Hopf Bifurcations and Horseshoes Especially Applied to the Brusselator

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Hopf Bifurcations and Horseshoes Especially Applied to the Brusselator Steven R. Jones Brigham Young University - Provo Follow this and additional works at: Part of the Mathematics Commons BYU ScholarsArchive Citation Jones, Steven R., "Hopf Bifurcations and Horseshoes Especially Applied to the Brusselator" (5). All Theses and Dissertations This Selected Project is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 HOPF BIFURCATIONS AND HORSESHOES ESPECIALLY APPLIED TO THE BRUSSELATOR by Steven Robert Jones A project submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Mathematics Department of Mathematics Brigham Young University August 5

3

4 Copyright c 5 Steven Robert Jones All Rights Reserved

5

6 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a project submitted by Steven Robert Jones This project has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Dr. Kening Lu, Chair Date Dr. Sum Chow Date Dr. Blake Peterson

7

8 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the project of Steven Robert Jones in its final form and have found that () its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; () its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Dr. Kening Lu Chair, Graduate Committee Accepted for the Department Dr. Tyler Jarvis Graduate Coordinator Accepted for the College G. Rex Bryce, Associate Dean College of Physical and Mathematical Sciences

9

10 ABSTRACT HOPF BIFURCATIONS AND HORSESHOES ESPECIALLY APPLIED TO THE BRUSSELATOR Steven Robert Jones Department of Mathematics Master of Mathematics In this paper we explore bifurcations, in particular the Hopf bifurcation. We study this especially in connection with the Brusselator, which is a model of certain chemical reaction-diffusion systems. After a thorough exploration of what a bifurcation is and what classifications there are, we give graphic representations of an occurring Hopf bifurcation in the Brusselator. When an additional forcing term is added, behavior changes dramatically. This includes the introduction of a horseshoe in the time map as well as a strange attractor in the system.

11

12 ACKNOWLEDGMENTS I would like to thank Dr. Lu for allowing me to work with him on his research. I respect him as a mathematician and found it enlightening to assist him. I want to thank all the wonderful teachers I have had including Drs. Williams, Chow, etc. I also thank Dr. Peterson for being willing to participate in my committee.

13

14 Table of Contents Introduction Background. Ordinary Differential Equations Partial Differential Equations Bifurcation Theory 8 3. Saddle-Node Bifurcation Pitchfork Bifurcation Transcritical Bifurcation Hopf Bifurcation The Brusselator on the Unit Interval 6 5 Hopf Bifurcation in the Brusselator 3 6 Numerical Analysis 3 6. Discretization Using Crank-Nicolson Finite Difference Approximations Implicit, Explicit and Crank-Nicolson Fortran Numeric Results of the Brusselator 43 References 57 xiii

15

16 List of Figures Phase Portrait for dx dt = x + µ, µ = Phase Portrait for dx dt = x + µ, µ = Phase Portrait for dx dt = x + µ, µ = Phase Portrait for dx dt = x + µ, µ = Bifurcation Diagram for dx dt = x + µ Bifurcation Diagram for dx dt = x µy, dy dt = x y Phase Portrait for dx dt = x µy, dy dt 8 Phase Portrait for dx dt = x µy, dy dt 9 Bifurcation Diagram for dx dt = µx + x, dy dt Phase Portrait for dx dt = µx + x, dy dt Phase Portrait for dx dt = µx + x, dy dt = x y, µ = = x y, µ = = y = y, µ= = y, µ = Solution curves for the simplified Brusselator, µ = Solution curves for the simplified Brusselator, µ = Solution curves for the simplified Brusselator, µ = Solution curves for the simplified Brusselator, µ = Components u and v with no forcing, µ = Components u and v with no forcing, µ = Components u and v with no forcing, µ =, and A = Components u and v with no forcing, µ =, and A = Varied Periodic Orbits within the System Varied Periodic Orbits within the System Varied Periodic Orbits within the System Varied Periodic Orbits within the System An Equilibrium Point in the System Components u and v with forcing, µ =, and A = 3.75, T= Components u and v with forcing, µ =, and A = 3.75, T= 54 xv

17

18 Introduction The Brusselator is a system of equations that frequently appears in certain reactiondiffusion systems in chemistry. These types of systems were first studied by Boris Belousov [] after he witnessed a chemical reaction happening in a periodic fashion. We use the Brusselator as an example of a Hopf bifurcation, a horseshoe and a strange attractor. Bifurcations occur when a system undergoes a radical change of behavior as a parameter passes through a critical value. Several types of bifurcations have been classified including saddle-node, pitchfork, transcritical, and Hopf. When forcing is added to the Brusselator, further changes take place in the solutions. With no forcing, the Brusselator will display a stable periodic orbit which is destroyed when forcing is added. Also, infinitely many other periodic orbits will be created. This is the basic idea of a horseshoe. Furthermore, the solutions can feature occasional chaotic behaviors. These seem to occur randomly and are evidences of what is called a strange attractor. The purpose of this project is to numerically verify theoretical results produced by Lu et al [8]. In a presently unpublished paper, they explore bifurcations, horseshoes and strange attractors for systems of partial differential equations. It is our purpose therefore to provide visual examples confirming the existence of these phenomena is the Brusselator. We will explore bifurcations in general, presenting examples of several types. Afterward, we will use numeric methods to graph several solutions of the Brusselator with and without forcing.

