scholarly journals Shape Coherence and Finite-Time Curvature Evolution

2015 ◽  
Vol 25 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Tian Ma ◽  
Erik M. Bollt
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Coherent Structures ◽  
Finite Time ◽  
Large Curvature ◽  
Nonautonomous Dynamical Systems ◽  
Finite Time Lyapunov Exponent ◽  
Close Proximity ◽  
Definition Of ◽  
Curvature Growth

We introduce a definition of finite-time curvature evolution along with our recent study on shape coherence in nonautonomous dynamical systems. Comparing to slow evolving curvature preserving the shape, large curvature growth points reveal the dramatic change on shape such as the folding behaviors in a system. Closed trough curves of low finite-time curvature (FTC) evolution field indicate the existence of shape coherent sets, and troughs in the field indicate the most significant shape coherence. Here, we will demonstrate these properties of the FTC, as well as contrast to the popular Finite-Time Lyapunov Exponent (FTLE) computation, often used to indicate hyperbolic material curves as Lagrangian Coherent Structures (LCS). We show that often the FTC troughs are in close proximity to the FTLE ridges, but in other scenarios, the FTC indicates entirely different regions.

scholarly journals Local finite-time Lyapunov exponent, local sampling and probabilistic source and destination regions

2015 ◽  
Vol 22 (6) ◽  
pp. 663-677 ◽  
Author(s):  
A. E. BozorgMagham ◽  
S. D. Ross ◽  
D. G. Schmale
Keyword(s):  
Lyapunov Exponent ◽  
Coherent Structures ◽  
Finite Time ◽  
Large Scale ◽  
Field Experiments ◽  
Local Velocity ◽  
Finite Time Lyapunov Exponent ◽  
Independent Observations ◽  
Large Scale Flow ◽  
Local Sampling

Abstract. The finite-time Lyapunov exponent (FTLE) is a powerful Lagrangian concept widely used for describing large-scale flow patterns and transport phenomena. However, field experiments usually have modest scales. Therefore, it is necessary to bridge the gap between the concept of FTLE and field experiments. In this paper, two independent observations are discussed: (i) approximation of the local FTLE time series at a fixed location as a function of known distances between the destination (or source) points of released (or collected) particles and local velocity, and (ii) estimation of the distances between the destination (or source) points of the released (or collected) particles when consecutive release (or sampling) events are performed at a fixed location. These two observations lay the groundwork for an ansatz methodology that can practically assist in field experiments where consecutive samples are collected at a fixed location, and it is desirable to attribute source locations to the collected particles, and also in planning of optimal local sampling of passive particles for maximal diversity monitoring of atmospheric assemblages of microorganisms. In addition to deterministic flows, the more realistic case of unresolved turbulence and low-resolution flow data that yield probabilistic source (or destination) regions are studied. It is shown that, similar to deterministic flows, Lagrangian coherent structures (LCS) and local FTLE can describe the separation of probabilistic source (or destination) regions corresponding to consecutively collected (or released) particles.

scholarly journals SUPREME LOCAL LYAPUNOV EXPONENTS AND CHAOTIC IMPULSIVE SYNCHRONIZATION

2013 ◽  
Vol 23 (10) ◽  
pp. 1350169 ◽  
Author(s):  
SHENGYAO CHEN ◽  
FENG XI ◽  
ZHONG LIU
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Finite Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
Local Instability ◽  
Impulsive Synchronization ◽  
Local Lyapunov Exponent ◽  
Expansion Behavior

Impulsively synchronized chaos with criterion from conditional Lyapunov exponent is often interrupted by desynchronized bursts. This is because the Lyapunov exponent cannot characterize local instability of synchronized attractor. To predict the possibility of the local instability, we introduce a concept of supreme local Lyapunov exponent (SLLE), which is defined as supremum of local Lyapunov exponents over the attractor. The SLLE is independent of the system trajectories and therefore, can characterize the extreme expansion behavior in all local regions with prescribed finite-time interval. It is shown that the impulsively synchronized chaos can be kept forever if the largest SLLE of error dynamical systems is negative and then the burst behavior will not appear. In addition, the impulsive synchronization with negative SLLE allows large synchronizable impulsive interval, which is significant for applications.

Sign in / Sign up

Export Citation Format

Share Document