Phase Transition Dynamics

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1 Phase Transition Dynamics

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3 Tian Ma Shouhong Wang Phase Transition Dynamics 123

4 Tian Ma Department of Mathematics Sichuan University Chengdu, People s Republic of China Shouhong Wang Department of Mathematics Indiana University Bloomington, IN, USA ISBN ISBN (ebook) DOI / Springer New York Heidelberg Dordrecht London Library of Congress Control Number: Mathematics Subject Classification: 35B, 35Q, 37L, 76E, 82B, 82C, 86A10, 92B Springer Science+Business Media, LLC 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 For Wenyuan Chen, Ciprian Foias, Louis Nirenberg, Roger Temam For Li and Ping

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7 Preface Most problems in the natural sciences are described by either dissipative or conservative systems. Phase transition dynamics for both types of systems is of central importance in the nonlinear sciences. In this book, the term phase transition is to be understood in a broad sense, including both classical phase transitions and general transitions as found in nature. This book is an introduction to a comprehensive and unified dynamic transition theory for dissipative systems and to applications of the theory to a wide range of problems in the nonlinear sciences. The main objectives of this book are to derive a general principle of dynamic transitions for dissipative systems, to establish a systematic dynamic transition theory, and to explore the physical implications of applications of the theory to a wide range of problems in the nonlinear sciences. As a general principle, dynamic transitions of all dissipative systems are classified into three categories: continuous, catastrophic, and random. In comparison with the classical classification scheme for equilibrium phase transitions, whereby phase transitions are identified by the lowest-order derivative of the free energy that is discontinuous at the transition, the dynamic classification scheme is suitable for both equilibrium and nonequilibrium phase transitions. Once the type of a dynamic transition is determined for a given equilibrium system, the order of a transition in the classical sense immediately becomes transparent. The philosophical basis of dynamic transition theory is to search for a system s full set of transition states, giving a complete characterization of stability and transition. The set of transition states is represented by a local attractor near or away from the basic state. Following this approach, dynamic transition theory is developed to identify the transition states and to classify them both dynamically and physically. The theory is strongly motivated by phase transition problems in the nonlinear sciences. Namely, the mathematical theory is developed with close links to physics, and in return, the theory is applied to physical problems, leading to physical predictions and new insights into both theoretical and experimental studies for the underlying physical problems. vii

8 viii Preface The study of equilibrium phase transitions presented in this book involves a combination of modeling, mathematical analysis, and physical predictions. We adopt the general idea of the Ginzburg Landau phenomenological approach, and introduce a unified time-dependent Ginzburg Landau model. Mathematically, this model is a quasigradient flow. Physically, the model is proposed using the Le Châtelier principle. A few typical and important equilibrium phase transition problems are addressed in this book, including the physical vapor transport (PVT) system, ferromagnetism, binary systems (Cahn Hilliard equation), and the superconductivity and superfluidity of liquid helium (helium-3, helium-4, and their mixture). The main focus for nonequilibrium transitions is on typical problems in classical and geophysical fluid dynamics, in climate dynamics, in chemical reactions, and in biology, including in particular Rayleigh Bénard convection, the Taylor problem, Taylor Couette Poiseuille (TCP) flow, and the rotating convection problem in classical fluid dynamics; the El Niño Southern Oscillation (ENSO) and wind-driven and thermohaline circulation in geophysical fluid dynamics and climate dynamics; the Belousov Zhabotinsky chemical reactions; and chemotaxis and population models in biology. Our intent has been to address the interests and backgrounds not only of applied mathematicians but also of sophisticated students and researchers working on nonlinear problems including those in physics, meteorology, oceanography, biology, chemistry, and the social sciences. The audience for this book includes second-year or more-advanced graduate students and researchers in mathematics and physics as well as in other related fields. Chapter 1 is designed so that most readers who have completed it will be able to jump directly to the related applications in Chaps. 3 6, after going quickly through the main ideas in this chapter as well as in the general introduction. The mathematical background for physicists and other researchers in other fields is a good graduate-level mathematical physics course. For example, any student who is capable of studying graduate-level quantum mechanics or statistical physics should be able to follow Chap. 1 and the applications in Chaps Most graduate students in applied mathematics nowadays have to learn some basic physics related to the applied problems they are working on, and they should have no additional difficulty in reading the related topics of their interest in this book. Sichuan, China Bloomington, IN Tian Ma Shouhong Wang Acknowledgments We have greatly benefited from illuminating discussions with Jerry Bona, Mickael Chekroun, Henk Dijkstra, Ciprian Foias, Susan Friedlander, Michael Ghil, Chun- Hsiung Hsia, Paul Newton, Hans Kaper, Louis Nirenberg, Benoit Perthame, Jie Shen, Roger Temam, Xiaoming Wang, and Kevin Zumbrun. We owe a special debt