19 Background Here we lay some foundation for the theory of ordinary and partial differential equations. Throughout the entire paper, we without further comment assume all functions are sufficiently smooth. A smooth function is one that has derivatives of every order n N. A sufficiently smooth function is one that has derivatives of whatever order we need. That is to say, when we speak of a function s derivative, we assume that it actually exists.. Ordinary Differential Equations We begin by discussing the nature of ordinary differential equations. This is a necessary foundation to later discuss partial differential equations, as many terms and definitions directly apply. The theory of ordinary differential equations is also used in solving partial differential equations as the latter can sometimes be reduced to an equation similar to the former. The following information is found in many textbooks, though here we draw primarily from Epstein [4] and Hale and Koçak [6]. Consider the system of ordinary differential equations x = f (x, x,..., x n ). (.) x n = f n (x, x,..., x n ) We summarize this simply as x (t) = f(x) (.) where it is understood that x, f R n are vector functions. Also, we have an associated initial condition x(t o ) = x o = (x o,..., x on ) (.3)

20 We speak of functions f which are locally or globally Lipschitz. The Lipschitz condition means that a function will not grow too fast. If f is globally Lipschitz then for some constant K, f(x ) f(x ) K x x for any x, x in the domain of f. Here, absolute value is taken to be the Euclidean norm on R n. A slightly weaker condition is that f be only locally Lipschitz. For f to be locally Lipschitz at a point x o means that there is a neighborhood U around x o where the above equation holds for all x, x U. We say that f is locally Lipschitz in a region R if f is locally Lipschitz for every x R. The Lipschitz condition is essential to guaranteeing that there is a unique solution. Otherwise, the behavior might become too wild at a point where f is not Lipschitz. In the study of ordinary differential equations, we want to know that a solution is well-posed. This means that we want to know if the solution exists, is unique, and is stable. If f is locally Lipschitz at x o then we are guaranteed a unique solution in the neighborhood U of x o. This is called a local solution to the differential equation. If f is globally Lipschitz as well, then a solution exists and is unique for the entire domain, making it a global solution. We state a fundamental theorem of existence and uniqueness. Theorem.. Let x(t) and f(x) be as in equation.. Suppose f is continuous and locally Lipschitz in an open region R R n and let (t o, x o ) R R. Then there is an interval t t o < h in which there exists a unique solution φ(t) satisfying equation. with initial conditions.3. The behavior of the solutions is of great importance to us. There are a few terms that explain the general patterns of the solutions. We call the overall picture of the solution curves the phase portrait, the phase plane, or orbit structure. In them are characteristics such as equilibrium points, periodic orbits, point and orbit stability, 3

21 instability, and asymptotic behavior. An equilibrium point x o is one for which any solution that has an initial condition x(t o ) = x o will be identically equal to that point. In other words, it is a stationary solution, never moving from the point x o. A periodic orbit is a solution that will eventually come back and repeat itself thus constituting a cycle or loop in the phase portrait. Both equilibrium points and periodic orbits have associated stabilities. This refers to the tendency of nearby solutions to be attracted to it. A stable equilibrium point x o is one for which solutions passing through any point in a neighborhood of x o will stay close to x o as t. If a solution curve passes near a stable periodic orbit φ(t) then it will always remain within a certain distance of it. A stronger condition of stability is that of asymptotic stability. This means that not only will a solution stay close to the equilibrium point or orbit, but it will converge to it as t. Since stability is of particular importance in this project, we give a formal definition. Definition.. Let x (t) = f(x) be a system of differential equations and x o be an equilibrium point. Suppose that for any neighborhood U of x o there is an open V U containing x o where if a solution φ(t) passes through V at a time t o then φ(t) U for all t t o. Then x o is a stable equilibrium point. Furthermore, if φ(t) x o as t then it is an asymptotically stable equilibrium point. Definition.3. Consider x (t) = f(x) and let φ(t) be a periodic solution and denote Γ = {φ(t) t R}. Suppose that for every ɛ > there is a δ > where if x (t) is a solution such that dist ( Γ, x (t o ) ) < δ for some t o then dist ( Γ, x (t) ) < ɛ for all t t o. Then φ(t) is a stable periodic orbit. Suppose further that dist ( Γ, x (t) ) as t. Then φ(t) is also said to be an asymptotically stable periodically orbit. If an equilibrium point or periodic orbit is not stable then it is called unstable. 4

22 Also, we can see that if all nearby solutions converge to an equilibrium point or periodic orbit, that it must be stable. This means that asymptotic stability implies ordinary stability. The converse, of course, is not always true.. Partial Differential Equations Partial differential equations are much more complex than ordinary differential equations and rarely have analytic methods for solving entire classes of them. Much work is still occurring in solving these types of equations. We restrict our discussion to functions of two variables. Since later discussion entails the use of a system with one spatial variable r and one time variable t, we begin now to use this notation. Eaton and Flaherty [5] help classify partial differential equations of two variables. Consider the differential equation a(r, t)x rr + b(r, t)x tr + c(r, t)x tt + d(r, t)x r + e(r, t)x t + f(r, t)x = g(r, t) (.4) where a, b, c, d, e, f, g are all continuous functions in t and r. Considering the equation λ, λ = b ± b ac then we classify equation.4 as a hyperbolic if λ, λ are real and distinct, i.e. b ac > elliptic if λ, λ are complex, i.e. b ac < parabolic if λ = λ, i.e. b ac = If a system has more than one equation, similar definitions hold for the associated matrix of the system, as opposed to merely the coefficients. The separate categories of partial differential equations have quite different analytic approaches to solving them. Since the hyperbolic equations have two distinct roots λ λ, they have two characteristics, which can cause weak discontinuities. The parabolic 5

