Dynamical Systems with Unbounded Time Interval in Engineering, Ecology and Economics

1987 ◽  
pp. 1-21
Author(s):  
D. A. Carlson ◽  
A. Haurie
Keyword(s):  
Dynamical Systems ◽  
Time Interval

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 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 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 COMPUTATIONAL MODELING OF DYNAMICAL SYSTEMS

2005 ◽  
Vol 15 (03) ◽  
pp. 471-481 ◽  
Author(s):  
JOHAN JANSSON ◽  
CLAES JOHNSON ◽  
ANDERS LOGG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Computational Modeling ◽  
Time Scales ◽  
Time Scale ◽  
Short Note ◽  
Multiple Time Scales ◽  
Time Interval ◽  
Multiple Time ◽  
Fast Time

In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10–9 computed over the time interval [0,100], and a lattice consisting of large and small point masses.

On the Differential Geometry of Flows in Nonlinear Dynamical Systems

2007 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Small Time ◽  
Time Interval ◽  
Reference Surface ◽  
Nonlinear Dynamical ◽  
Reference Flow ◽  
Contact Flows ◽  
Order Contact

In this paper, in order to investigate the relation between two flows given in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. For a small time interval, the change rate of the normal distance between the reference and compared flows on the normal direction of the reference surface is measured by a new function (i.e., G-function). Based on the surface of the reference flow, the kth -order G-functions for the non-contact and lth-order contact flows in two different dynamical systems are introduced. Through the new functions, the geometric relations between two flows in two dynamical systems imposed in the same phase space are investigated without contact between the reference and compared flows. The compared flow passing through, returning back from and paralleling to the surface of the reference flow is discussed first. The tangency and passability for a compared flow to the surface of a reference flow with the lth-order contact are presented. The dynamics for the compared flow with such a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given. The contact and tangential singularity between two flows are different. The kth-order contact of the compared flow to the reference surface indicates that the compared flow to the reference surface is of the kth-order singularity. However, the compared flow with the kth-order singularity to the reference surface does not mean that the compared flow to the reference surface is of the kth-order contact.

Dynamical Systems with Unbounded Time Interval in Engineering, Ecology and Economics

1991 ◽  
pp. 1-19
Author(s):  
Dean A. Carlson ◽  
Alain B. Haurie ◽  
Arie Leizarowitz
Keyword(s):  
Dynamical Systems ◽  
Time Interval

Numerical analysis of dynamical systems

Acta Numerica ◽  
1994 ◽  
Vol 3 ◽  
pp. 467-572 ◽  
Author(s):  
Andrew M. Stuart
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Equilibrium Points ◽  
Invariant Sets ◽  
Phase Portraits ◽  
Existence Theory ◽  
Time Interval ◽  
Time Dynamics ◽  
Long Time

This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration.Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors.Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method.

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