Linearization and Invariant Manifolds

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1 CHAPTER 4 Linearization and Invariant Manifolds Coordinate transformations are often used to transform a dynamical system that is analytically intractable with respect to its native coordinates, into a new coordinate framework in which the analysis is tractable. These transformations, therefore, are very useful tools in the analysis of nonlinear dynamical systems. One of the most commonly used transformations is based on the Taylor series expansion of the native system s vector field about an equilibrium point. When the higher order terms in this expansion are neglected, one obtains a linearization of the original system whose analysis and control are well understood. This chapter investigates the sense in which a nonlinear system is locally equivalent to its linearization. To do this, we discuss various notions of equivalence used in relating different dynamical systems. We then introduce the system s linearization and prove a basic theorem that characterizes when a nonlinear system is locally equivalent to its linearization. The main finding from this study will be that this equivalence is well defined at hyperbolic equilibria of the nonlinear system and for such equilibria it is possible to locally map flows on the stable and unstable invariant manifolds of the system onto flows of the stable/unstable eigensubspaces of the linearization. This result, therefore, provides considerable insight into when linear control methods can be used to stabilize nonlinear systems. An important consequence of our study, however, is that it also identifies how one should handle non-hyperbolic equilibria. In particular, we show that about a non-hyperbolic equilibrium it is possible to reduce the original system to a lower dimensional system defined on a center manifold. We will later see that the stabilizability of this reduced system on the center manifold is critical for the stabilization of the entire system. 1. Mathematical Preliminaries The systems we ll be studying are multivariate continuous-time systems evolving over Euclidean n-space, R n. In general, however, the motions generated by these systems will not cover the entire space. They are often confined to a subset of R n that is often called the system s domain, D. The restriction of a physical system s states to a domain D occurs in many physical systems. In a simple pendulum of length ` shown in the left plane of Fig. 1, for example, the position of the pendulum bob in Euclidean 2-space is denoted by the ordered pair (x, y). That position, however, is constrained to lie on a circle that satisfies x 2 + y 2 = `2. As another example, a spacecraft s attitude is often described by a direction cosine matrix, R 2 R 3 3. The elements of this matrix equal the inner product of the vehicle s body axes with the inertial frame s axes. The dynamics governing R are described by the following set of differential equations, Ṙ = S(!)R where R 2 R 3 3 is the direction cosine matrix,! 2 R 3 is the vector of body angular velocities, and S(!) is a skew 71

2 72 4. LINEARIZATION AND INVARIANT MANIFOLDS antisymmetric matrix whose components are the vehicle s angular velocities. These equations describe how R changes over time for a given set of angular velocities, but R is not free to range over all of R 3 3. It is constrained to represent transformations of an orthogonal set of coordinate frames and as such is restricted to a 3D group of rotation matrices called SO(3). A final example is shown on the right pane of Fig. 1. This pane shows the attractor for a tritrophic food web based on the Rosenzweig-MacArthur model [Ros71]. The particular attractor shown in the figure exhibits the bursting phenomenon discussed earlier in chapter 1. What may be of greater interest in this case is the topological nature of the attractor, rather than the system s behavior over the entire state space. 0-5 l (1,0,0) ( , ,0) (0,0,0) (0.8861,0.125, ) (x,y) FIGURE 1. (left) pendulum bob location constrained to a circle - (center) direction cosines for spacecraft orientation confined to 3D symmetric group (SO(3)) of rotations - (right) asymptotic behavior of Rosenzweig-MacArthur Food Web in R 3 We need, therefore, to establish the notational conventions and results used in studying the motion of dynamical systems constrained to subsets of the state space. In particular, let f : U! R be a function mapping an open connected subset (domain) U of R n onto R. The value of f at x =(x 1,...,x n ) will be denoted as f(x) =f(x 1,...,x n ). The function f is said to be smooth or C 1 if its partial derivatives with respect to all orders exist and are continuous. The function f is said to be analytic if it is C 1 and at each point p 2 U there exists a neighborhood, N (p), in which the Taylor series expansion of f at p converges to f(x) for any x 2 N (p). A mapping F : U! R m is a collection, {f 1,...,f m }, of functions f i : U! R for i =1,...,m. Let U, V R n be open connected sets in R n. We say the mapping F : U! V is a diffeomorphism if it is bijective and both F and F 1 are smooth. We say F is a homeomorphism if it is bijective and both F and F 1 are continuous functions (i.e. C 0 ). Consider a domain U R n and define the collection, x = {x 1,...,x n }, of functions x i : U! R as a coordinate system for U. Let {e i } n i=1 be a collection of elementary basis vectors for Rn. A system of coordinates is said to be admissible if the mapping x : U! R n for any p 2 U is a diffeomorphism that takes values x(p) =x 1 (p)e x n (p)e n Essentially this means the coordinates are locally equivalent to Euclidean n-space.