23 and elliptic equations have one root and no roots, respectively, which guarantees no discontinuities. If we have a parabolic or elliptic equation, it will have an analytic solution (provided it is a well-posed problem). As the Brusselator is a well-posed elliptic equation, this tells us that it will have an analytic solution. In partial differential equations, most of the important terminology of ordinary differential equations still carry the same meaning. We continue to use equilibrium points, periodic orbits, stability, and asymptotic behavior with the same definitions. In fact, the definitions as defined in section. need no modification for partial differential equations with the simple exception that instead of a single variable t, we have additional spatial variables. However, t is still often used to denote time in application problems, and thus we can still use the definitions requiring t. Analytic solutions to PDE s can be difficult and are limited to a small number. However, we outline a couple of basic techniques. This includes writing solutions as the infinite sum of functions we know. Consider the interval [,] and the class of L functions. Then basic computation shows that the functions sin(nπt) for n N form an orthonormal basis. This means that sin(nπt) sin(mπt)dt = δ m,n where if m = n δ m,n otherwise An orthonormal set always spans the entire space through linear combinations. This gives that for any L function x(t), x(t) = α n sin(nπt) (almost everywhere) n= where α n is called the coefficient. Furthermore, to obtain a solution, we assume a property called separation of variables. This entails taking a multi-variable function 6

24 x(r, t) and splitting up the function into the product of two single-variable functions as x(r, t) = g(t)h(r). Let me demonstrate how this is useful. Consider the simple heat conduction equation Example.4. x t = x rr We assume that x(r, t) = g(t)h(r) to get x(r, t) = g(t)h(r) = g n (t) sin(nπr) n= where g n (t) = α n g(t). Substituting and differentiating term by term, g n(t) sin(nπr) = n= g n (t) ( (nπ) sin(nπr) ) n= Now for each m, we can multiply through by sin(mπr) and integrate with respect to r from to and use the orthogonality principle to obtain where β m = β m g m (t) = β m(mπ) g m (t) sin (mπr)dr. This last equation has reduced to a simple ordinary differential equation for which we have a unique solution for each m. We now know each g m and thus have a candidate for an analytical solution of x(t). This technique is used quite often in partial differential equations and will be considered in the Brusselator. The existence and uniqueness of a partial differential equation is a tough subject to tackle. General existence proofs typically do not exist. It is often a subject that goes case by case proving existence and uniqueness of specific problems. Street, in his partial differential equations book [9], discusses existence and uniqueness. A problem is said to be well-posed if there is a solution to the model (existence), there is at most one solution (uniqueness) and the solution depends continuously on 7

25 all the data given (stability). There are several definitions of solutions to differential equations, though I consider only the classical definition. In each problem posed, one must decide if these three conditions are met. In the example of the heat equation.4 that would entail guaranteeing that the Fourier series converges uniformly to make sure the resulting solution exists and is stable. Then one needs to prove it is the only possible solution. 3 Bifurcation Theory A bifurcation deals with the change of dynamics of a system of equations as a parameter in the equations changes. In essence, a bifurcation occurs when the solutions behavior changes in a radical way. This can happen through the appearance of new stable orbits, changing asymptotic behavior, breaking stability, or in several other ways. These changes transpire when a parameter in the equation passes through a critical value. That is to say, as one varies a particular constant in the equation, there is a certain value which will cause a qualitative change to take place in the solutions. Since a bifurcation deals with a change in the phase portrait, we need to establish rigorously what that entails. Hale and Koçak [6] explain when a bifurcation happens. First we decide that the essential characteristics of a phase portrait are given by the number of equilibrium points, the number of periodic orbits, the stability of each of these and where the equilibrium points are in relation to the periodic orbits. Therefore, if two phase portraits are equal in these attributes, we consider them to have the same orbit structure. Now we use a topological relation called a homeomorphism. A function h : X Y is a homeomorphism if it is continuous, bijective and has a continuous inverse. 8

26 Next suppose we have two differential systems x = f(x) and x = g(x) These are said to be topologically equivalent if there is a homeomorphism h that maps the orbits of one system to the orbits of the other system while preserving direction in time. This means that the essential characteristics of the orbits are preserved. We present this as a theorem. Theorem 3.. Two differential equations x = f(x) and x = g(x) each with a finite number of equilibrium points are topologically equivalent if and only if they have the same orbit structure. This allows us to define properly what it means for a system to have a bifurcation for a parameter µ. Consider the system x = f(x, µ) where µ = (µ,..., µ k ) is a group of parameters in the equation, which can be varied. Definition 3.. For a fixed value µ o, the phase portrait of x = f(x, µ) is called structurally stable if there is an ɛ > such that x = f(x, µ) is topologically equivalent to x = f(x, µ o ) for all values µ µ o < ɛ. If we have a critical value µ o where x = f(x, µ) is not structurally stable, then we have a bifurcation and µ o is called a bifurcation point. Thus, at a bifurcation we expect to see the introduction or destruction of periodic orbits, equilibrium points and/or a change in asymptotic behavior. The next important question is what kind of change takes place in a given system. For this, we often sketch bifurcation diagrams. These keep track of the existence of equilibrium points which help show what changes occur in the orbit structure. Several types of bifurcations have been classified and precisely defined. A few basic ones include the saddle-node, pitchfork, transcritical, and Hopf bifurcations. 9

27 Hale and Koçak [6] continue their discussion of bifurcations with definitions and examples of each of these. We assume that µ R is a parameter in the system of equations and also use the notation B ɛ (x o ) = {x x x o < ɛ} 3. Saddle-Node Bifurcation Definition 3.3. Let µ o be a bifurcation point, x o R n and I be an interval containing µ o. Let ɛ be a constant such that the following conditions are met: i) to one side of µ o in I there are no equilibrium points in B ɛ (x o ). ii) to the other side of µ o in I there are two equilibrium points in B ɛ (x o ) and one of them is stable and the other is unstable. Then this type of bifurcation is a saddle-node. We now turn our attention to a detailed look at an example of a saddle-node bifurcation. For reasons of introduction, the first equation we encounter is a simple single variable scalar function x(t). Consider the equation dx dt = x + µ (3.) In order to analytically understand this particular equation, let s solve it directly. Afterward, it will be important to highlight certain restrictions on µ and what happens as µ passes beyond those restrictions. At that point we will display graphical depictions of the solution curves for µ =, µ =.5, µ =, and µ =. For now, let us assume that µ > and solve the differential equation straightforwardly. dx dt = x + µ dx x + µ dt = dx ( µ µ x ) + dt = dx ( µ x ) + dt = µ

