Acknowledgments.- I Equilibrium Solutions of Evolution Problems.- I.I One-Dimensional, Two-Dimensional, n-Dimensional, and Infinite-Dimensional Interpretations of (I.1).- I.2 Forced Solutions; Steady Forcing and T-Periodic Forcing; Autonomous and Nonautonomous Problems.- I.3 Reduction to Local Form.- I.4 Equilibrium Solutions.- I.5 Equilibrium Solutions and Bifurcating Solutions.- I.6 Bifurcating Solutions and the Linear Theory of Stability.- I.7 Notation for the Functional Expansion of F(t, µ, U).- Notes.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- II.1 The Implicit Function Theorem.- II.2 Classification of Points on Solution Curves.- II.3 The Characteristic Quadratic. Double Points, Cusp Points, and Conjugate Points.- II.4 Double-Point Bifurcation and the Implicit Function Theorem.- II.5 Cusp-Point Bifurcation and Characteristic Quadratics.- II.6 Triple-Point Bifurcation.- II.7 Conditional Stability Theorem.- II.8 The Factorization Theorem in One Dimension.- II.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation.- II.10 Exchange of Stability at a Double Point.- II.11 Exchange of Stability at a Double Point for Problems Reduced to Local Form.- II.12 Exchange of Stability at a Cusp Point.- II.13 Exchange of Stability at a Triple Point.- II.14 Global Properties of Stability of Isolated Solutions.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- III.1 The Structure of Problems Which Break Double-Point Bifurcation.- III.2 The Implicit Function Theorem and the Saddle Surface Breaking Bifurcation.- III.3 Examples of Isolated Solutions Which Break Bifurcation.- III.4 Iterative Procedures for Finding Solutions.- III.5 Stability of Solutions Which Break Bifurcation.- III.6 Isolas.- Exercise.- Notes.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions.- IV.I. Eigenvalues and Eigenvectors of a n × n Matrix.- IV.2 Algebraic and Geometric Multiplicity-The Riesz Index.- IV.3 The Adjoint Eigenvalue Problem.- IV.4 Eigenvalues and Eigenvectors of a 2 × 2 Matrix.- 4.1 Eigenvalues.- 4.2 Eigenvectors.- 4.3 Algebraically Simple Eigenvalues.- 4.4 Algebraically Double Eigenvalues.- 4.4.1 Riesz Index 1.- Riesz Index 2.- IV.5 The Spectral Problem and Stability of the Solution u = 0 in ?n.- IV.6 Nodes, Saddles, and Foci.- IV.7 Criticality and Strict Loss of Stability.- Appendix IV. 1 Biorthogonality for Generalized Eigenvectors.- Appendix IV.2 Projections.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- V.l The Form of Steady Bifurcating Solutions and Their Stability.- V.2 Classification of the Three Types of Bifurcation of Steady Solutions.- V.3 Bifurcation at a Simple Eigenvalue.- V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue.- V.5 Bifurcation at a Double Eigenvalue of Index Two.- V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue of Index Two.- V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a Double Eigenvalue of Index One (Semi-Simple).- V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple Double Eigenvalue.- V.9 Examples of Stability Analysis at a Double Semi-Simple (Index-One) Eigenvalue.- Appendix V.l Implicit Function Theorem for a System of Two Equations in Two Unknown Functions of One Variable.- Exercises.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VI.1 The Evolution Equation and the Spectral Problem.- VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude.- VI.3 ?1 and ?1 in Projection.- VI.4 Stability of the Bifurcating Solution.- VI.5 The Extra Little Part for ?1 in Projection.- VI.6 Projections of Higher-Dimensional Problems.- VI.7 The Spectral Problem for the Stability of u = 0.- VI.8 The Spectral Problem and the Laplace Transform.- VI.9 Projections into ?1.- VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory).- VI.11 The Method of Projection at a Double Eigenvalue of Index Two.- VI.12 The Method of Projection at a Double Semi-Simple Eigenvalue.- Appendix VI.1 Examples of the Method of Projection.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VII.1 The Structure of the Two-Dimensional Problem Governing Hopf Bifurcation.- VII.2 Amplitude Equation for Hopf Bifurcation.- VII.3 Series Solution.- VII.4 Equations Governing the Taylor Coefficients.- VII.5 Solvability Conditions (the Fredholm Alternative).- VII.6 Floquet Theory.- 6.1 Floquet Theory in ?1.- 6.2 Floquet Theory in ?2 and ?n.- VII.7 Equations Governing the Stability of the Periodic Solutions.- VII.8 The Factorization Theorem.- VII.9 Interpretation of the Stability Result.- Example.- VIII Bifurcation of Periodic Solutions in the General Case.- VIII.1 Eigenprojections of the Spectral Problem.- VIII.2 Equations Governing the Projection and the Complementary Projection.- VIII.3 The Series Solution Using the Fredholm Alternative.- VIII.4 Stability of the Hopf Bifurcation in the General Case.- Examples.- Notes.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- Notation.- IX.1 Definition of the Problem of Subharmonic Bifurcation.- IX.2 Spectral Problems and the Eigenvalues ?(µ).- IX.3 Biorthogonality.- IX.4 Criticality.