3 1. MATHEMATICAL PRELIMINARIES 73 Let U be a domain in R n with an admissible coordinate system, {x 1,...,x n }. Let I be an open interval in R. Consider any point p 2 U, then we say the differentiable function : I! U is a curve leaving point p if (0) = p. The velocity vector, v =(v 1,...,v n ), of at p in coordinate system {x i } is defined in a component-wise manner as (40) v i = dx i( (t)) dt t=0 for i =1, 2,...,n. Given two curves 1, 2 : I! U that leave the same point p 2 U (i.e. 1(0) = 2 (0) = p), one says these curves are tangent to each other if the distance between points satisfies 1(t) 2(t) t! 0 as t! 0. A function that satisfies this limit is sometimes said to be o(t) (little o ). It essentially asserts that the differences goes to zero faster than a linear rate of decrease. A necessary and sufficient condition for two curves to be tangent at p is that their velocity vectors at point p be equal. This can be proven using a multivariate version of Taylor s theorem [TM55], which we state below without proof. To state this result it will be convenient to introduce some notational conventions. In particular a multi-index is an n-tuple of non-negative integers. If is a multi-index, we define = n,!= 1! 2! n! We refer to as the order of the multi-index. Multi-indices are used to represent multivariate monomials in a more compact form. Let x =(x 1,x 2,...,x n ) 2 R n, then x = x 1 1 x 2 2 x n n These indices are also valuable in simplifying how we write the partial derivative of multivariate functions. In particular, we can use the f f 1 2 n n With these notational conventions we can now write down the Taylor series expansion for a multivariate function in a more compact form. In particular, let f : R n! R be a C k+1 function on an open convex set U. Let p 2 U and p + h 2 U, then (41) f(p + h) = X applek f(a)h + R a,k (h) where the summation is done over all multi-indices of order less than or equal to k. Taylor s theorem gives explicit equations for the remainder term, R a,k (h), but these equations may be bounded as R a,k (h) apple M (k + 1)! h k+1 f(x) applem for x 2 U. The key point to note is that if one considers R a,k (h) h k apple M (k + 1)! h which goes to 0 as h goes to zero. This means that the remainder term is of order o( h k ), or rather that the remainder goes to zero faster than h k.