28 This gives us a solution for x(t) explicitly as x(t) = µ tan( µt + C) (3.) In figure we have a graph for µ =. Maple was used to generate these general phase portraits. - - x(t) t - - Figure : Phase Portrait for dx dt = x + µ, µ = In the figure for µ = we can see the family of tangent functions from equation 3.. With regard to the parameter µ, we see an immediate restriction. It obviously must be non-negative or else µ would be undefined. In our process we have considered, which also means µ. Therefore, to confine ourselves within µ the real numbers, we must have µ >. We now consider graphical representations of the solution curves for dx dt = x + µ with µ =.5, µ =, and µ =. We should expect a radical shift of behavior when we traverse.

29 - - x(t) t - - Figure : Phase Portrait for dx dt = x + µ, µ = x(t) t - - Figure 3: Phase Portrait for dx dt = x + µ, µ =

30 - - x(t) t - - Figure 4: Phase Portrait for dx dt = x + µ, µ = We immediately see a definite change of behavior as µ passes from the right side of to the left side. Again, when µ = and µ =.5, the solution curves form a family of tangent functions. Exactly when µ = (figure 3), we see a horizontal asymptote at x =. We can see this from an explicit look at the equation when µ =, which makes equation 3. become dx dt = x and we can definitely see that if x =, dx = which denotes an equilibrium value. dt Below the line x = we see the solution curves asymptotically approaching x = as t increases. Above the line x = we have solutions diverging away. This behavior can be shown since for all x, we have x > dx >. So the solutions are dt strictly increasing for all values of t such that x. Thus as µ passes from being positive to being, we introduced a previously non-existent equilibrium value for 3

31 x. Next, as µ became negative, we ended up with two equilibrium values for x. These can be seen in figure 4. Where will these points be? Explicitly looking at the equation again, let µ < and consider dx dt = x + µ If we want an equilibrium value, we must set dx dt x + µ = x = ± µ =. This occurs when Thus we have the two locations of the equilibrium values when µ <. In addition, if x (, µ) ( µ, ) then x +µ > dx dt > which gives us the increasing behavior outside of the two lines x = µ and x = µ. But if x ( µ, µ) then x + µ < dx < which causes the solution dt curves to be decreasing there. We then have x = µ is a stable equilibrium value and that x = µ is an unstable equilibrium value, as in figure 4. Let s consider a graph of the equilibrium value s position, or in other words, the bifurcation diagram. By the equation x = ± µ we have figure 5. Notice that for µ > there are no equilibrium values. For µ < the upper branch denotes the unstable equilibrium value and the lower branch denotes the stable equilibrium value. In our bifurcation diagrams we show the unstable points in red and the stable points in black. This graph is done on a µ, x-axis system. 4

32 x Figure 5: Bifurcation Diagram for dx dt = x + µ 3. Pitchfork Bifurcation Definition 3.4. Let µ o be a bifurcation point, x o R n and I be an interval containing µ o. Let ɛ be a constant such that the following conditions are met: i) to one side of µ o in I there is only one equilibrium point x (µ) in B ɛ (x o ). ii) to the other side of µ o in I there are three equilibrium points in B ɛ (x o ), one of which is still x (µ). This type of bifurcation is called a pitchfork bifurcation. The example we consider to illustrate the pitchfork bifurcation is given by the system of equations dx dt = x y µy (3.3) dy dt = x y First note that for all values of µ we have that is an equilibrium point. We 5

33 set dx dt = dy dt = to find the other equilibrium points. Assuming x, y x y µy = x = ± µ x y = y = x = ± µ Thus when µ < we have no additional equilibrium points and when µ > we have two more equilibrium points at ( µ, µ) and ( µ, µ). In the following bifurcation diagram (figure 6), though the equilibrium points are in R, we still use a µ, (x, y) plane to represent the positions of the equilibrium points x Figure 6: Bifurcation Diagram for dx dt = x µy, dy dt = x y The name pitchfork is given due to the bifurcation diagram, which resembles a pitchfork. It is interesting to point out that in most known cases of pitchfork bifurcations, two properties hold. The equilibrium point x (µ) that shows up on both sides of the bifurcation point usually reverses stability as µ passes through µ o. 6

34 Also, the other two equilibrium points created on one side of µ o are usually of the same stability. The following two figures represent what happens to the phase portrait as µ passes through. This example follows the guidelines set forth in the preceding paragraph. In particular figure 7 shows when µ = we have one unstable equilibrium point at. Figure 8 shows that when µ = we have three equilibrium points. The stability of has reversed to stable and the other two equilibrium points are both unstable. - - y x - - Figure 7: Phase Portrait for dx dt = x µy, dy dt = x y, µ = 7

35 - - y x - - Figure 8: Phase Portrait for dx dt = x µy, dy dt = x y, µ = 3.3 Transcritical Bifurcation Definition 3.5. Let µ o be a bifurcation point and I be an interval containing µ o. Let x o R n and ɛ be a constant such that the following conditions are met: i) to one side of µ o in I we have exactly two equilibrium points x (µ), x (µ) of opposite stability in B ɛ (x o ) that converge to x o as µ µ o. ii) as we pass to the other side of µ o, x (µ) and x (µ) continue to exist, diverging away from x o with their stabilities reversed. This gives a transcritical bifurcation. We take a rather simple system of equations to exemplify the transcritical bifurcation. It is seen as the reversal of stability of two equilibrium points. Consider dx = µx + x dt dy dt = y 8 (3.4)