- IX.5 The Fredholm Alternative for J(µ) -?(µ) and a Formula Expressing the Strict Crossing (IX.20).- IX.6 Spectral Assumptions.- IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality.- IX.8 The Operator J and its Eigenvectors.- IX.9 The Adjoint Operator J*, Biorthogonality, Strict Crossing, and the Fredholm Alternative for J.- IX.10 The Amplitude ? and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions.- IX.11 The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to? at ? = 0.- IX.12 Bifurcation and Stability of T-Periodic and 2T-Periodic Solutions.- IX.13 Bifurcation and Stability of nT-Periodic Solutions with n > 2.- IX.14 Bifurcation and Stability of 3T-Periodic Solutions.- IX.15 Bifurcation of 4T-Periodic Solutions.- IX.16 Stability of 4T-Periodic Solutions.- IX.17 Nonexistence of Higher-Order Subharmonic Solutions and Weak Resonance.- IX.18 Summary of Results about Subharmonic Bifurcation.- IX.19 Imperfection Theory with a Periodic Imperfection.- Exercises.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- X.1 The Biorthogonal Decomposition of the Solution and the Biorthogonal Decomposition of the Equations.- Exercise.- X.2 Change of Variables.- X.3 Normal Form of the Equations.- X.4 The Normal Equations in Polar Coordinates.- X.5 The Torus and Trajectories on the Torus in the Irrational Case.- X.6 The Torus and Trajectories on the Torus When ?0T/2? is a Rational Point of Higher Order(n ? 5).- X.7 The Form of the Torus in the Case n = 5.- X.8 Trajectories on the Torus When n = 5.- X.9 The Form of the Torus When n > 5.- X.10 Trajectories on the Torus When n ? 5.- X.11 Asymptotically Quasi-Periodic Solutions.- X.12 Stability of the Bifurcated Torus.- X.13 Subharmonic Solutions on the Torus.- X.14 Stability of Subharmonic Solutions on the Torus.- X.15 Frequency Locking.- Appendix X.l Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Rational Points of Higher Order (n ? 5) by the Method of Power Series Using the Fredholm Alternative.- Appendix X.2 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Irrational Points Using the Method of Two Times, Power Series, and the Fredholm Alternative.- Appendix X.3 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times.- Notes.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions of (Hopf's Type) in the Autonomous Case.- Notation.- XI.1 Spectral Problems.- XI.2 Criticality and Rational Points.- XI.3 Spectral Assumptions about J0.- XI.4 Spectral Assumptions about J in the Rational Case.- XI.5 Strict Loss of Stability at a Simple Eigenvalue of J0.- XI.6 Strict Loss of Stability at a Double Semi-Simple Eigenvalue of J0.- XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two.- XI.8 Formulation of the Problem of Subharmonic Bifurcation of Periodic Solutions of Autonomous Problems.- XI.9 The Amplitude of the Bifurcating Solution.- XI.10 Power-Series Solutions of the Bifurcation Problem.- XI.11 Subharmonic Bifurcation Whenn= 2.- XI.12 Subharmonic Bifurcation When n > 2.- XI.13 Subharmonic Bifurcation When n = 1 in the Semi-Simple Case 265 XI.14 "Subharmonic" Bifurcation When n = 1 in the Case When Zero is an Index-Two Double Eigenvalue of J0.- XI.15 Stability of Subharmonic Solutions.- XI.16 Summary of Results about Subharmonic Bifurcation in the Autonomous Case.- XI.17 Bifurcation of a Torus in Autonomous Nonresonant Cases.- XI.18 Asymptotically Quasi-Periodic Solutions on the Bifurcated Torus.- XI.19 Strictly Quasi-Periodic Solutions on the Bifurcated Torus.- Exercises.

In its most general form bifurcation theory is a theory of equilibrium solutions of nonlinear equations. By equilibrium solutions we mean, for example, steady solutions, time-periodic solutions, and quasi-periodic solutions. The purpose of this book is to teach the theory of bifurcation of equilibrium solutions of evolution problems governed by nonlinear differential equations. We have written this book for the broaqest audience of potentially interested learners: engineers, biologists, chemists, physicists, mathematicians, econom­ ists, and others whose work involves understanding equilibrium solutions of nonlinear differential equations. To accomplish our aims, we have thought it necessary to make the analysis 1. general enough to apply to the huge variety of applications which arise in science and technology, and 2. simple enough so that it can be understood by persons whose mathe­ matical training does not extend beyond the classical methods of analysis which were popular in the 19th Century. Of course, it is not possible to achieve generality and simplicity in a perfect union but, in fact, the general theory is simpler than the detailed theory required for particular applications. The general theory abstracts from the detailed problems only the essential features and provides the student with the skeleton on which detailed structures of the applications must rest. It is generally believed that the mathematical theory of bifurcation requires some functional analysis and some of the methods of topology and dynamics.