4 74 4. LINEARIZATION AND INVARIANT MANIFOLDS With Taylor series expansion for f in equation (41), we can now state a theorem that establishes necessary and sufficient conditions for two curves at p to be tangent. Essentially, this theorem confirms our intuition that tangent curves have the same derivative (i.e. velocity vector) at the point of tangency. THEOREM 37. (Tangent and Velocity Vector) Two curves, 1, 2 : I! U are tangent at p if and only if their velocity vectors at point p are equal. Proof: Since 1 and 2 are differentiable they have Taylor series expansions about h =0of the form, i(h) =p + for i =1, 2. Taking the norm of the difference shows that d dh i (h) t + o( h ) h=0 1(h) 2(h) t apple d dh 1 (0) d dt 2 (0) + o( h ) h and taking the limit as h!0 we can conclude that o( h )/ h and 1(h) 2(h) / h go to zero, which implies d dh 1 d (0) dh 2 (0) =0so that the velocity vector for both curves must be equal. } The tangent space of a set U R n at a point p 2 U is the set of velocity vectors for all curves leaving p and is denoted as TU p. The elements of this set are called tangent vectors. Now consider a mapping F : U! V where U R n and V R m. The differential of F at p 2 U is a mapping df p : TU p! TV F (p) such that for any curve (42) : I! U leaving p, the mapping takes values d (h) df ( (t)) df p = dh h=0 dt t=0 The differential of F is a linear transformation between the tangent spaces TU p and TV F (p). This assertion is stated and proven below. THEOREM 38. (Linearity of Differential) Given a mapping F : U! V, the differential of F at p 2 U defines a linear transformation between the tangent spaces TU p and TV F (p). Proof: Let ẋ i denote the components of the velocity vector of the velocity vector ẏ of f (42) yields, (43) at point x. Let ẏ j denote the components of at point y = f(x). Taking the derivative of the composite function in equation ẏ j = apple d dt (f ) j = nx i ẋ i where y j (x 1,...,x n ) for j =1,...,mare functions specifying the mapping f in the coordinates x i. Note that this representation for ẏ j is independent of the choice of the curve linear map from ẋ to ẏ. }. Also note that equation (43) is a Consider a mapping F : U! V with differential at p 2 U of df p : TU p! TV F (p). Both TU p and TV F (p) are clearly linear spaces and since df p is a linear transformation (theorem 38) it has a concrete representation

5 1. MATHEMATICAL PRELIMINARIES 75 as a matrix. If we let {x i } be an admissible system of coordinates for U, then this matrix realization of the differential is called the Jacobian matrix of F at a point x 2 U and is defined by the equation 2 @x = 6 1 The value of the Jacobian at a point p will be An extremely useful result in the study of constrained dynamical systems is the inverse function theorem. This theorem establishes conditions under which a mapping F : U! R n has an inverse function F 1. Essentially, this means that if the equation y = F (x) is written in component form as a system of n n p. 7 5 y i = F i (x 1,...,x n ), i =1,...,n then this system can be solved for x 1,...,x n in terms of y 1,...,y n, provided we restrict ourselves to a small enough neighborhood where the Jacobian is nonsingular. We now state and prove the inverse function theorem. Our purpose in showing the proof is that the inverse function theorem is a result characterizing the existence of a diffeomorphism. Diffeomorphism are, essentially, nonlinear coordinate transformations that preserve the underlying dynamics of a system. Since one of the main results of this chapter is the Hartman- Großman theorem; a result asserting the local equivalence of a nonlinear system to its linearization, we felt it would be useful to see how one actually establishes that such diffeomorphism (coordinate transformations) exist. THEOREM 39. (Inverse Function Theorem) Let U be an open connected subset of R n and consider the C 1 mapping F : U! R n. If the Jacobian is nonsingular at some point p 2 U, then there exists an p open neighborhood N (p) of p 2 U such that V = F (N (p)) is open in R n and the restriction of F to U is a diffeomorphism