy (LME) is introduced and used as a measure of local instability of chaotic dynamical systems. The predictability time scale of a dynamical system during a given period of time can also be estimated with high accuracy by using the LME. It is shown that LME, at any time during the evolution of a dynamical system, can be calculated as the sum of all the positive local Lyapunov exponents (LEs). This conclusion implies that the positive local LEs represent the rates of local information changes along the directions of their respective Lyapunov vectors. LME does not depend upon the amplitudes nor the configurations of initial perturbations; it depends on the positive local LEs which are intrinsic properties of dynamical systems. In addition, the sum of all the local LEs is proven to be equal to the divergence of phase space. Thus for a general chaotic system at any time, the sum of all the local LEs is equal to the sum of all the local growth rates of either instantaneous optimal modes or normal modes. In analyzing local instability, the performance of LME is evaluated by comparing an instability index with LME, the first local LE, locally largest LE, local growth rates of the dominant instantaneous optimal mode and normal mode. When LME is used to estimate the predictability time scales of systems over specified time periods, it is found that the time scales defined by LME are generally closer to the standard predictability times than the Lyapunov times and Kolmogorov–Sinai times for most cases in the two dynamical systems we have tested. Both the instability index and standard predictability time are defined and calculated through a large number of random errors.

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