Appendix A

An Overview of Bifurcation Theory

A.I Introduction

In this appendix we want to provide a brief introduction and discussion of the con­ cepts of dynamical systems and bifurcation theory which has been used in the pre­ ceding sections . We refer the reader interested in a more thorough discussion of the mathematical results of dynamical systems and bifurcation theory to the books of Wiggins (1990) and Kuznetsov (1995). A discussion intended to more econom­ ically motivated problems of dynamical systems and chaos can be found in Medio (1992) or Day (1994). It is noteworthy here that chaos is only possible in the non­ autonomous case; that is, the dynamical system depends explicity on time; but not for the autonomous case where time does not enter directly in the dynamical sys­ tem (cf. Wiggins (1990». Since both models we have considered in the preceding sections are autonomous they cannot reveal any chaos . However, as the reader has seen, the dynamical systems we have derived reveal some bifurcations. Roughly spoken, bifurcation theory describes the way in which dynamical sys­ tem changes due to a small perturbation of the system-parameters. A qualitative change in the phase space of the dynamical system occurs at a bifurcation point, that means that the system is structural unstable against a small perturbation in the parameter space and the dynamic structure of the system has changed due to this slight variation in the parameter space. We can distinguish bifurcations into local, global and local/global types, where only the pure types are interesting here. Local bifurcations such as Andronov-Hopfor Fold bifurcations can be detected in a small 182 An Overview of Bifurcation Theory neighborhood ofa single fixed point or stationary point. In a similar way we can find bifurcations in limit cycles by the construction of a Poincare Map. Bifurcations that cannot be detected by an analysis of the properties of the fixed points as parameter values vary are called global (cf. Wiggins (1990) ; Kuznetsov (1995); Glendinning (1996)) .

A.2 Some Definitions and Results

Let us now introduce some important definitions and results for dynamical systems with continuous time T = IR with t E T. In this section we use the same notation and definitions as Kuznetsov (1995) . Since we are just interested in dynamical sys­ tems with a finite-dimensional state space, we define the state space ofthe dynamical system as X = R' of some finite dimension l. The evolution law that determines I the state of the variable Xt E IR at time t under the condition that the initial state value X o is known is defined by

(A.l)

which transforms the initial state X o E Rl into some state Xt E ]Rl at time t > 0, i.e. Xt = cr:/xo. We call the map cpt evolution operator. The formal definition of a dynamical system is given by

I I Definition A.t. A dynamical system is a pair {IR , cpt}, where IR is the state space I I and cpt : IR ---+ IR is a family ofevolution operators satisfying the properties

and (A.2)

I where the first property in (A.2) means the identity map on IR , that is idx = x for all x E ]Rl and the second property means that cpt+s = cpt (cp sx) for all x E ]Rl. The second property states that we can evolve the dynamical system in the course of t + s units of time starting at x and get the same final state value as if we had evolved the system first over s units of time and then over the next t units of time starting on the same initial value x.

Definition A.2. An orbit starting at Xo is an ordered subset ofthe state space ]Rl,

(A .3)

By an orbit through a point Xo we mean a curve in the state space ]Rl passing through the point Xo E ]Rl that is parameterized by the time t. We use here the term A.2 Some Definitions and Results 183

orbit and trajectory as synonyms. Note that in the literature of bifurcation theory sometimes both terms are distinguished (cf. (Wiggins, 1990, p. 2)). The simplest solutions we might expect for a dynamical system are solutions that remain constant over time. Such solutions are called equilibria, fixed points or stationary points. A formal definition is given below

l O Definition A.3. A point X O E IR is called an equilibrium (fixed point) ifcptxo = X for all t > 0. A second solution type is a periodic orbit.

