We present examples of familiar phenomena found in nonequilibrium systems, including oscillatory phenomena, order-formation processes, and pattern formation. In particular, we introduce commonly used mathematical methods to analyze their characteristics. First, we present oscillations described by the Lotka–Volterra and van der Pol equations, the Brusselator, the Oregonator, and relaxation oscillations as examples of oscillatory phenomena. Second, we investigate the order-formation process in colloidal crystals and present an experimental observation of 2D array formation. Third, we demonstrate pattern formation in crystals on the basis of the Mullins–Sekerka instability, and in chemical and biological systems on the basis of the Turing instability. In particular, we describe the optical properties and development of sophisticated structural patterns that directly interact with light. Finally, we briefly describe a theoretical phase-transition analogy that might clarify the concept of order formation in nonequilibrium systems.

Patterns and their formations appear throughout nature, and are studied to analyze different problems in science and make predictions across a wide range of disciplines including biology, physics, mathematics, chemistry, material science, and nanoscience. With the emergence of nanoscience and the ability for researchers and scientists to study living systems at the biological level, pattern formation research has become even more essential. This book is an accessible first of its kind guide for scientists, researchers, engineers, and students who require a general introduction to this research area, in order to gain a deeper analytical understanding of the most recent observations and experiments by top researchers in physics. Pattern Formations describes the most up-to-date status of this developing field and analyzes the physical phenomena behind a wide range of interesting topics commonly known in the scientific community. The study of pattern formations as a research field will continue to grow as scientists expand their understanding of naturally occurring patterns and mimic nature to help solve complex problems. This research area is becoming more highly recognized due to its contributions to signal processing, computer analysis, image processing, complex networks development, advancements in optics and photonics, crystallography, metallurgy, drug delivery (chemotherapy) and the further understanding of gene regulation. - The only introductory reference book which places special emphasis on the theoretical analyses of experiments in this rapidly growing field of pattern formation - A wide range of physical applications make this book highly interdisciplinary - Explanations of observations and experiments deepen the readers understanding of this developing research field

Some bird and insect species exhibit iridescent colors that originate from submicron microstructures. One example is a periodic stack of thin layers that selectively reflects light wavelengths with high efficiency through multilayer optical interference. However, the periodicity of the microstructure is not a unique factor in the coloration mechanism; other physical factors such as structural irregularities, large-scale structures, and pigmentation also contribute greatly to the coloration in natural examples. This chapter describes basic experimental and theoretical methods of studying structural color, mainly for those who are unfamiliar with this subject. The chapter discusses two insect species, the jewel beetle and Morpho butterfly, repeatedly rather than introducing many examples. It is emphasized in discussing these examples that additional factors are as important for the coloration as the periodicity in the microstructures.

In this chapter, we consider the motion of a droplet and the surrounding flow accompanied by the motion. Our specific attention is on the spontaneous and autonomous motion of a droplet. Such a system has no applied external force and no asymmetry imposed a priori. Nevertheless, the droplet moves by consuming energy and by breaking the symmetry of the system. The phenomenon reminds us of biological systems that can also move spontaneously. These systems, which are called self-propulsive systems, have recently been extensively studied after several model experiments were proposed using chemical reactions. The mechanism of such motion is less clear, though theoretical and computational studies have revealed several novel aspects of the motion in contrast with the motion under a given asymmetry. We discuss recently developed experimental systems. Then, we focus on a suspended droplet that swims, and explain how the result can be analyzed in terms of hydrodynamics by using the concept of surface tension. Finally, we apply the method to the analysis of a swimming suspended droplet induced propelled by a chemical pattern generated inside the droplet.

In this chapter, temporal and spatial patterns observed in the Belousov–Zhabotinsky (BZ) reaction are described, which is a typical nonlinear chemical reaction system that exhibits oscillation and wave propagation. Some related phenomena in biological systems are presented. The underlying mechanism and modeling of oscillation based on chemical equations are then introduced. Reaction diffusion equations are derived and numerical simulations are performed based on the model to explain how the dynamical pattern appears. Finally, synchronous phenomena observed in the BZ reaction are described with an analytical method based on the phase model.

