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A Course in .. . APPLIED BIFURCATION THEORY A Course in .. APPLIED BIFURCATION THEORY VEMURI BALAKOTAIAH University ofHouston COURSE DESCRIPTION Houston, TX 77204-4792 Introduction to Applied Bifurcation Theory The main goal of the course is to expose chemical ifurcation theory deals with the solution of engineering graduate students to some important non­ nonlinear equations and is useful to chemi­ linear techniques and concepts. Table 1 gives an out­ B cal engineers studying nonlinear phenom­ line of the material that is covered in a fourteen-week ena. Most of the traditional courses on applied math­ semester. Although the course is for 3 credits, 28 ematics offered by chemical engineering departments two-hour lectures are necessary to cover the topics cover only linear analysis. While linear analysis is listed in Table 1. necessary, since it is the foundation of all nonlinear The course is organized into six topics and two in­ techniques, it does not prepare students to deal with troductory lectures. The introductory lectures give a the nonlinear problems that will be encountered later brief history of bifurcation theory, examples from vari­ in research. This is especially true for students work­ ous disciplines, and the usefulness and limitations of ing on stability problems in fluid flow, heat and bifurcation theory. Several chemical engineering ex­ mass transfer, catalysis, reaction engineering, con­ amples covering fluid flow, heat and mass transfer, trol, and separations. catalysis and reaction engineering, separations, con­ For many years we have sent our University of trol and multi-phase transport are selected as model Houston students to the mathematics department problems and are used throughout the course to for courses on differential equations and dynamical illustrate various concepts. All the examples are systems, bifurcation theory, nonlinear dynamics, sin­ deterministic models and vary from the following gularity theory, and group theory. We found, how­ simple (but non-trivial) two ordinary differential ever, that many of these courses were too special­ equation models ized, were abstract, and had a narrow focus (from an engineer's point of view). Typically, a student had to t =-Da +(1-x)exp{l+:/y} (la) take three or four of these courses to grasp a few Leda =-_JL+B(l-x)exp{-9-}-a(0-0c) (lb) useful nonlinear techniques. dt Da 1+0/y To address these problems, in 1989 the author describing the dynamic behavior of a CSTR in which a designed a new course on applied bifurcation theory first order exothermic reaction occurs to the follow­ as a sequel to the two-semester applied mathemat­ ing, somewhat complicated, model involving a set of ics course taught by Professor Neal R. Amundson. six partial differential equations in three spatial coor­ The course was well-received by the students and dinates and time was repeated in the Spring of 1991, and with some minor changes and updating is scheduled to be taught V-v=0· VIl=-v- Ra ye (2a) ' Peh z in the Spring of 1994 and regularly thereafter. ay +v-Vy=-1-V2y+~Daex.f__JJ_)c (2b) a-c P% P\,_l+y Vemurl Balakotaiah is professor of chemical engineering at the University of Houston. He 2 received his BTech degree from the Indian Insti­ cr~+v-Vc=p;m V c-Daexp(1':i\} (0 <z,r < l,0 < 0 < 21t) (2c) tute of Technology (Madras) in 1978 and his PhD from the University of Houston in 1982, boundary conditions: both in chemical engineering. He worked as a Research Engineer at Shell Development Com­ at the wall pany and is a consultant to Exxon, Shell, and (2d) the Western Company of North America. His at the inlet (z=O) main research interests are in the area of chemi­ y=0; C= 1; Vz =1 (2e) cal reaction engineering. at the exit (z=l) ay =0· ac =0· Il=Il1 (2f) © Copyright ChE Division ofASEE 1993 az ' az ' 154 Chemical Engineering Education describing flow maldistributions and hot algebraic systems). For example, for Eqs. (1), spots in a down-flow cylindrical packed bed (x(t)J t reactor. After a brief discussion of model C= 1 0), U= , Y= 2, p =(Le,y,B, Da,a,ec) {4a) (0 Le 0{t) formulation and the origins of various nonlinearities, we discuss the advantages of and using the function space formalism. It is 8 ---1L+(l-x)exp{- } shown that most of the models can be writ­ F(u )= Da 1+0/ y l (4b) ten in the abbreviated form ,P [ _.JL.