Definition A.4. A cycle is a periodic orbit for each point X o E L o which satisfies cpt+Toxo = cpt xo with some To > °for all t > O. We understand under a periodic orbit the smallest possible amount of time To > 0 under which a system placed at a point X o will return exactly to this point. In this case the solution is periodic of period To and the system exhibits periodic oscillations. Now let us introduce a formal definition of a phase space plot which contains a lot of information about the dynamical behavior of a system under consideration. Definition A.5. The phase portrait ofa dynamical system is a partitioning ofthe state space into orbits. l Definition A.6. A (positively) invariant set ofa dynamical system {IR , cpt} is a l subset S C IR such that Xo E S implies cpt xo E S for all t > O. In the Definition A.6 we restrict ourselves to positive times. The definition means that a system which starts on the set S with initial state X o remains on the set S as time moves forward. Obviously, the orbit defined in Definition A.2 is an invariant set. Other examples of invariant sets are equilibria and cycles which are l l closed in IR , since the space IR is endowed with the Euclidean metric. Notice that the equilibria and the cycles are both orbits, the former are equilibria orbits whereas the latter are periodic orbits due to the Definition A.2. Definition A.7. An invariant set S C IRl is said to be a C", (r 2: 1) invariant man­ ifold iff S has the structure ofa C" differentiable manifold. Similarly, a positively invariant set S C IRl is said to be a C"; (r 2: 1) positively invariant manifold ifS has the structure ofa C" differentiable manifold. The term C" means continuously differentiable at every xES for every integer r 2: 1. Roughly spoken, Definition A.7 states that an invariant set S where the sur­ face is r-times continuously differentiable at every xES is called an O" invariant manifold. 184 An Overview of Bifurcation Theory

To avoid the technical details concerning our understanding of the term "mani­ fold" we give here only a intuitive description of a manifold. A manifold is a set that has locally the structure of Euclidean space. The most familiar examples of mani­ l folds are the sphere or the torus which are embedded in IR • Furthermore, we obtain manifolds by the solution of a system of nonlinear equations, the so-called solution manifold. In contrary the cone is not a manifold. Although each point but one on the cone has a small Euclidean neighborhood but no neighborhood of the vertex has Eu­ clidean structure. For a more thorough treatment of a manifold we refer the reader to Guillemin and Pollack (1974) or Brocker and Janich (1990). In the case of dy­ namical systems we get two types ofmanifolds. Firstly, a linear vector subspace and secondly, a smooth surface embedded in IRl which can be represented locally with the implicit function theorem as a graph. Since the surface has no singular points if the surface is smooth and the derivative of the function representing the surface has maximal rank, the surface can be locally represented by a graph using the implicit function theorem (for more details see (Wiggins, 1990, p.14-15)). Once we have found a solution such as an equilibrium orbit or periodic orbit we want to find out if the solution is stable. An invariant set So must attract nearby solutions to classify as a stable orbit.

Definition A.S. An closed invariant set So is called stable if

(i) for any sufficient small neighborhood U ::J So there exists a neighborhood V ::J So such that <.pt x E U for all x E V and all t > 0;

(ii) there exists a neighborhood U ::J So such that <.pt x ...... Sofor all x E U as t ...... +00.

Property (i) in Definition A.8 is the so-called Lyapunov stability. This property states that an invariant set Soin order to be Lyapunov stable must assure that nearby orbits cannot leave its neighborhood. Property (ii) is the so-called asymptotic stabil­ ity which states that an invariant set So attracts nearby orbits as time goes by. Definition A.8 gives us a clear designation of stability. Now, we provide a suffi­ cient condition for an equilibrium X o to be stable by the following theorem .

l Theorem A.I. Consider a dynamical system {IR , <.pt} defined by

x = j(x), (A.4) where j is smooth. Suppose that it has an equilibrium xO,i.e.f(xO) = 0, and denote by A the Jacobian matrix of j(x) evaluated at the equilibrium, A = j ",(XO). Then XOis stable ifall eigenvalues J.Ll, J.L2, ••• , J.L l ofA satisfy Re J.L < O. A.2 Some Definitions and Results 185

Proof Hint : The theorem can be proved by construction of a Lyapunov function. Compare with Wiggins (1990) 0