Submicron-sized colloidal particles dispersed in a liquid medium self-assemble to form ordered “crystal” structures in which the particles are regularly arranged in body-centered or face-centered cubic lattice structures. Colloidal crystals have been useful models for studying crystallization in general. They have also attracted considerable attention as novel materials. In this chapter, we describe the physical basis and experimental studies of colloidal crystallization, in particular those of charged colloids. We first present a general introduction to the physical background on the stability and interaction of colloids. Then we report our studies on the crystallization of charged colloids, including charge-induced crystallization, unidirectional crystallization, gel immobilization, and exclusion of impurities. Finally, we review recent research topics in this field, including those on opal crystals and complex structures.

Systems exhibiting spontaneous regular rhythms abound in nature, and several that are characterized by more than two different time scales are known as relaxation oscillators. The density oscillator is an excellent model system for investigating the fundamental mechanisms of relaxation oscillators. It is a system consisting of an inner container, with a thin pipe in its bottom and filled with heavy fluid, inside an outer container filled with light fluid; the fluids alternately exhibit upflow and downflow through the pipe between the two containers. Although the density oscillator is a simple system, its oscillation mechanism is nontrivial and clarifying it is a challenging task. We have recently clarified the mechanism by constructing a simple model on the basis of detailed experiments. In this chapter, we review studies of this topic and introduce relevant work.

This book is a survey of basic oscillatory concepts with the aid of Mathematica® computer algebra system to represent them and to calculate with them. It is written for students, teachers, and researchers needing to understand the basic of oscillatory motion or intending to use Mathematica® to extend their knowledge. All illustrations in the book can be replicated and used to learn and discover oscillatory motion in a new and exciting way. It is meant to complement the analytical skills and to use the computer to visualize the results and to develop a deeper intuitive understanding of oscillatory motion by observing the effects of varying the parameters of the problem.

The book introduces the oscillatory reaction and pattern formation in the Belousov-Zhabotinsky (BZ) reaction that became model for investigating a wide range of intriguing pattern formations in chemical systems. So many modifications in classic version of BZ reaction have been carried out in various experimental conditions that demonstrate rich varieties of temporal oscillations and spatio-temporal patterns in non- equilibrium conditions. Mixed-mode versions of BZ reactions, which comprise a pair of organic substrates or dual metal catalysts, have displayed very complex oscillating behaviours and novel space-time patterns during reaction processes. These characteristic spatio-temporal properties of BZ reactions have attracted increasing attention of the scientific community in recent years because of its comparable periodic structures in electrochemical systems, polymerization processes, and non-equilibrium crystallization phenomena. Instead, non-equilibrium crystallization phenomena which lead to development of novel crystal morphologies in constraint of thermodynamic equilibrium conditions have been investigated and are said to be stationary periodic structures. Efforts have continued to analyze insight mechanisms and roles of reaction-diffusion mechanism and self-organization in the growth of such periodic crystal patterns. In this book, non-equilibrium crystallization phenomena, leading to growth of some novel crystal patterns in dual organic substrate modes of oscillatory BZ reactions have been discussed. Efforts have been made to find out experimental parameters where transitions of the spherulitic crystal patterns take place. The book provides the scientific community and entrepreneurs with a thorough understanding and knowledge of the growth and form of branched crystal pattern in reaction-diffusion system and their morphological transition.

This monograph presents teaching material in the field of differential equations while addressing applications and topics in electrical and biomedical engineering primarily. The book contains problems with varying levels of difficulty, including Matlab simulations. The target audience comprises advanced undergraduate and graduate students as well as lecturers, but the book may also be beneficial for practicing engineers alike.