9 Preface ix of gratitude to all of them. We are grateful to Honghu Liu and Taylan Sengul for their critical reading, several times, of an earlier draft of the book and for their many constructive comments. We extend our wholehearted thanks to the anonymous referees; their careful reading and thoughtful remarks have led to a more effective book. Also, we are grateful to Wen Masters and Reza Malek-Madani, of the Office of Naval Research, for their constant support and encouragement. We express our sincerest thanks to Achi Dosanjh, of Springer-Verlag, for her great effort and support on this book project. The research presented in this book was supported in part by grants from the ONR, the NSF, and the Chinese NSF. The authors owe a debt beyond our ability to express it to Li and Ping for their kindness, love, patience, and dedications to our families. This book could never have been written without their devoted support from both Li and Ping.

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11 Contents Introduction...xvii 1 Introduction to Dynamic Transitions First Principles and Dynamic Models Physical Laws and Mathematical Models Rayleigh Bénard Convection Mathematical Formulation of Physical Problems Introduction to Dynamic Transition Theory Motivation and Key Philosophy Principle of Exchange of Stability Equation of Critical Parameters Classifications of Dynamic Transitions Structure and Characterization of Dynamic Transitions General Features of Dynamic Transitions Examples of Typical Phase Transition Problems Rayleigh Bénard Convection El Niño Southern Oscillation Dynamic Transition Versus Transition in Physical Space Andrews Critical Point and Third-Order Gas Liquid Transition Binary Systems Dynamic Transition Theory General Dynamic Transition Theory Classification of Dynamic Transitions Characterization of Transition Types Local Topological Structure of Transitions Continuous Transition Finite-Dimensional Systems S 1 -Attractor Bifurcation S m -Attractor Bifurcation xi

12 xii Contents Structural Stability of Dynamic Transitions Infinite-Dimensional Systems Transition from Simple Eigenvalues Real Simple Eigenvalues Transitions from Complex Simple Eigenvalues Computation of b Transition from Eigenvalues with Multiplicity Two Index Formula for Second-Order Nondegenerate Singularities Bifurcation at Second-Order Singular Points The Case ind (F,0)= The Case ind (F,0)= The Case ind (F,0)= Indices of kth-order Nondegenerate Singularities Structure of kth-order Nondegenerate Singularities Transition from kth-ordernondegeneratesingularities Bifurcation to Periodic Orbits Application to Parabolic Systems Singular Separation General Principle Saddle-Node Bifurcation Singular Separation of Periodic Orbits Perturbed Systems General Eigenvalues Simple Eigenvalues Complex Eigenvalues Notes Equilibrium Phase Transition in Statistical Physics Dynamic Models for Equilibrium Phase Transitions Thermodynamic Potentials Time-Dependent Equations Classification of Equilibrium Phase Transitions Third-Order Gas Liquid Phase Transition Introduction Time-Dependent Models for PVT Systems Phase Transition Dynamics for PVT Systems Physical Conclusions Ferromagnetism Classical Theory of Ferromagnetism Dynamic Transitions in Ferromagnetism Physical Implications Asymmetry of Fluctuations Phase Separation in Binary Systems Modeling...154