+B{l-x)exp{- 8- }-a(e-e ) Da 1+0/ y c c~~ =F(u, p) (3) while for Eqs. (2) where p is a vector of parameters, C is a 0 0 0 0 v(z,r,0,t)} capacitance matrix, and the vector of state 0 0 0 0 TI(z,r,0,t) t variables u may be expressed in terms of the C= u= p ={Ra,Peh, Pem ,Y,~, Da, cr,Il1,a) (5a) elements of a function space having certain 0 0 1 0 c(z,r,e, t) properties, e.g., satisfying differentiability 0 0 0 cr y(z,r,e, t) conditions and the appropriate boundary con­ ditions. The function spaces of interest are V -v usually Banach or Hilbert spaces. The ca­ -VIT-v- Ra ye pacitance matrix C, the parameters vector Peh z F(u,p)= (5b) p, and the nonlinear operator F on the func­ - v -Vy+--V1 2 y+~Daex{yy) -- c Peh l+y tion space Y are identified for some selected - v -Vc+--V1 2 c-Daex{yy --)c examples. Some of these include cases in Pem l+y which C is not invertible (differential- TABLE 1 Course Outline for Applied Bifurcation Theory Introduction 1. Definition and examples from different disciplines 4. Floquet theory; degenerate Hopf bifurcations 2. Behavior of nonlinear systems; uses and limitations of 5. Bifurcation theory for maps; normal forms of codimension bifurcation theory one bifurcations ;attractors and basins of attraction Nonlinear Functional Analysis 6. Poincare maps; averaging method; Melnikov theory I. Operators on Banach spaces; Frechet derivatives 7. Characterization of attractors; attractor dimensions, K­ 2. Contraction mapping theorem; iterative methods for entropy, L-exponents; analysis of experimental data nonlinear operator equations; uniqueness criteria 8. Poincare-Bendixson theory; degree and index theory; 3. Implicit function theorem; necessary and sufficient condi­ group theory and normal forms; Hamiltonian chaos; tions for bifurcation; determination of stationary stability fractals boundary (Bifurcation Set) Nonlinear Partial Differential Equations Steady-State Bifurcation Theory I. Linear stability analysis of coupled PDEs 1. Liapunov-Schmidt reduction; elementary catastrophe theory 2. Center-manifold reduction of coupled PDEs; amplitude 2. Singularity theory with a distinguished variable; classifica­ equations tion of bifurcation diagrams; construction of phase diagrams 3. Mode interactions; bifurcation with symmetry , ) 4. Bifurcation in large systems (continuous spectrum); 3. Effects of discrete symmetry (Z2 D3 4. Shooting technique with sensitivity functions; determination Landau and Ginzberg-Landau equations; phase and of singular points of two-point boundary value problems; amplitude turbulence singular points of elliptic PDEs 5. Energy stability and Liapunov functions 5. Effects of symmetry on boundary value problems 6. Bifurcation theory for delay-differential, integral, and , (Z2 0(2)) integro-differential equations Branching equations with symmetries Nonlinear Wave Phenomena Dynamical Systems I. Review of basic concepts; physical examples 1. Invariant manifolds; Hartman-Grohman theorem; stable and 2. Analysis of traveling waves and pulses center manifold theorems, applications Computational Methods in Bifurcation Theory 2. Amplitude equations; codimension 1, 2, 3 singularities 1. Arc length continuation technique; continuation of steady­ 3. Poincare-Birkhoff normal form; local codimention 1, 2 state and periodic branhces bifurcations 2. Review of software on nonlinear dynamics and chaos Fall 1993 155 and Y is the space of 6-tuples of continuous func­ ing details, equations, and commentary is available tions in the variables (z,r,0) satisfying the appropri­ from the author. ate differentiability conditions and homogeneous Nonlinear Functional Analysis• The course starts with boundary conditions. nonlinear functional analysis which introduces the notation Next, an overview of bifurcation theory, and its and forms the basis for all later topics. First, the concept of potential uses and limitations, are reviewed by a completeness and convergence in normed linear (Banach) discussion of the following frequently asked ques­ spaces is reviewed in a non-abstract manner. This is fol­ tions about nonlinear models: 1) Given a nonlinear lowed by the definition of a Frechet derivative (or local model of a physical system, what are the different linearization) of a nonlinear operator, chain rule, partial and types of behaviors that are possible (for different higher-order Frechet derivatives, and Taylor's theorem in choices of the parameters vector p )? 2) What are the function spaces. The model problems are used for illustra­ regions in the parameter space in which the behav­ tion with formulas such as ior of a model may be described by a lower dimen­ sional simplified model containing fewer parameters (6a) and/or a lower dimensional state space? What is the simplified form of the model? 3) How does one con­ D!uF(u0 ,p)•(v,w)=-aa _aa [F(u 0 +s1v+s2w,p)~ (6b) 5 1 5 2 1s 1 =s2=0 struct phase diagrams in the parameter space which classify the p space into regions, in each of which a for determining the Frechet derivatives of the nonlinear op­ different type of behavior exists? 4) How do the pre­ erators (such as those in Eqs. 4b and 5b). Next, the concept dicted features of a model change when it is sub­ of a nonlinear operator being a contraction is introduced and jected to small perturbations (or equivalently, is the the contraction mapping theorem is stated.
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