Theorem A.l provides us with a method to answer the question whether a fixed point is stable. Moreover, Theorem A.l declares that we can analyze the stability of an equilibrium without explicity solving the system (AA). It suffices to evaluate the Jacobian matrix A at the equilibrium point and to determine if for all eigenvalues the real part is negative. In this section we will see that the inspection ofthe eigenvalues calculated as the solution of the characteristic equation det(A - J-LI) = 0), where I is the n x n identity matrix, reveals additional local information about the dynamical behavior at the fixed point. Now, let us introduce a definition which help us to classify the behavior of dy­ namical systems. This is useful, since very often we want to investigate a dynamical system which is not tractable. Thus, if we are able to classify certain dynamical types according to their dynamical structure, then perhaps we can define an equiva­ lence relation between a dynamical system whose dynamical properties we already know and a dynamical system we want to study. Therfore, we have to specify when we consider two dynamical system as being equivalent. In bifurcation theory the is­ sue whether two dynamical systems are similar is answered in terms ofconjugacies or topological equivalence. Both terms provide us with a characterization when two dynamical systems have the same qualitatively dynamics.

I Definition A.9. A dynamical system {IR ,

I I We get the following dynamical consequences for {IR ,

I I (i) fixed points of {IR ,

I I (ii) To-periodic orbits of {IR ,

(iii) The eigenvalues are the same in both systems except for a positive multiplica­ tive.

Note that both orbits in (ii) have the same periods since the systems are conju­ gate . But if both systems are just topologically equivalent then the periods of both orbits in (ii) need not to be equal. 186 An Overview of Bifurcation Theory

For getting an idea what happens under a continuous transformation of dynam­ ical systems, consider Figure A.l where both phase portraits show qualitatively the same dynamics around the same spiral sink but in different coordinate systems.

Figure A.I. Topological equivalence

Definition A.I0. Let L, l+, l« be the numbers of eigenvalues ofJacobian matrix A with negative. positive and zero real part. respectively. An equilibrium is called hyperbolic if lo = O. that is, if there are no eigenvalues on the imaginary axis. A hyperbolic equilibrium is called a hyperbolic saddle ifL, l; #- O. Note that hyperbolicity is a generical property of an equilibrium. That means that the system preserves it topological classification against perturbations at the equilibrium. For making the point clear let us consider an example which we have drawn from (Kuznetsov, 1995, p.63).

Example A.l (Persistence of a hyperbolic equilibrium). We assume that Xo is an l hyperbolic equilibrium of the dynamical system {IR , tpt} defined by x = f(x) , (A.5) where f is smooth . For an equilibrium we get f(xO) = O. Now let us introduce a one-parameter perturbation of system (A.5) by x= f(x) + e g(x) , (A.6) where the function g is smooth and e is a small parameter, with the property that for e = 0 in (A.6) we obtain our original system (A.5). The system (A.6) has an equilibrium x(e) for all sufficiently small lei such that x(O) = x" . We can write the equation defining an equilibria for (A.6) by A.2 Some Definitions and Results 187

F(x, e) = f(x) + e g(x) = 0, (A.7)

with F(xO,0) = O. Additionally, we get Fx(xO ,0) = fx(xO) = Ao, thus, Ao is the Jacobian matrix of (A.5) at the equilibrium point a", Moreover, we obtain det Ao f= 0, since the equilibrium XO is hyperbolic . Therefore, the Implicit Function Theorem guarantees the existence of a smooth function x = x(e), x(O) = z", satisfying

F(x(e), e) = 0 (A.8)

for small values of 14 Now, the Jacobian matrix of x(e) in system (A.6) is

A. = (df(X) + e d9(X)) I' (A.9) dx dx x=x(€)

which depends smoothly on e and coincides with Ao in (A.5) at e = o. Note that the eigenvalues of the matrix A. depend smoothly on the parameter. Thus, the eigen­ values will vary continuously under the variation of the parameter e. Hence, x(e) has no eigenvalues on the imaginary axis for all sufficiently small lei, since it has no such eigenvalues at e = O. Furthermore, the number of stable and unstable eigen­ values of A€ are the same for all lei small enough . According to the next theorem we can conclude that both systems are topologically equivalent and that hyperbolic equilibria are persistent or generic. o

Theorem A.2. The phase portraits ofa dynamical system (A.5) neartwo hyperbolic equilibria XO and yO, are locally topologically equivalent iff these equilibria have the same number ofLand1+ eigenvalues with Re f.L < 0 and with Re f.L > O.