36 For all values of µ, we have the equilibrium point. To discover other equilibrium points, we again set the derivatives to to get µx + x = x = µ y = y = So the other equilibrium point is at ( µ, ). This is shown in the bifurcation diagram x Figure 9: Bifurcation Diagram for dx dt = µx + x, dy dt = y Now we consider stability. For all initial positions for y, y because if y > then y < and if y < then y >. For x, we have x = x(µ + x). This plots a parabola that has zeros at and µ. Since it is constantly concave up, we have that x > on (, µ) (, ) if µ > and (, ) ( µ, ) if µ <. We also have x < on ( µ, ) if µ > and (, µ) if µ <. Consider µ >. To the right of µ, x is decreasing and to the left of µ, x is 9

37 increasing. And, again, since y then all nearby solutions converge to ( µ, ) making it an asymptotically stable equilibrium point. Now we consider µ <. Between and µ, x decreases away from µ and to the right, x is increasing away from it. Therefore, the nearby solution diverge away from ( µ, ), making it unstable. From the discussion of the behavior of x, we also see that is unstable for µ > and stable for µ <. The following figures are for µ = and µ =. - - y x - - Figure : Phase Portrait for dx dt = µx + x, dy dt = y, µ=

38 - - y x - - Figure : Phase Portrait for dx dt = µx + x, dy dt = y, µ = 3.4 Hopf Bifurcation The Hopf bifurcation not only deals with equilibrium points as do the other three types we have considered, but it also involves the creation of a periodic orbit. The periodic orbit is a special kind, called a limit cycle. For this, we need the notions of α- and ω-limit points. Hale and Koçak [6] explain these concepts. Suppose we have an orbit φ(t) of a differential equation with initial condition φ(t o ) = x o. Let y R n such that there is a sequence of time {t n } where φ(t n ) y. Then y is called an ω-limit point. The set of all ω-limit points is called the ω-limit set and is denoted ω(x o ). If we consider the same definition except reversing time so that {t n } then y is called an α-limit point. The α-limit set is denoted α(x o ). If a limit set comprises a periodic orbit, then this periodic orbit is given the special name limit cycle. We now proceed to the definition of the Hopf bifurcation.

39 Definition 3.6. Let µ o be a bifurcation point and I be an interval containing µ o. Suppose on one side of µ o in I, there is one stable equilibrium point x (µ) in the ball B ɛ (x o ). Suppose further that as we pass to the other side of µ o in I, x (µ) loses stability and a limit cycle is created around it. This is called a Hopf bifurcation. We will use the Brusselator as an example of a Hopf bifurcation. We examine the Brusselator in greater detail in later sections, while here we define it and then consider only a simpler version of it to illustrate the Hopf bifurcation. Following deeper discussion of the Brusselator, we also present the Poincaré-Andronov-Hopf theorem which shows how to detect a Hopf bifurcation in a system. Considering functions x(r, t) and y(r, t), the Brusselator is given by the system of equations x t = A (B + )x + x y + x π r y t = Bx x y + θ (3.5) y π r For a more simplistic example at this point, we only consider the case where the solution is constant with respect to r. With this requirement x r = y =. With r x and y being constant with respect to r, we can consider the functions x(r, t) and y(r, t) as their one-dimension counterparts x(t) and y(t). Equation 3.5 becomes dx dt = A (B + )x + x y (3.6) dy dt = Bx x y In this section, where my purpose is to illustrate a Hopf bifurcation, let A = and let B = µ be the parameter of bifurcation. Making these changes in equation 3.6 we now we explore several values of µ in the system dx dt = (µ + )x + x y dy dt = µx x y (3.7)

40 Since the Hopf bifurcation entails the creation of a limit cycle around an equilibrium point, we wish to have greater detail to display the creation of the orbit. We will graph this system in Matlab to see the detailed behavior of a few solutions in lieu of the general phase portrait graphed by Maple Figure : Solution curves for the simplified Brusselator, µ =.5 We can see from figure that we have an asymptotically stable equilibrium point. For µ =.5 it is around the point (,.5). This leads to one quick observation for equation 3.6. Fact 3.7. For equation 3.6 with A we have an equilibrium point at (A, B A ). Proof. The proof is merely an explicit look at equation 3.6. Set x = A and y = B A. Substituting into equation 3.6, we have for dx dt dx dt = A (B + )A + A B A 3 = A AB A + AB =

41 Similarly for dy dt, dy dt = BA A B A = AB AB = By the definition of an equilibrium point, (A, B ) is an equilibrium point. A We turn our attention back to equation 3.7. Let s see what happens when we use the values µ =.75, µ = and µ = Figure 3: Solution curves for the simplified Brusselator, µ =.75 4

42 Figure 4: Solution curves for the simplified Brusselator, µ = Figure 5: Solution curves for the simplified Brusselator, µ =.5 We have a very evident bifurcation. For µ =.5 and µ =.75 all solutions asymptotically approached the equilibrium point (A, B A ). As µ passed near the 5

