scholarly journals Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

2014 ◽  
Vol 21 (1) ◽  
pp. 165-185 ◽  
Author(s):  
S. Wiggins ◽  
A. M. Mancho
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Fluid Transport ◽  
Invariant Tori ◽  
Point Of View ◽  
Time Dependent ◽  
Time Interval ◽  
Two Dimensional ◽  
Kam Theorem ◽  
Nekhoroshev Theorem

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

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.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

scholarly journals Square of Brownian motion

1975 ◽  
Vol 59 ◽  
pp. 1-8
Author(s):  
Hisao Nomoto
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Correlation Function ◽  
Finite Time ◽  
Present Note ◽  
Point Of View ◽  
Time Interval ◽  
Sample Function ◽  
Finite Time Interval ◽  
Zero Crossings

Let Xt be a stochastic process and Yt be its square process. The present note is concerned with the solution of the equation assuming Yt is given. In [4], F. A. Grünbaum proved that certain statistics of Yt are enough to determine those of Xt when it is a centered, nonvanishing, Gaussian process with continuous correlation function. In connection with this result, we are interested in sample function-wise inference, though it is far from generalities. A glance of the equation shows that the difficulty is related how to pick up a sign of . Thus if we know that Xt has nice sample process such as the zero crossings are finite, no tangencies, in any finite time interval, then observations of these statistics will make it sure to find out sample functions of Xt from those of Yt (see [2]). The purpose of this note is to consider the above problem from this point of view.

FLUCTUATION OF SURFACE IN POLISHING PROCESS

Fractals ◽  
1996 ◽  
Vol 04 (02) ◽  
pp. 213-218 ◽  
Author(s):  
Y. HASEGAWA ◽  
S. MIYAZIMA ◽  
Y. NAMBA
Keyword(s):  
Point Of View ◽  
Time Dependent ◽  
Surface Pattern ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Polishing Process ◽  
Two Dimensional Model

A change of surface pattern of materials during polishing is simulated and analyzed from the fractal point of view. We have proposed a simple two-dimensional model for microscopic processes of polishing. We have obtained a result that the time dependent fluctuation of surface scales with the momentum of the polishing particle.

scholarly journals Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets

2002 ◽  
Vol 9 (3/4) ◽  
pp. 237-263 ◽  
Author(s):  
K. Ide ◽  
D. Small ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Vector Fields ◽  
Analytical Formula ◽  
Velocity Fields ◽  
Dynamical Systems Theory ◽  
Time Dependent ◽  
Time Interval ◽  
Data Set ◽  
Hyperbolic Trajectories

Abstract. In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.

scholarly journals Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

2012 ◽  
Vol 468 (2146) ◽  
pp. 2915-2938 ◽  
Author(s):  
Michael Higley ◽  
Michael Siegel ◽  
Michael R. Booty
Keyword(s):  
Analytical Solutions ◽  
Stokes Flow ◽  
Finite Time ◽  
Capillary Number ◽  
Time Dependent ◽  
Two Dimensional ◽  
Elastic Interfaces ◽  
Critical Capillary Number ◽  
Field Velocity ◽  
Elastic Capsules

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modelled, but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytical solutions for the steady-state response and time-dependent evolution of two-dimensional elastic capsules with an inviscid interior in Stokes flow. This provides a complete picture of the steady response of initially circular capsules in linear strain and shear flows as a function of the capillary number Ca . The analysis is complemented by spectrally accurate numerical computations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor's four-roller mill is found to lead to cusped steady shapes at a critical capillary number Ca c for Hookean capsules. Numerical simulation of the time-dependent evolution for Ca > Ca c shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This is compelling evidence of finite-time singularity formation in fluid flow with elastic interfaces.

STABILITY ANALYSIS OF DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT DELAY

2006 ◽  
Vol 16 (02) ◽  
pp. 465-472 ◽  
Author(s):  
WEIHUA DENG ◽  
YUJIANG WU ◽  
CHANGPIN LI
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Finite Time ◽  
Time Dependent ◽  
Time Interval ◽  
Finite Time Interval ◽  
Predator Prey ◽  
Gauss Type ◽  
Predator Prey Models ◽  
The Stability

In this Letter, we study the stability of differential equations with time-dependent delay. Several theorems are established for stability on a finite time interval, called "interval stability" for simplicity, and Liapunov stability. These theorems are applied to the generalized Gauss-type predator–prey models, and satisfactory results are obtained.

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