Evidently, Theorem A.2 gives us necessary and sufficient conditions under which phase portraits near hyperbolic equilibria have the same topological clas­ sification. We have used the expression "bifurcation" serval times so far without giving the exact meaning now let us introduce its definition.

Definition A.n. The appearance ofa topologically nonequivalent phase portrait under variation ofparameters is called a bifurcation.

1 If we consider a region which depends on a parameter Q, i.e. U", C IR , then two

things can happen in the phase plot when we let the parameter Q vary. The system does not change its dynamical structure in Uo or it changes its topological type. Such a critical point in the parameter space where we observe a change in the topological structure is called a bifurcation. This means that the system is structural unstable at 188 An Overview of Bifurcation Theory the critical value. Thus, the study of bifurcation theory can be viewed as the study of structural unstable dynamical systems. But before we can study structural unstable systems we need to say when we consider two dynamical systems as close. For this purpose we introduce the definition of G'e-close or just G'-close. First, consider the fol1owing two dynamical continuous-time systems

± = f(x) , (A .10) and

± = g(x) , (A.ll) where both f and 9 are smooth.

Definition A.12. The distance between (A.lO) and (A.II) in a closed region U C IR1 is a positive number d, given by

d, := ~~e {lIf(X) - g(x)1I + IId~~) _ d~~) II} · (A.l2)

The system are e-close in U ifd, :::; e.

The appearance of the first derivatives in the G'-distance in Definition A.I2 ensures that close systems have nearby equilibria of the same topological1y type.

Definition A.13 (Andronov's structural stability). A system (A.lO) defined in a region Dc IR1 is called structurally stable in a region Do C D ifforany sufficiently G'-close in D system (A. 11) there are regions U, V cD, Do c U such that (A. 10) is topological equivalent in U to (A.11) in V .

The next theorem states that for 2-dimensional dynamical systems structural stability is a generic property. This result provides us with precise conditions under which dynamical systems are structurally stable.

Theorem A.3. A smooth dynamical system

± = f(x) , (A.l3) is structurally stable in a region Do C IR2 iff

(i) the number offixed points and periodic orbits in Do is finite and all ofthem are hyperbolic;

(ii) there are no orbits connecting saddle points. A.2 Some Definitions and Results 189

Notice that a similar result for l-dimensional systems with 1 > 2 does not exist.

Example A.2. As an example for structural stability and for a bifurcation point let us investigate the following coupled system of differential equations. x=y, (A. 14) if = -x - ey.

We obtain for this system a hyperbolic fixed point at the origin for e > 0 and e < O. Notice, that for both parameter regions the system is structurally stable . In the former case we have a spiral sink fixed point and in the latter case we have a spiral source fixed point (see the discussion below about equilibria types). At e = 0 the system has a nonhyperbolic fixed point surrounded by a family ofperiodic orbits with frequency 1. At this point the system radically changes its dynamical structure. Thus, e = 0 is the critical parameter value where a bifurcation occurs.

Figure A.2. Left: Structurally stable for e > O. Middle: Bifurcation point at e = O. Right: Structurally stable for e < O.

Now let us introduce and discuss the well known Hartman-Grobman Theorem Theorem A.4 (Hartman-Grobman). Assume that the dynamical system x = f(x) , x E ffil has a hyperbolic fixed point x", i.e. the Jacobian matrix A = f.(xO) has no eigenvalues on the imaginary axis (10 = 0). The flow

The Hartman-Grobman Theorem states that in a sufficiently small neighborhood of a hyperbolic fixed point of a nonlinear system its exhibits the same dynamic be­ havior as the associated linearized one. This theorem has the following consequence in studying the dynamic behavior for nonlinear systems. If we linearize a nonlinear system at its hyperbolic fixed point then we can conclude that the dynamics in a 190 An Overview of Bifurcation Theory sufficiently small neighborhood of the fixed point XO is similar to the dynamics in the nonlinear case. Such a conclusion is not permissible if we linearize the nonlin­ ear system at its nonhyperbolic fixed point. In this case the dynamic structure of the linearized system can be totally different from the dynamics of the original system.