onto V. Proof: There are three things we need to establish in this proof. We first need to show that for any y in a suitably small ball, W, about F (p) that there is a unique x such that F (x) =y, thereby establishing the existence of an inverse F 1. We next need to show that this set W is open to ensure the continuity of F 1 and then we need to establish that F 1 is differentiable. We can use the contraction mapping principle to establish the existence of F 1. For notational convenience define the function L(v) =df p (v). Since df x is continuous in x at x = p, there is a sufficiently small neighborhood V about p such that sup df x (v) L(v) apple 1 v =1 2 inf L(v) = 1 v =1 2sup w =1 L 1 (w) and by the linearity of the differential of F is a linear transformation (theorem 38) we can deduce that for all v 2 R n and x 2 V that (44) df x (v) L(v) apple v

6 76 4. LINEARIZATION AND INVARIANT MANIFOLDS To each y 2 R n, we now associate a function A y : R n! R n with each y 2 R n that takes values (45) A y (x) =x + L 1 (y F (x)) and note that y = F (x) if and only if x is a fixed point of A y. This suggests that for any y in a ball W about F (p), if we can prove A y is a contraction, then there is a unique x such that y = F (x). From the equation (46) it should be apparent that da y = I L 1 (df x )=L 1 (L df x ). The preceding inequalities imply that da y x(v) apple 1 2 v for any x 2 V and v 2 Rn. So for any w, x 2 V (46) A y (w) A y (x) = apple Z 1 0 Z 1 0 d dt Ay (x + t(w x))dt da y x+t(w x) (w x) dt apple 1 w x 2 which implies that A y is a contraction mapping in V and so for any y 2 F (V )=W, we expect to have at most one fixed point x for which F (x) =y. In this neighborhood, therefore, F has a unique inverse, F 1. To establish continuity of F 1, we recall from theorem 19 in chapter 2, that the map F 1 is continuous if and only if W = F (V ) is open for any open set V. We again do this using the contraction mapping principle. In particular, choose any point p 2 V such that there is an >0 for which the open ball B = N (p) has a closure B V. Let q = F (p) and consider the open ball B (q) in the range space of F. If we can show that B (q) W, then this will mean that W is open. So consider any y 2 B (q) and use the function A y to look at For any x 2 B it then follows that A y (p) p = L 1 (y q) < 1 2 = 2 A y (x) p apple A y (x) A y (p) + A y (p) p < 1 2 x p + 2 apple and so A y (x) 2 B. By equation (46) we can therefore see that A y is a contraction of B and so A y has a fixed point x in B and y = F (x) 2 F (B) F (V )=W, thereby proving B (q) W. This establishes the continuity of F 1. The differentiability of F 1 is noted by recognizing that since df x is a linear transformation, it can be represented as a matrix. The differential of that inverse is the inverse of that matrix and since matrix inversion is a smooth function of the entries of the matrix, we can deduce that F differentiable. } 1 is continuously The Implicit Function Theorem is another useful technical result that follows from the Inverse Function Theorem. It provides conditions under which one can solve a system of equations f(x, y) =0for y as a function of x. It has a number of applications, one of which we will later use in characterizing structurally stable systems. THEOREM 40. (Implicit Function Theorem) h Suppose i f : R m R n! R n is a C 1 function near a point (a, b) 2 R m R n with f(a, b) (a, b) 6= 0. Then {(x, y) 2 W : f(x, y) =0} = {(x, g(x)) x 2 X}

7 2. EQUIVALENCE CONCEPTS 77 for some open neighborhood W of (a, b) in R m R n and some C 1 function g : X! R n where X is a neighborhood of a 2 R (x) = and g is smooth when f is smooth. apple (x, (x, Proof: Define F (x, y) =(x, f(x, y)) and compute df (a,b) =det( (a, b)) 6= The Inverse function theorem therefore gives a C 1 inverse F of (a, b) and W of (a, 0) in R m R n. The set X = {x 2 R m point x 2 X we can see that F 1 : W! V for some open neighborhoods V : (x, 0) 2 W } is open in R m and for each 1 (x, 0) = (x, g(x)) for some point g(x) 2 R n. Moreover, {(x, y) 2 W : f(x, y) =0} = (F 1 F )(W \ f 1 {0}) One can readily