43 point, the equilibrium point became suddenly unstable. But this is not all, an entire limit cycle was created around that point! Furthermore, as µ increased, the solutions more quickly gravitated to the limit cycle. 4 The Brusselator on the Unit Interval The Brusselator is a system of equations that deals with the reaction-diffusion system of certain chemical reactions. According to Shaun Ault [], it is an example of autocatalytic, oscillating reactions. Boris Belousov was studying the Kreb s cycle when he discovered a mixture of citric acid, bromate, and cerium in a sulfuric acid solution that changed colors in a periodic fashion. This encouraged the study of elliptic systems, of which the Brusselator is a product. We now consider in greater detail the Brusselator equations. Recall that the Brusselator consists of the system of equations x t = A (B + )x + x y + x π r y t = Bx x y + θ (4.) y π r where A, B and θ are prescribed parameters with constraints A and < θ <<. In the previous section we calculated that equation 4. has an equilibrium point at (A, B ). For simplicity, we perform a change in variables to shift this equilibrium A point to (, ). We define the change of u and v in the following manner and u(r, t) = x(r, t) A (4.) v(r, t) = y(r, t) B A (4.3) From equations 4. and 4.3 we calculate the partial derivatives of u and v to get u rr = x rr u t = x t (4.4) 6

44 and v rr = y rr v t = y t (4.5) Making direct substitutions into equation 4. we obtain for the first part u t = x t = A (B + )x + x y + x π r ) u + π r = A (B + )(u + A) + (u + A) ( v + B A = A Bu AB u A + (u + Au + A )(v + B A ) + u π r = Bu AB u + u v + B A u + Auv + Bu + A v + AB + π u r = (B )u + A v + B A u + Auv + u v + π u r For the second part of the equation name v t = y t = Bx x y + θ y π r = B(u + A) (u + A) (v + B A ) + θ π v r = Bu + AB (u + Au + A )(v + B A ) + θ π v r = Bu + AB (u v + B A u + Auv + Bu + A v + AB) + θ π v r = Bu A v B A u Auv u v + θ π v r With the occurrence of B A u + Auv + u v in both parts we give it a special h(u, v) B A u + Auv + u v (4.6) Our final result, the Brusselator equation with equilibrium point (, ), can be written as u t = (B )u + A v + u + h(u, v) π r v t = Bu A v + θ v h(u, v) π r 7 (4.7)

45 As we did in section 3.4, consider just the ordinary differential equation of the Brusselator with the equilibrium point at (, ) du dt = (B )u + A v + h(u, v) dv dt = Bu A v h(u, v) (4.8) which in matrix form is expressed u = B A u h(u, v) + v B A v h(u, v) (4.9) For later convenience, we create a governing parameter µ = B A. We show two important facts from equation 4.9 pertaining to an important Hopf bifurcation theorem discussed in section 5. Fact 4.. The matrix M = B A has eigenvalues α(µ) ± iβ(µ) such B A that α() =, β() and dα (). dµ Furthermore, the nonlinear function h(u, v) F (u, v) = is such that F (, ) = and D [u,v] F (, ) = where D [u,v] is h(u, v) the Jacobian. Proof. We directly compute the eigenvalues of M. B λ A = (B λ)( A λ) + BA B A λ = BA λb + A + λ + λa + λ + BA = λ + (A + B)λ + A Setting this equal to and using the parameter µ, we have λ = α(µ) ± iβ(µ) = µ ± µ 4A. This means that when µ =, α() = and β() = A. Also, 8

46 dα dµ = for all µ. For the second part, recall that h(u, v) = B A u + Auv + u v so h(, ) =. The Jacobian of h is given by B u + Av + uv Au + uv A B u Av + uv Au uv A which has no terms free of u and v. So when we set u = v =, D [u,v] F (, ) =. These requirements will actually guarantee the existence of a Hopf bifurcation at µ =. This is known as the Poincaré-Andronov-Hopf theorem and will be discussed in the next section. We are now ready to include a forcing term in the Brusselator. Forcing is to take the form of a periodic delta function P T (t) n= n= δ(t nt ) (4.) This forcing will then be given magnitude ρ sin(πr). We choose to have forcing of the form u forcing = ρ sin(πr)p T (t). Combining this with equation 4.7 we now v forcing have u t = (B )u + A v + u π r + h(u, v) + ρ sin(πr)p T (t) v t = Bu A v + θ (4.) v h(u, v) π r Furthermore, we want to consider equation 4. on the unit interval r [, ] with insulation on the boundaries. This means we intend to hold the edges at the constant value of, keeping in line with having shifted the equilibrium to the point (, ). Thus we include the following boundary conditions u(, t) = u(, t) = v(, t) = v(, t) = (4.) The Brusselator equation 4. together with the boundary conditions 4. lay the foundation for the form of the Brusselator with which we will work. 9

47 5 Hopf Bifurcation in the Brusselator Here we look at an essential theorem in detecting Hopf bifurcations. In the last section, we showed that the ordinary part of the Brusselator had certain characteristics. The matrix had eigenvalues with non-zero imaginary part and zero real part with non-zero speed at µ =. We also showed that F and its Jacobian were zero for u = v =. Hopf, based upon earlier work by Poincaré and expounded upon by Andronov, compiles this information into an important theorem on how to detect a Hopf bifurcation in a system. Hale and Koçak [6] describe the result. Theorem 5.. The Poincaré-Andronov-Hopf Bifurcation Theorem. Let x = M(µ)x+F (µ, x) be a planar vector field depending on a scalar parameter µ with M C m,n and F C k for some k 3. Assume that F (µ, ) = and D x F (µ, ) = for all sufficiently small µ (D x is the Jacobian). Assume that the linear part M(µ) at the origin has eigenvalues α(µ)±iβ(µ) with α() =, β() and dα dµ (). Then in any neighborhood U of the origin and for any given µ o > there is a µ with µ < µ o such that the differential equation has a nontrivial periodic orbit in U. Now we analytically show that the Brusselator has a Hopf bifurcation. Instead of considering only the ordinary part of the Brusselator as in the last section, we show how to apply the Poincaré-Adronov-Hopf theorem to the entire Brusselator. First we must perform some manipulations. Lu et al [8] show how to set up the Brusselator in terms of theorem 5.. As in section. we can write u(r, t) = a m (t) sin(mπr) v(r, t) = b m (t) sin(mπr) m= m= Substituting into equation 4.7, multiplying through by sin(nπr) and integrating 3