In the previous discussion we have seen that the dynamic structure of the lin­ earized system i = A s , e E IRI of Theorem AA depends on the eigenvalues of A. According to Definition A.10 we have also seen that we can distinguish equilibria in hyperbolic and nonhyperbolic types. Now, let us identify for the hyperbolic and nonhyperbolic system certain equilibria types. We restrict our investigation of the eigenvalues to 2-dimensional continuous-time dynamical systems of (AA).

Classification of hyperbolic equilibria

(1) Stable equilibria

(i) Eigenvalues are real, equal and negative with two independent eigenvec­ tors. The equilibrium is calledfoeus sink.

(ii) Complex eigenvalues with negative real part. The equilibrium is called spiral sink.

(iii) Eigenvalues of opposite sign and real. The equilibrium is called saddle. See Figure A.3.

(iv) Eigenvalues are real, unequal and negative. The equilibrium is called nodal sink. See Figure A.3.

(v) Eigenvalues are real, equal and negative with one eigenvector. The equi­ librium is called improper nodal sink.

(2) Unstable equilibria

(i) Eigenvalues are real, equal and positive with two independent eigenvec­ tors. The equilibrium is calledfoeus source.

(ii) Complex eigenvalues with positive real part. The equilibrium is called spiral source.

(iii) Eigenvalues are real, unequal and positive. The equilibrium is called nodal source.

(iv) Eigenvalues are real, equal and positive with one eigenvector. The equi­ librium is called improper nodal source A.3 Local and Global Bifurcations 191

Figure A.3. Left: Phase plane of a saddle. Right: Phase plot of a nodal sink

Classification of nonhyperbolic equilibria

(i) Nonzero purely imaginary eigenvalues. The equilibrium is called center or spiral equilibrium.

(ii) For saddle nodes we have a single zero eigenvalue.

A.3 Local and Global Bifurcations

Now, we are at the point where we can introduce and discuss the definitions of a simple type of local bifurcation and two types of global bifurcations which are rel­ evant for studying the dynamic resource management problem. Recall that by the term "local" we mean a bifurcation occurring in a small neighborhood of a fixed point. The simplest type of local bifurcations in n-dimensional systems are the fold and the Andronov-Hopfbifurcation or just Hopf-bifurcation. We have already met a Hopf-bifurcation while we have studied structural stability in Example A.2 (see Fig­ ure A.2). Compared to local bifurcations the so-called global bifurcation cannot be detected by the study of the orbit structure in the neighborhood of a fixed point. For global bifurcation we observe a change of the dynamical structure in the whole sys­ tem. Global bifurcation appears when there are orbits connecting hyperbolic equi­ libria in smooth dynamical systems. These orbits types are called homoclinic and heteroclinic. In case where we observe a homoclinic or respectively, a heteroclinic orbit in a phase portrait the topological classification has changed due to the param­ eter variation and we call the resulting bifurcations a homoclinic or respectively, heteroclinic bifurcation. 192 An Overview of Bifurcation Theory

Here, we give the definition of a Hopf-bifurcation.

Definition A.I4. The bifurcation corresponding to the presence ofJ.LI.2 = ±wo, Wo > O. is called an Andronov-Hopfbifurcation.

Resume example A.2. For the critical value e = 0 we obtain a fixed point with two complex eigenvalues J.LI' J.L2 having zero real part which indicates a nonhyper­ bolic fixed point. Thus, due to Definition A.14 we conclude that example A.2 reveals an Andronov-Hopfbifurcation for the value e = O. Before we study global bifurcations in more detail we introduce two invariant sets for an equilibrium:

WS(Xo) := {x :

x = f(x), (A.16) where f is a smooth function.

Definition A.IS. WS(xo) is called the stable set ofxo• while WS(xo) is called the unstable set ofx o.