verify that g is C 1 with = F 1 (W \ (R m {0})) = {(x, g(x)) : x 2 X} 1 ] i (x) (x, j for i =1,...,n, j =1,...,mand x 2 W. The follows from differentiating the identity f(x, g(x)) 0 on W and using the chain rule. The smoothness of g follows from the smoothness of f by repeatedly differentiating this identity. } 2. Equivalence Concepts To introduce the way in which two different dynamical systems might be equivalent, we start from a simpler problem in which we simply ask the sense in which two linear transformations are equivalent or similar. This notion of similar linear systems should be familiar to most readers since we ve assumed they have taken courses in linear algebra or linear systems theory. So consider two linear spaces, X and Y over the same field and consider the linear transformations F 2 L(X, X) and G 2 L(Y,Y ). Recall that any finite dimensional linear transformation may be represented concretely as a matrix, so without loss of generality we can assume that F and G are matrices defined over the same field. We say these linear transformations (i.e. matrices) are similar if there exists a nonsingular linear transformation T 2 L(X, Y ) such that TF = GT The matrix T is called a similarity transformation Let y be an eigenvector of F with eigenvalue (i.e. F[y] = y). Note that for any similarity transformation, T, between F and G that Ty = TFy = GTy

8 78 4. LINEARIZATION AND INVARIANT MANIFOLDS This implies that Ty is an eigenvector of G with the same eigenvalue. The linear transformations F and G, therefore have the same eigenvalues with eigenvectors related through the nonsingular similarity transformation T. Since, T is nonsingular, this means there is a one-to-one and onto mapping between the eigenvectors of G and F, and one can say that the eigenvectors of G are simply a relabeling of the eigenvectors of F in a new coordinate frame. Now let us apply these ideas to a pair of linear dynamical systems of the form ẋ(t) =Fx(t), ẏ(t) =Gy(t) If we assume that F and G are similar through the similarity transformation T, then we can think of T as a coordinate transformation between the system states x and the system states y. Moreover, we have just seen that the eigenvalues of F and G = TFT 1 are the same. So these two dynamical systems are similar in the sense that the eigenvalues of their modes is unchanged. This notion of similarity between linear dynamical system is frequently used to transform a linear dynamical system to a canonical form, from which one can more easily deduce system stability. Our objective is to generalize this notion to nonlinear dynamical systems. In particular, consider a function f : R n! R n.we say f is a C k diffeomorphism if f is invertible and both f and f 1 are C k functions. Let us now consider two C r functions, f : R n! R n and g : R n! R n. These two functions are said to be C k -conjugate (where k apple r) if there exists a C k -diffeomorphism h : R n! R n such that h f = g h where represents function composition. If k =0so that h and its inverse, h 1 are simply continuous, then we say f and g are topologically conjugate. If the transformation also preserves the orientation of time, then we say the systems are flow equivalent or topologically equivalent. The preceding example of similar matrices is a special case of conjugancy in which f and g are linear transformations. The following theorem shows in which sense the notion of conjugancy extends our notion of similar linear systems to nonlinear dynamical systems. THEOREM 41. Consider the dynamical systems ẋ = f(x) and ẏ = g(y) where f and g are C k conjugate and k 1. Then the trajectories of the f system map to trajectories of g system under h. Moreover, if x is an equilibrium point of f, then the eigenvalues of Df(x ) are equal to the eigenvalues of Dg(h(x )). Proof: We have two ODEs ẋ = f(x) and ẏ = g(y) with flows systems are topologically conjugate we know that t and t, respectively. Since these two h (t, x) = (t, h(x)) Differentiate this equation with respect to t to obtain Dh(x)f(x) =g(h(x)) and differentiating one more time yields, D 2 h(x)f(x)+dh(x)df(x) =Dg(h(x))Dh(x)