13 Contents xiii Phase Transition in General Domains Phase Transition in Rectangular Domains Spatial Geometry, Transitions, and Pattern Formation Phase Diagrams and Physical Conclusions Superconductivity Ginzburg Landau Model TGDL as a Gradient-Type System Phase Transition Theorems Model Coupled with Entropy Physical Conclusions Liquid Helium Dynamic Model for Liquid Helium Dynamic Phase Transition for Liquid 4 He Superfluidity of Helium Dynamic Model for Liquid 3 He with Zero Applied Field Critical Parameter Curves and PT -Phase Diagram Classification of Superfluid Transitions Liquid 3 He with Nonzero Applied Field Physical Remarks Mixture of He-3 and He Model for Liquid Mixture of 3 He and 4 He Critical Parameter Curves Transition Theorems Physical Conclusions Fluid Dynamics Rayleigh Bénard Convection Bénard Problem Boussinesq Equations Dynamic Transition Theorems Topological Structure and Pattern Formation Asymptotic Structure of Solutions for the Bénard Problem Structure of Bifurcated Attractors Physical Remarks Taylor Couette Flow Taylor Problem Governing Equations Narrow-Gap Case with Axisymmetric Perturbations Asymptotic Structure of Solutions and Taylor Vortices Taylor Problem with z-periodic Boundary Condition Other Boundary Conditions Three-Dimensional Perturbation for the Narrow-Gap Case Physical Remarks...315

14 xiv Contents 4.3 Boundary-Layer and Interior Separations in the Taylor Couette Poiseuille Flow Model for the Taylor Couette Poiseuille Problem Phase Transition of the TCP Problem Boundary-Layer Separation from the Couette Poiseuille Flow Interior Separation from the Couette Poiseuille Flow Nature of Boundary-Layer and Interior Separations Rotating Convection Problem Rotating Boussinesq Equations Eigenvalue Problem Principle of Exchange of Stabilities Transition from First Real Eigenvalues Transition from First Complex Eigenvalues Physical Remarks Convection Scale Theory Geophysical Fluid Dynamics and Climate Dynamics Modeling and General Characteristics of Geophysical Flows El Niño Southern Oscillation Walker Circulation and ENSO Equatorial Circulation Equations Walker Circulation Under Idealized Conditions Walker Circulation Under Natural Conditions ENSO: Metastable Oscillation Theory Thermohaline Ocean Circulation Boussinesq Equations Linear Analysis Nonlinear Dynamic Transitions Convection Scales and Dynamic Transition Arctic Ocean Circulations Model Linear Theory Transition Theorems Revised Transition Theory Physical Conclusions Large-Scale Meridional Atmospheric Circulation Polar, Ferrel, and Hadley Cells β -Plane Assumption Meridional Circulation Under Idealized Conditions Physical Implications...445

15 Contents xv 6 Dynamical Transitions in Chemistry and Biology Modeling Dynamical Equations of Chemical Reactions Population Models of Biological Species Belousov Zhabotinsky Chemical Reactions: Oregonator The Field Kőrös Noyes Equations Transition Under the Dirichlet Boundary Condition Transitions Under the Neumann Boundary Condition Phase Transition in the Realistic Oregonator Belousov Zhabotinsky Reactions: Brusselator Prigogine Lefever Model Linearized Problem Transition from Real Eigenvalues Transition from Complex Eigenvalues Bacterial Chemotaxis Keller Segel Models Dynamic Transitions for a Rich Stimulant System Transition of Three-Component Systems Biological Conclusions Biological Species Modeling Predator Prey Systems Three-Species Systems Appendix A A.1 Formulas for Center Manifold Functions A.2 Dynamics of Gradient-Type Systems References Index...553