Definition A.I6. An orbit I'°starting at a point x E IR.t is called homoclinic to the equilibrium point Xo ofsystem (A. 16) if

The phase plot in Figure A.4 on the facing page shows an example for a homo­ clinic orbit connecting the saddle point to itself. The phase plot in Figure A.5 on the next page is an example for a heteroclinic orbit that connects a saddle point with a spiral sink. A homoclinic orbit r°to the equilibrium X o belongs to the intersec­ tion of its stable and unstable sets, that is, r o c WS(xo) n WU(xo). Analogous, heteroclinic orbit I', to the fixed point Xl and X2 belongs to WU(x l ) n WS(x2 ). It is noteworthy that homoclinic or heteroclinic orbits can also connect nonhyperbolic equilibria. Due to Theorem A.3 the next result is not very surprising.

Lemma A.!. A homoclinic orbit to a hyperbolic equilibrium of (A.16) is struc­ turally unstable. A.3 Local and Global Bifurcations 193

Lemma A.I states that it is possible to disturb the dynamical system with an orbit I'0 that is homoclinic to the equilibrium Xo such that the phase plane in a neighborhood of I'0 U Xo changes its topological structure compared to the original system and therefore a bifurcation will be observable in the phase portrait.

x '= y 0 =0 Y'=x -il-oy

" '~ -s, ~.-=..-;;x,-",-_,,· -s, ~...,

o 0.5 1.5 x

Figure A.4. Homoclinic orbit in the plane.

x ' =y 0=2 y' =x -il-o y 0.3 , , , -, " \ , \ 0.2 .. " ~ ' '. . 1\ : ~ . "" , '. \ , 0.1 .'\" , , \:y . , . ' ; >0 0 , I ; , ,0: , " , , , I , , , -0.1 i . 1 . . , " ; • i 1 " -0.2 \ 1 \ ~ \ , ; J \ • " .. , -0.3 o 0.5 1.5 x

Figure A.S. Heteroc1inic orbit in the plane.

Notice that an important aspect of heteroclinic and homoclinic orbits is that this bifurcation drastically changes the topological structure of the dynamical system. List of Figures

1.1 HSW payoffdata set ...... 7

2.1 Optimal state and control time path for the centralized model 21 2.2 Saddle path equilibrium ...... 23 2.3 Phase portrait for the centralized approach ...... 25 2.4 Optimal state and control time path for the differential game 31 2.5 Phase portrait for the differential game with n = 8 33 2.6 Phase portrait for the differential game with n = 2 34 2.7 Saddle connection...... 37 2.8 Structurally stable for n > 1 and B = 1/10. . 37 2.9 Saddle-node bifurcation...... 38 2.10 Structurally stable forn = 1 and C E (0,9.5) 38

4.1 Best reply function a(b) with increasing part . 67 4.2 Nonempty o-core . 77 4.3 Different a- and (3-core ...... 82 4.4 Empty ,-core for the example 4.6 92 4.5 Empty ,-core for the example 4.7 93

5.1 Symmetric common pool game. . 109 5.2 Core for a symmetric common pool game 112 5.3 Asymmetric common pool game ... .. 115 5.4 Core for a asymmetric common pool game . 116

6.1 Core for a 2-convex common pool game 161

A.1 Topological equivalence 186 A.2 Structural stability. ... 189 196 List of Figures

A.3 Left: Phase plane of a saddle. Right: Phase plot of a nodal sink 191 A.4 Homoclinic orbit in the plane. 193 A.5 Heteroclinic orbit in the plane...... 193 List of Tables

4.1 Coalitional values for an o-common pool TU-game 73 4.2 Coalitional values for a ;3-common pool TU-game . 79 4.3 Coalitional values for a ,-common pool TU-game . 91

5.1 Coalitional values for an asymmetric case .... 114 5.2 Coalitional values for a common pool TU-game . 116

6.1 Coalitional values for a l -convex common pool TU-game 159 6.2 Coalitional values for a non k-convex common pool TU-game 159 6.3 Coalitional values for a 2-convex common pool TU-game 160 6.4 Coalitional values for the bankruptcy game ofexample 6.10 . 167 6.5 Coalitional values for the common pool TU-game of example 6.10 167 6.6 Coalitional values for the bankruptcy game of example 6.11 170 6.7 Values for the total claims of example 6.11 171 6.8 A not representable common pool TU-game . 171 6.9 Values for the corresponding new TU-game . 172 6.10 Values for the new TU-Game of example 6.13 174 Bibliography

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