48 from to we have a n b n = B n A B A θn a n + Considering theorem 5., M = B n A F = B A θn b n h(u, v) sin(nπr)dr (5.) h(u, v) sin(nπr)dr h(u, v) sin(nπr)dr h(u, v) sin(nπr)dr But by a quick observation, h(, ) = which means that F (µ, ) =. Furthermore, both h u (, ) = and h v (, ) = as well, so D [u,v] F (µ, ) =. h(u, v) and sin(nπr) are analytic functions with derivatives of all orders, so F is also C k for all k. This completes all the requirements for F. Next, we compute the eigenvalues of M. B n λ A = (B n λ)( A θn λ) + BA B A θn λ = λ + (θn + A + n + B)λ + (θn 4 + A n + θn + A θbn ) and we see then that the eigenvalues are ( µ + ( + θ)( n ) (µ ± + ( + θ)( n ) ) 4 ( A (n + ) θn (µ + + θ + A n ) )) where, similarly to before, we define the parameter µ = (B A θ). It is µ that fulfills the requirements in theorem 5. and is hence the value that we refer to from here on. We can see that when n = and µ =, we have eigenvalues α(µ) ± iβ(µ) where α() =, β() and dα =. This fulfills the remaining dµ hypothesis of theorem 5.. Corollary 5.. The Brusselator has a Hopf bifurcation at µ =. 3

49 6 Numerical Analysis Before we can continue the study of the Brusselator and the bifurcation that takes place, we must lay down several computing tools that we use. We discretize the Brusselator equation to make it ready for computer calculation in order to better explore its properties. In these computations we use the Fortran 9 and Matlab 6.5 languages. Many numeric methods are based on the Taylor s expansion. In essence, the Taylor series expansion takes a function and rewrites it as an infinite degree polynomial. If the function u is smooth in an interval I containing r o and every derivative u (n) is bounded in I, then there exists a constant c such that u(r) = k= u (k) (r o ) (r r o ) k for every r [r o c, r o + c] (6.) k! Under similar conditions, we can also expand multi-variable functions by the Taylor s expansion with respect to one of the variables. We can expand u(r) in the r i th direction around r o by the formula u(r) = k= k u (r k! ri k o )(r i r oi ) k (6.) 6. Discretization Using Crank-Nicolson In working with partial differential equations, one of the most crucial elements is the ability to use computer algorithms to show the behavior of the system. Many partial differential equations are unsolvable by common manipulation and even if one can manually solve a PDE, the equation can often times be too complicated to properly understand without the aid of a computer. Discretizing a problem is a common tool which transforms a continuous problem into a discrete one from which approximations can be calculated that closely resemble the actual solutions. 3

50 The method of choice for this project is the Crank-Nicolson discretization. It is a mixture of both explicit and implicit methods using central difference and forward difference approximations. It is our intent, therefore, to explain these approximations and how they combine to form the Crank-Nicolson discretization. 6.. Finite Difference Approximations Since a finite difference approximation turns a continuous process into a discrete process, we use the notation rz n. By that notation we mean the space ( r Z) ( r Z) ( r n Z) where r i is a scalar and r i Z is just a rescaling of Z. Suppose u : R n R m has a kth partial (or ordinary) derivative k u r k. Let U : rzn R m such that U(i) u(i) for all i rz n and k u (i) is approximated by surrounding rk points of U near i, then U is called a finite difference approximation for k u. It is rk written as k U. If U is a finite difference approximation of u, then the space rzn rk is called the grid of U. Flaherty and Eaton [5] provide constructions of pertinent finite difference approximations for solutions to partial differential equations. Let u : R n R m be a smooth function and for convenience write u as u(r, t) where r R n and t R. We now write the Taylor s series expansion (6.) for u with respect to t at t o. u(r, t+ t) = u(r, t)+ t u t (r, t)+ u! t t (r, t)+ + k u k! tk (r, t)+ (6.3) tk Here we keep the first two terms of 6.3 as an approximation of u(r, t + t) to get This can now be solved for u t u(r, t + t) u(r, t) + t u t u t u(r, t + t) u(r, t) t 33

51 We replace u by U and thus obtain the first forward finite difference approximation U t (r, t) = U(r, t + t) U(r, t) t (6.4) Returning to the Taylor s Expansion of u in equation 6.3, we replace t+ t with t t. We now have u(r, t t) = u(r, t) t u t (r, t) + u ( )k! t (r, t) + + t k k u (r, t) + t k! tk (6.5) which, when truncated at two terms, gives u(r, t t) u(r, t) t u t Solving for u and replacing u with U as before, we get the first backward finite t difference approximation U t (r, t) = U(r, t) U(r, t t) t (6.6) We combine the forward and backward methods to create a more powerful and commonly used finite difference method. Consider now the Taylor s Expansion for both t + t and t t in equations 6.3 and 6.5. If we truncate after the third term in each expansion we obtain a higher degree of guaranteed accuracy. Doing so gives u(r, t + t) u(r, t) + t u t (r, t) + u t (r, t) t u(r, t t) u(r, t) t u t (r, t) + u t (r, t) t Next, subtract the second equation from the first u(r, t + t) u(r, t t) t u t (6.7) (r, t) + t u(r, t) (6.8) t As before, we solve the equation for u and exchange u for U. This is called the t first centered finite difference approximation. U t (r, t) = U(r, t + t) U(r, t t) t 34 (6.9)