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17 Introduction Transitions are to be found throughout the natural world. The laws of nature are usually represented by differential equations, which can be regarded as dynamical systems both finite- and infinite-dimensional. There are two types of dynamical systems: dissipative and conservative. The term dissipative structure was coined by Ilya Prigogine. A conservative system is often described by a Hamiltonian structure, and a dissipative structure is closely associated with the concept of absorbing sets and global attractors; see among many others (Lorenz 1963a; Prigogine and Lefever 1968; Foiaş and Temam 1979; Ladyzhenskaya 1982; Temam 1997). In thermodynamics, a phase transition is understood as the transformation of a thermodynamic system from one phase or state of matter to another. However, many problems in nature involve the study of transitions of one state to new states and the stability/robustness of the new states, and so in this book, the term phase transition will be understood in a broad sense, to include both the classical phase transitions in physics and the general transitions found in nature. The study of dynamic transitions also involves the dynamic behavior of transitions and transition states. This main objectives of this book are threefold. The first is to derive, in Theorem 2.1.3, the following general principle on dynamic transitions for dissipative systems: Principle 1 Dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic, and random. The second objective is to develop a systematic dynamic transition theory to study the types and structure of dynamic transitions for dissipative systems. The third is to apply dynamic transition theory to explore the physical implications of dynamic transitions in a wide range of scientific problems. The applications include most of the typical types of dissipative systems, including (1) gradient-type systems for equilibrium phase transitions in statistical physics, (2) Navier Stokes-type equations for incompressible fluid flows, (3) Boussinesq-type equations for geophysical fluid dynamics and climate dynamics, and (4) systems of reaction diffusion equations modeling chemical reactions, chemotaxis, and population dynamics in biology. xvii

18 xviii Introduction The set of states of a dissipative system can be a complicated object in the phase space, and may include steady states, periodic states, and transients. Hence the starting point of our study is the following key philosophy: Philosophy 2 The key philosophy of dynamic transition theory is to search for the full set of transition states, giving a complete characterization of stability and transition. The set of transition states is a local attractor, representing the physical reality after the transition. It is natural to understand this philosophy from the physical point of view. For example, according to John von Neumann (1960), the motion of the atmosphere can be divided into three categories depending on the time scale of a prediction about its behavior, namely the motions corresponding to the short-, medium-, and long-term behavior of the atmosphere. Climate, which corresponds to the long-term behavior, should be represented by an attractor containing not only climate equilibria, but also the transients. The general principle for dynamic transitions for dissipative systems given by Principle 1, along with Philosophy 2, provides a global viewpoint for the study of dynamic transitions. To illustrate the basic motivation and ideas behind this point of view, we consider the following simple example. Example 3 For x =(x 1,x 2 ) t R 2, the system ẋ 1 = λ x 1 x 3 1, ẋ 2 = λ x 2 x 3 2, (1) undergoes a dynamic transition from a basic stable state x = 0 to a local attractor Σ λ = S 1,asλ crosses λ 0 = 0. This local transition attractor is as shown in Fig. 1. f a b Fig. 1 The states after the transition are given by a bifurcated attractor containing four nodes (the points a, b, c, and d), four saddles (the points e, f, g, h), and orbits connecting these eight points e d h c g It contains exactly four nodes (the points a, b, c, and d), four saddles (the points e, f, g, h), and orbits connecting these eight points. The connecting orbits represent transient states. From the physical transition point of view, as λ crosses 0, the new transition states are represented by the whole local attractor Σ λ, rather than by any of the steady states or any of the connecting orbits.

19 Introduction xix To demonstrate this point of view further, we consider another example: Example 4 For x =(x 1,x 2 ) t R 2, consider the system ẋ 1 = λ x 1 + x x 1x 2 10x 3 1, ẋ 2 = λ x 2 2x 1 x 2 x x 3 2. (2) In this example, the lowest-order nonlinear terms are quadratic, given by F(x) = ( x x 1 x 2, 2x 1 x 2 x 2 ) t 2. As we shall prove in Chap. 2, the index of the vector field F( ) at x = 0is 2, leading to a dynamic transition as shown in Fig. 2. For λ > 0 and near 0, there are seven steady-state solutions: the origin O =(0,0) t and six nonzero points A, B, C, d, e,andf such that the three fixed points A, B,andC are close to the origin O and converge to O as λ 0, and the other three fixed points are away from 0 and do not converge to the origin O as λ 0. The three line segments OA, OB, andoc divide a neighborhood of O into three regions, and for any initial value in one of the regions, the solution converges to one of the three fixed points d, e, and f, away from O. Hence as λ crosses 0, the transition is catastrophic, also called in this book a jump (Type II) transition. The three points A, B,andC near the origin are the three bifurcation points, which do not represent the transition state in any way. We note that the six nonzero fixed points are connected by heteroclinic orbits, and the region enclosed by these orbits is the global attractor of the system. a b c A B C Fig. 2 Transition structure from a singular point with index ind(f,0)= 2: (a) λ < 0; (b) λ = 0; (c) λ > 0 The above two simple examples clearly demonstrate the necessity of the key philosophy, which will become more transparent in the wide range of applications of the theory in Chaps Dissipative systems are governed by differential equations both ordinary and partial which can be written in the following unified abstract form: du dt = L λ u + G(u,λ ), u(0)=u 0, (3)

