A synthesis of elementary computation and dynamical system theories leads to a constructive approach to discovering coherent structures in spatial systems and to quantifying a pattern’s complexity. The basic technique reviewed here builds probabilistic automata from temporal and spatial data series generated by a simple nonlinear spatial system. In this way, a given pattern’s unpredictability a...

Leo H. O. Hellström, a) Bharathram Ganapathisubramani, b) and Alexander J. Smits c) Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, U.S.A. Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, U.K. Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, U.S.A., and Mechanical and Aerospace Engineer...

The topic of Lagrangian coherent structures (LCS) has been a rapidly growing area of research in nonlinear dynamics for almost a decade. It provides a means to rigorously define and detect transport barriers in dynamical systems with arbitrary time dependence and has a wealth of applications, particularly to fluid flow problems. Here, we give a short introduction to the topic of LCS and review ...

For a nonuniformly stratified layer of fluid, internal gravity waves propagate at varying angles depending on the local buoyancy and Coriolis (in geophysical applications) frequencies. Relatively confined geometries, such as multiple submarine ridges, can support internal wave attractors, which can be viewed as Lagrangian coherent structures for the energy density flux. Since traditional approa...

In this article we introduce generalizations of several well known reliability bounds. These bounds are based on arbitrary partitions of the family of minimal path or cut sets of the system and can be used for approximating the reliability of any coherent structure with iid components. An illustration is also given of how the general results can be applied for a specific reliability structure (...

It is thought that the small-scale magnetic fields observed in accretion discs, galaxies and galactic clusters are generated by a dynamo process in which the turbulent plasma amplifies small initial magnetic fluctuations. Numerical simulations of turbulence have revealed that turbulence consists of filamentlike vortex structures superimposed on an incoherent background, which carry a considerab...

It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characteriza...

A systematic method is proposed for the determination of the statistical properties of a field consisting of a coherent structure interacting with turbulent linear waves. The explicit expression of the generating functional of the correlations is obtained, performing the functional integration on a neighborhood in the function space around the soliton. The results show that the non-Gaussian flu...

The inherent difficulty of understanding turbulence has led to researchers attacking the topic in many different ways over the years of turbulence research. Some approaches have been more successful than others, but most only deal with part of the problem. One approach that has seen reasonable success (or at least popularity) is that of attempting to deconstruct the complex and disorganised tur...