52 We now turn our attention to the discretizing of second partial derivatives. The approach will be similar. Taking equations 6.3 and 6.5 we truncate them each at the fourth term, yielding a good degree of accuracy. u(r, t + t) u(r, t) + t u t (r, t) + u t t (r, t) + 3 u 6 t3 (r, t) t3 u(r, t t) u(r, t) t u t (r, t) + u t t (r, t) 3 u 6 t3 (r, t) t3 Adding the second equation to the first gives (6.) u(r, t + t) + u(r, t t) u(r, t) + u(r, t) + u t t (r, t) + u t (r, t) t u(r, t + t) + u(r, t t) u(r, t) + t u t (r, t) (6.) We are looking for u tt, so we solve for u t approximation to get and substitute U in place of u as an U tt (r, t) = U(r, t + t) U(r, t) + U(r, t t) t (6.) which is called the second centered finite difference approximation. Notation: Since our finite difference approximations are in the discrete space rz n, we can simplify the notation used to write U. Often we are in the space of R R in which case r = r R. In our discretizations, this means that we can let r = r i and t = t j and write U(r i, t j ) U i,j. Since we are dealing with the Brusselator (see equation 4.7), we will from this point assume the domain R R unless specifically stated otherwise. Furthermore, since r is used for the spatial variable and t is used for the time variable, according to the preceding notation, i refers to the spatial variable r and j refers to the time variable t. These approximations are used in determining the behavior of a solution u of a partial (or ordinary) differential equation to which we have no analytical solution. The finite difference approximations are substituted in the equation whenever 35

53 a derivative appears. This eliminates the need to calculate any derivatives, and instead our entire equation is written directly in terms of certain points of u. Example 6.. Consider the simple heat equation u rr = u t. Although this equation can, in fact, be determined analytically, we use it as an example of using the finite difference approximations. Suppose for the second partial derivative in r, u r we choose the centered difference approximation and for the first partial in t, we use the forward difference approximation. Then the equation becomes U i+,j U i,j + U i,j r = U i,j+ U ij t Note that now all the terms are simply points of the approximating function U and we no longer have need of computing derivatives. Depending on what we seek, we can solve this equation for various points (i, j) and use a computer to approximate them. 6.. Implicit, Explicit and Crank-Nicolson The Crank-Nicolson method is a specialized finite difference approximation using forward, backward, and centered type finite differences. It is, in fact, half way between what are called implicit and explicit methods. Kincaid and Cheney [7] explore the differences between these methods and how they can be combined to form the Crank-Nicolson method. Suppose we are attempting to approximate u i,j+ using a finite difference method. We construct a finite difference approximation for u i,j+ using a grid of size (n + ) (m + ) (the + is simply to include the th step). If every term of the approximation is of the form U i l,j k for k j and i n l i then we call this an explicit method. 36

54 Let us justify the name explicit method. In most problems, we are given an initial set of data (t = j = ). Suppose we are trying to find u i,. Using only terms of the form U i l,j k with k j, means j k must equal and thus we have terms only as U i l,. But we know every point at j = (given) and so we can explicitly compute U i,. This means we can figure out U i, for every i and so we have a full approximation for the j = step. Now if we want to compute U i,, we have every term in the approximation either U i l, or U i l,. But again, we know U i,j for every point at j = and j =. Therefore, we can again explicitly compute U i,. In example 6., if we were to solve for U i,j+ (on the right hand side) we would have an explicit method because all other time steps are less than j +. Suppose we construct a finite difference approximation for u i,j+ using a grid size of (n + ) (m + ). If there is a least one term of the approximation of the form U i l,j k where k < then we have have an implicit method. What makes this type of method implicit? When approximating u i,j+, we may have to use points that are on that same time level or a higher time level. But since we haven t yet computed those time levels, we have no expression in which we explicitly know every term. We have to rely on other computing techniques (i.e. implicit techniques) to solve an approximation for u i,j. Setting up systems of equations and using matrix inverses is a common method. If we were to solve example 6. for U i,j then we would have an explicit method because we have terms at time steps bigger than j. It is now possible to combine the aforementioned explicit and implicit definitions into a broader category. Let U i,j represent an explicit finite difference approximation for u i,j and let V i,j represent an implicit finite difference approximation for u i,j. Let θ [, ]. Then we can mix these two approximations in the following format θu i,j + ( θ)v i,j (6.3) 37

55 Immediately we see that if θ = then we have our implicit method and if θ = we have our explicit method. Kincaid and Cheney [7] use this formula to define the Crank-Nicolson. Definition 6.. When θ = in equation 6.3 we have the Crank-Nicolson method. This is the method of choice for our use. The program code we use is set up by applying the Crank-Nicolson method to the Brusselator. 6. Fortran One of the earlier, powerful languages of computer programming is Fortran. It is reputed for its relative speed against other computer languages. Therefore, for longtime computations such as this project requires, Fortran is a good choice. Here we explain Fortran programming in relation to the code we use in our computations of the Brusselator. The information regarding the Fortran language comes from the Digital Equipment Corporation s manual [3]. The following is the main program we work with in this project. The basic code we use was set up by Guowei Wei of Michigan State University using the Brusselator equations together with the Crank-Nicolson discretization. It calls several subroutines, such as setting up the initial data, performing the numeric methods and recalculating the next time step. PROGRAM FPE! IMPLICIT DOUBLE PRECISION(A H,M, O Z) PARAMETER ( nn=5,a=3.9d,mu=.d, prd=5. d ) DIMENSION u( nn ), v ( nn ), r r ( nn ), aa ( nn ), bb( nn ), cc ( nn ), u ( nn)! pi=dacos(. d ) d=.d/ pi ; 38