20 xx Introduction where u : [0, ) X is the unknown function, λ R 1 is the system control parameter, and X is a Banach space. We shall always consider u the deviation of the unknown function from some equilibrium state u. Hence, L λ : X X is a linear operator, and G : X R 1 X is a nonlinear operator. For example, for the classical incompressible Navier Stokes equations, u represents the velocity function, and for the time-dependent Ginzburg Landau equations of superconductivity, we have u =(ψ,a),whereψ is a complexvalued wave function, and A is the magnetic potential. To address the dynamic transition of a given dissipative system, the first step is to study the linear eigenvalue problem for system (3), which is closely related to the principle of exchange of stability (PES), leading to precise information on linear unstable modes. The precise statement of the PES is as follows. Let {β j (λ ) C j N} be the eigenvalues (counting multiplicity) of L λ, and assume that < 0 ifλ < λ 0, Re β i (λ ) = 0 ifλ = λ 0, 1 i m, (4) > 0 ifλ > λ 0, Re β j (λ 0 ) < 0 j m + 1. (5) Much of the linear theory on stability and transitions involves establishing the PES. There is a vast literature devoted to linear theory, including, among many others, (Chandrasekhar 1981; Drazin and Reid 1981) for classical fluid dynamics and (Pedlosky 1987) for geophysical fluid dynamics. With the linear theory and the PES at our disposal, we can immediately show that the system always undergoes a dynamical transition at the critical threshold and that Principle 1 holds; see Theorem for details. The details of the transition behavior are then dictated by the nonlinear interactions of the system. The dynamic transition theory presented in Chap. 2 provides a systematic approach to the study of nonlinear interactions, leading to detailed information on the types of transitions and the structure of transition states and their physical implications. The general principle of dynamic transitions for dissipative systems, Principle 1, classifies all dynamic transitions of a dissipative system into three categories, continuous, catastrophic, and random, which are also called Type I, Type II, and Type III in this book. To say that a transition is continuous amounts to saying that the control parameter crosses the critical threshold and the transition states stay in a close neighborhood of the basic state. The transition at λ = 0inExample3 is a continuous transition. In fact, continuous transitions are essentially characterized by the attractor bifurcation theorem, Theorem 2.2.2, first proved in Ma and Wang (2005c,b). The attractor bifurcation theorem states that when the PES holds and the basic state is asymptotically stable at the critical parameter value λ 0, the system undergoes a continuous dynamic transition, which is described by a bifurcated attractor.