9 3. LINEARIZATION OF DYNAMICAL SYSTEMS 79 Evaluating at x yields Dh(x )Df(x )=Dg(h(x ))Dh(x ) ) Df(x )=[Dh(x )] 1 [Dg(h(x ))] [Dh(x )] All of the above objects are real-valued matrices and so the above equations show that [Df(x )] and [Dg(h(x ))] are related through a similarity transformation and so have the same eigenvalues. } 3. Linearization of Dynamical Systems Consider a time-invariant dynamical system whose state trajectory x : R! U with U R n satisfies the ordinary differential equation ẋ(t) =f(x(t)) where f : U! R n is assumed to be Lipschitz. We say that a point x 2 U is an equilibrium point for this system if f(x )=0. It is also common to say that x is a critical point of f. If x is any point such that f(x) 6= 0, then we say x is a regular or nonsingular point of f. Without loss of generality, we will assume the equilibrium x =0. This can be justified through a change of variables. In particular, let y = x x. Then y satisfies the differential equation ẏ = d dt (x x )=ẋ = f(x) =f(y + x ) We let g : U! R n take values g(y) =f(y + x ), so that the motions of the differential equation ẏ = g(y) are also the motions of ẋ = f(x) through a translation x = y + x. If f has a critical point at x then clearly g has its critical point at 0, so we can assume without loss of generality that we are studying a system with an equilibrium at 0. So let f(0) = 0, then from elementary calculus, one can use the mean value theorem [TM55] to express the components of f(x) for all x in a neighborhood, N (0), of the origin f i (x) =f i (0) zi x where z i is some vector in R n that lies on the line segment connecting x to the origin. Since f(0) = 0 this implies that for all x 2 N (0) and i =1, 2,...,n f i (x) zi x 0 x + zi i 0 which in matrix-vector form becomes f(x) = Ax + g(x) where we define the matrix A 2 R n n as j 0 = Df x (0)

10 80 4. LINEARIZATION AND INVARIANT MANIFOLDS The mapping g =(g 1,...,g n ) consists of functions g i : U! R for i =1, 2,...,nthat take the value " i g i (x) = 0 From the Taylor series with remainder equation (41) we know that g i (x) is o( x ) function which means g(x) x! 0 as x!0 This means that in a small enough neighborhood of the equilibrium point, the vector field f is dominated by the linear term Ax, rather than the nonlinear term g(x). So the conjecture is that in this neighborhood one can approximate the original nonlinear system by its linearization (47) ẋ 0 x := Ax The question to be investigated now is in what sense this linearization will be equivalent to the original nonlinear system. In particular, we will answer this question in terms of the topological conjugacy of the respective system s flows. The linearized system ẋ = Ax is well understood [AM06]. We know that solutions can be written as where the matrix exponential e At is x(t) =e At x 0 (48) e At = I + At + 1 2! A2 t ! A3 t 3 + We also know that the state space R n can be decomposed as the direct sum of three subspaces denoted as E s, E u, and E c. These subspaces are invariant in the sense that if x(0) lies in one of them then x(t) remains in the subspace for all future time. These three invariant subspaces are defined as E s = span{e 1,...,e s } E u = span{e s+1,...,e s+u } E c = span{e s+u+1,...,e s+u+c } where s + u + c = n and where {e 1,...,e s } are the generalized eigenvectors of A corresponding to the eigenvalues of A having negative real (i.e. stable) parts, {e s+1,...e s+u } are the generalized eigenvectors of A whose associated eigenvalues have positive real (i.e. unstable) parts, and {e s+u+1,...,e s+u+c } are the generalized eigenvectors of A whose associated eigenvalues have zero real (also known as center ) parts. We refer to E s, E u, and E c as the stable, unstable, an center eigensubspaces of the system, respectively. As mentioned above these subspaces are invariant with respect to the linear system s flow. Moreover solutions starting in E s asymptotically approach the origin as t!1(hence the name stable) and solutions starting in E u asymptotically approach the origin as t! 1(hence the name unstable).

