We review a variety of control methods which are capable of enhancing the chaoticity and mixing properties of chaotic flows and also methods which work towards promoting the coherence properties of such flows. We discuss a parameter control method which can enhance the chaoticity and the rate of mixing of dissipative as well as conservative flows and outline methods which promote global mixing by the addition of noise and by preventing island formation. As the inverse side of this problem, we summarize methods which can create coherent structures in chaotic dynamical flows. We also discuss the utility of these methods from the point of view of applications as well as for understanding phenomena which occur in natural systems.

Neelima Gupte

Department Of Physics

Ashutosh Sharma

Computational methods

Control

Inverse problems

Mixing

Problem solving

Coherent structures

EXTENDED SYSTEMS

LAGRANGIAN CHAOS

Chaos theory

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Control methods for problems of mixing and coherence in chaotic maps and flows

Pramana - Journal of Physics

The problem of synchronization of coupled Hamiltonian systems presents interesting features due to the mixed nature (regular and chaotic) of the phase space. We study these features by examining the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Caroll method. The master stability function approach is used to study the stability of the synchronous state and to identify the percentage of synchronizing initial conditions. The transient to synchronization shows intermittency with an associated power law. The mixed nature of the phase space of the studied map has notable effects on the synchronization times as is seen in the case of the standard map. Using finite-time Lyapunov exponent analysis, we show that the synchronization of the maps occurs in the neighborhood of invariant curves in the phase space. The phase differences of the coevolving trajectories show intermittency effects, due to the existence of stable periodic orbits contributing locally stable directions in the synchronizing neighborhoods. Furthermore, the value of the nonlinearity parameter, as well as the location of the initial conditions play an important role in the distribution of synchronization times. We examine drive response combinations which are chaotic-chaotic, chaotic-regular, regular-chaotic, and regular-regular. A range of scaling behavior is seen for these cases, including situations where the distributions show a power-law tail, indicating long synchronization times for at least some of the synchronizing trajectories. The introduction of coherent structures in the system changes the situation drastically. The distribution of synchronization times crosses over to exponential behavior, indicating shorter synchronization times, and the number of initial conditions which synchronize increases significantly, indicating an enhancement in the basin of synchronization. We discuss the implications of our results. © 2016 American Physical Society.

Physical Review E

Synchronization in area-preserving maps: Effects of mixed phase space and coherent structures

We propose a new method for the creation of coherent structures in two-dimensional area-preserving maps. The coherent structures are created by applying a parameter perturbation in the neighbourhood of the periodic points of the map. We demonstrate the success of our method in the case of the standard map and in two cases of maps which represent realistic fluid flows, the blinking vortex map of Aref and the journal bearing map of Chaiken. We also find the signatures of the coherent structures in quantifiers like the local exit time distribution and of the local Lyapunov exponents of the system. Our method may provide useful insights into the ways in which coherent structures are created in natural systems. © 2007 Elsevier B.V. All rights reserved.

Physics Letters, Section A: General, Atomic and Solid State Physics

Creation of coherent structures in area-preserving maps

Pramana

Chaos in Josephson junctions

Preserving chaos: Control strategies to preserve complex dynamics with potential relevance to biological disorders

Nature

Controlling chaos in the brain

Chaos: An Interdisciplinary Journal of Nonlinear Science

Jupiter’s Great Red Spot and zonal winds as a self‐consistent, one‐layer, quasigeostrophic flow

Anisotropic geophysical vortices

Journal	Pramana - Journal of Physics
Publisher	Indian Academy of Sciences
ISSN	03044289
Open Access	No