21 Introduction xxi A key assumption in the attractor bifurcation theorem is the asymptotic stability of the basic solution at the critical parameter value λ 0. One important idea of the proof is a refined upper semicontinuity of attractors, which also plays an important role in analyzing the other two types of transitions. There are many physical systems that can undergo a continuous transition. For example, consider the classical Bénard convection. As the Rayleigh number crosses the critical Rayleigh number, the system undergoes a continuous transition to an attractor, homeomorphic to an (m 1)-dimensional sphere S m 1, that consists of steady states and transients. Here m is the number of unstable modes of the linearized eigenvalue problem at the critical Rayleigh number, dictated by the spatial geometry, which also defines the pattern formation mechanism of the problem. As we shall see in Sect. 4.1, steady states occupy only a zero-measure subset of the sphere S m 1, and the transients occupy most the sphere. When there is no longer asymptotic stability of the basic state at the critical parameter, the system undergoes either catastrophic or random transitions, as dictated by the nonlinear interactions. The dynamic transition theory presented in this book gives a systematic approach to distinguishing these transitions. Intuitively speaking, catastrophic transition corresponds to the case in which the system undergoes a more drastic change as the control parameter crosses the critical threshold. The transition given in Example 4 and the transitions in many examples given in Chaps. 3 6 are typical catastrophic transitions. A random transition corresponds to the case in which a neighborhood (fluctuations) of the basic state can be divided into two regions such that fluctuations in one of them lead to continuous transitions, and those in the other lead to catastrophic transitions. Given the above observations, it is then crucial to determine the asymptotic behavior of a system at the critical threshold. For this purpose and for determining the structure of the local attractor representing the transition states, the most natural approach is to project the underlying system to the space generated by the most unstable modes, preserving the dynamic transition properties. This is achieved with center manifold reduction. An important part of dynamic transition theory deals with the analysis of reduced systems with close links to physics and with the philosophy of searching for a complete set of transition states. First, the transition type is completely dictated by the flow structure of the reduced system at the critical threshold, leading to precise information on the phase transition diagram. Also, for a continuous (Type I) transition, the reduced system provides a complete description of the set of transition solutions. It is worth mentioning that the far-field solutions in the catastrophic and random cases will have to be derived from the original partial differential equations, although the transition types and phase diagrams are completely determined by the reduced system. In a nutshell, this book establishes a general principle for dynamic transitions for dissipative systems. Namely, all transitions of a dissipative system can be dynamically classified into one of three types: continuous, catastrophic, and random. A systematic dynamic transition theory is then developed to identify the types and

22 xxii Introduction structure of transitions, and is further applied to a wide range of problems in physics, fluid dynamics, climate dynamics, chemistry, and biology. It is believed that the methods and ideas presented in this book can be readily applied to many scientific problems in related fields as well, as evidenced by the wide range of applications presented in this book. The main outline of this book is as follows. The philosophy and general principles of dynamic transition theory with applications to a few typical problems are addressed in Chap. 1. Mathematical aspects of dynamic transition theory are addressed in Chap. 2. Chapter 3 consists of applications of the general theory to a few typical equilibrium phase transition problems, including the PVT system, ferromagnetism, binary systems, superconductivity, and superfluidity. As mentioned earlier, the study of these equilibrium phase transition problems involves a combination of modeling, mathematical analysis, and physical predictions. Chapter 4 focuses on transition problems in classical fluid dynamics. The transition of the classical Bénard convection problem is studied in Sect. 4.1, leading to a continuous (Type I) transition at the first critical Rayleigh number. For the Taylor problem, the TCP flow problem, and the rotating convection problem, the associated linear operators are not symmetric, leading to the existence of all three types of transitions as well as the existence of periodic solutions. This chapter gives a detailed account of these transitions and their physical implications. In the TCP flow case, for example, we show that the Taylor vortices do not appear immediately after the first transition, and that they appear only after the system has undergone a transition in its structure in the physical space as we increase the Taylor number to another critical number. This result is derived using a combination of the dynamic transition theory presented in this book and the geometric theory of incompressible flows developed recently in Ma and Wang (2005d). Chapter 5 addresses dynamic transitions in geophysical fluid dynamics and climate dynamics. One important aspect of the study is the introduction of turbulent frictional terms in the model, leading to a transition to circulations over the tropics, consistent with the Walker circulation. Another important component of this chapter is the introduction of a new mechanism of ENSO that is consistent with observations. Dynamic transitions corresponding to typical meridional circulations and the thermohaline circulations are examined as well. Chapter 6 can be considered an introduction to transition problems in chemistry and biology, focusing on Belousov Zhabotinsky chemical reaction equations, the chemotactic model, and the population model. Application-oriented readers may go directly to the study of the scientific problems presented in Chaps. 3 6 after reading through Chap. 1.