Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)


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The manuscript demonstrates with a few selected examples how the nonlinear phenomena can change the way in which heterogeneous reactions proceed. Firstly it is shown that the kinetic oscillations and synergetic effect can originate due to the diffusion coupling of two different catalysts.


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Secondly it is demonstrated that due to the nonlinearity of the system and the ability to form spatial structures two catalysts can produce the same catalytic activity as one catalyst. The dynamics of coupled catalyst wafers in a tube reactor was studied and traced by examining the oxidation of CO on a supported palladium zeolite catalyst under normal pressure in a continuous flow of reactants. An X-type zeolite loaded with 0.

Lakshmanan, Muthusamy

The overall conversion rate exhibits a dynamic behavior with self-affine patterns of excursions to smaller conversion rates on a time scale in the order of some seconds. Similar patterns were observed for the temperature time series of the catalyst wafers. The influence of the flow rate on catalyst wafer coupling was studied. Increasing the flow rate causes decoupling of the catalyst wafers, increased frequency of maximum excursions, increasing rate of smaller excursions, increasing pattern complexity and a decrease in maximum conversion baseline.

They appear as banded formations, layered and folded structures, diapirs, or cockade ores and in numerous other shapes and forms. Their size ranges from less than one centimeter up to kilometers. In many cases ordered structures develop in self-organized systems, making this concept a useful tool for better understanding complex and diverse processes in geology. In essence, self-organized systems are autonomously shaped units resulting from their inner determination under the influence of environmental conditions. Conversely, self-organized geological systems that have ordered structures contain valuable information about their genesis that is preserved within the structure.

Because geological objects typically preserve only one picture for a very long time-frame, it is important to evaluate the ordered structures from the perspective of self-organization to gain access to the information. For this reason, applying the self-organization concept to geology is also important for the society as a whole.

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Accordingly, patterns can be found on animal skins or in chemical reactive systems. The formation of these structures is governed by gradients of chemical reactants and ions and thus, of electric fields. Here, an electric field is applied to a chemical compartmentalized reaction i. In this system, percolation can occur when the volume of water is large compared to that of the oil. Thus, water droplets generate a network of water channels.


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Due to the presence of ions, this formation can be manipulated by an electric field. Turing patterns show a spatial drift, caused by the electric field. The strength of the field resolves the resulting drift velocity of the patterns. Additionally, a reorientation of the patterns is induced by a gradient generated by the electric field.

Spiral waves are propagating self-organized structures commonly found in excitable media.

Dynamics of Nonlinear Time-Delay Systems by Muthusamy Lakshmanan

Spiral waves of electrical excitation in cardiac systems connect to some arrhythmias, such as tachycardia and fibrillations, potentially leading to sudden cardiac death so that they should be eliminated. Such waves may drift and eventually annihilate at the boundary. However, they can be stabilized, when they are pinned to obstacles, that are weakly excitable or unexcitable regions in the medium. Recently, we used the Belousov-Zhabotinsky solutions, the well-known excitable chemical systems, to study the propagation of spiral waves pinned to obstacles and applied electrical forcing to unpin them in different situations of obstacle size and excitability.

We employed simulations with the Oregonator model, a realistic scheme for the Belousov-Zhabotinsky reaction, to confirm the experimental findings as well as to reveal the detailed motions of the spiral waves under some specific conditions that are difficult to be realized in the experiments.

A rechargeable battery can be considered as a complex system which is self-organizing under charging, discharging and relaxation. Depending of the type of battery and the kind of external influences, the battery is stressed in many different ways. This leads to an alteration of the internal self-organization processes which are indicative for the present internal state and the future development of the battery. To get detailed information about the State-Of-Charge SOC and the State-Of-Health SOH of such electrochemical systems we carried out on the one hand internal spatio-temporal measurements of the half-cell potentials within lead-acid batteries.

By this, significant structure formation can be observed and is represented using several methods of synergetics. The fits of these curves are comprehended in a small set of significant parameters. Excitable systems can sustain different kinds of wave forms like target patterns, two-dimensional spiral waves or their three-dimensional counterparts, the scroll waves. The dynamics of these excitation patterns and their responses to different kinds of internal and external perturbations are being looked into.

These waves interact with neighboring vortices, that could lead to either attraction or repulsion and sometimes even their merging. Thermal gradients and electric fields can be used to control the motion of spiral and scroll waves. Scroll waves anchor to unexcitable heterogeneities and external field gradients can be used to unpin them from such obstacles. Our experiments with the Belousov—Zhabotinsky reaction are explained on the basis of numerical simulations using the Barkley model. When a hydrophobic IL dissolves in water, it forms a droplet with a clear interface even though it is soluble to water.

The transition between the two liquids remains sharp throughout the dissolution process, and it seems that the diffusion occurs unidirectionally, only outward from the droplet. The dissolution dynamics can be described as an activation process in which IL molecules escape from the droplet with a probability proportional to the surface area of the droplet. Pattern formation processes in marine science are very versatile, the spatial scale of these patterns ranges from cm to about a km.

We demonstrate three mechanisms of pattern formation in the water column as well as in the sediment of the ocean. Plankton patterns result from an intricate interplay between biological growth and transport by ocean currents. Mesoscale hydrodynamic structures like vortices can act as incubators for plankton blooms, while transport barriers in the flow can lead to a segregation of species in certain spatial regions of the ocean characterized by the dominance of a particular plankton species.

Spatial patterns of chemicals and microorganisms in the sediment can emanate from a Turing instability leading to inhomogeneous distributions of nutrients and bacteria. Marine aggregates agglomerations of plankton, bacteria and inorganic substances form preferential concentrations, i. Several self-propelled objects have been investigated and used to add functionalities mimicking biological systems. One promising approach is the introduction of nonlinear chemical reactions such as the Belousov-Zhabotinsky BZ reaction. In this work we placed an aqueous droplet of the BZ solution into an oil phase composed of monoolein and squalane.

Time delay embedding for Koopman

The BZ droplet moved spontaneously, and its speed oscillated periodically in synchronization with the redox state of the aqueous solution. This finding and measurements of the interfacial tension between water and squalane reveal that the oscillatory motion of the BZ droplet originated from the oscillation of the Br 2 -concentration. Last but not least, the presentation as a whole strives for a balance between the necessary mathematical description of the basics and the detailed presentation of real-world applications. Dynamics of Nonlinear Time-Delay Systems.

Muthusamy Lakshmanan , Dharmapuri Vijayan Senthilkumar. Chapter 2 Linear Stability and Bifurcation Analysis. Amplitude Death and Other Effects. In this connection, time-delay systems described by delay differential equations have developed as particularly suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant.

This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics. Last but not least, the presentation as a whole strives for a balance between the necessary mathematical description of the basics and the detailed presentation of real-world applications.

This book would serve as a good starting point for anyone wanting to explore further the research topics presented. In particular, it is a good introduction to topics and questions in delay-differential equations mostly explored by physicists and the considerable literature in physics journals that has emerged in recent years. It is suitable for senior undergraduate and graduate students as well as practical engineers and researchers interested in dynamics of nonlinear time-delay systems and synchronization.

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Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)
Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics) Dynamics of Nonlinear Time-Delay Systems (Springer Series in Synergetics)

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