11 4. INVARIANT MANIFOLDS OF HYPERBOLIC SYSTEMS Invariant Manifolds of Hyperbolic Systems The eigensubspaces of the linearized system e At are the tangent spaces of their nonlinear counterparts, at the equilibrium. The nonlinear counterparts are sets known as manifolds. The objective of this section is to establish the existence of such manifolds for hyperbolic systems. Consider a topological space, M and define a coordinate chart on M by the pair (U, ) where U is an open set of M and is a homeomorphism of U onto an open set of R n. It is often convenient to think of as a collection, { i } n i=1, of functions i : U! R that represents the ith coordinate function. If p 2 U, the n- tuple of real numbers ( 1 (p),..., n(p)) is therefore called the set of local coordinates of p in the coordinate chart (U, ). Since these are maps to R n, one can think of these coordinate charts as straightening out the coordinates maps for U so they fit on a Euclidean coordinate grid as shown in Fig. 2. U p φ : U R n φ(p) M Rn FIGURE 2. Coordinate Chart Let (U, ) and (V, ) be two coordinates charts on a manifold N where U \ V 6= ;. Let { 1,..., n } be the set of coordinate functions associated with. The homeomorphism 1 : (U \ V )! (U \ V ) that takes (for each p 2 U \V ) the set of local coordinates ( 1 (p),..., n(p)) into the set of local coordinates ( 1 (p),..., n (p)) is called a coordinate transformation. The relationship between,, and 1 is shown in Fig. 3. on U \ V. We say these two coordinate charts are compatible if whenever U \ V 6= ;, the coordinate transformation 1 is a C k -diffeomorphism where k apple n. An atlas on M is a collection A = {(U i, i),i2 I} of pairwise compatible coordinate charts such that [ i2i U i = M. This atlas is said to be complete if it is not properly contained in any other atlas. The set M is said to be a C k -differentiable manifold if it can be equipped with a complete atlas whose coordinate transformations, C k -diffeomorphisms. 1 i j are all So what does the above formal definition actually say. Since the set M is a topological space, one knows which sets are open in that set. For a given open set U its coordinate chart (U, ) may be seen as mapping

12 82 4. LINEARIZATION AND INVARIANT MANIFOLDS ψ : U V R n V M φ : U V R n p R n U ψ(p) φ(p) R n ψ φ -1: φ( U V ) ψ( U V ) FIGURE 3. Coordinate Transformation elements of U onto a point in R n which already has a natural coordinate function. We may therefore view the action of the coordinate map as straightening out the coordinate system on U so that U looks like a part of R n as shown in Fig. 2. Euclidean n-space has a natural notion of differentiability and we want our set M to also have that structure. To make sure that is so we place the other requirement on our coordinate transformations, i 1 j, namely these transformations must be sufficiently smooth or compatible with each other. Essentially, what we are requiring is that the transformation of the coordinate frames impose on different open sets of M are sufficiently smooth that we can consider these sets to be smooth deformations of each other. Informally, we think of a differentiable manifold as a sufficiently smooth surface in Euclidean space such that if we confine ourselves to that surface that the surface locally looks another flat Euclidean space. So let us return to our earlier discussion regarding the invariant subspaces of a linearized system. We know these subspaces are essentially R s, R u, and R c, the question is whether, when we locally integrate the nonlinear system s vector field, the local trajectories leaving a neighborhood of the equilibrium can be viewed as forming invariant differentiable manifolds that map to these subspaces in a natural manner. The answer, of course, is yes and the following theorem states and proves the existence of the stable and unstable invariant manifolds about a hyperbolic equilibrium. There will also be a center manifold associated with the center eigensubspace of a non-hyperbolic equilibrium, but we postpone the statement and proof of this till a later section in this chapter. THEOREM 42. (Local Invariant Manifold - Hyperbolic Equilibrium) Consider the equation ẋ = Ax + g(x) where A 2 R n n has n eigenvalues j for j =1, 2,...,n with Re( j ) 6= 0, g : R n! R n is C k in a g(x) neighborhood of the origin and lim x!0 x =0. Then there exists a neighborhood U of the origin, a C k manifold W s and a C k function h s : P s (U)! E u where P s (U) is the projection of U onto E s such that h s (0) = s (0) = 0, and W s is the graph of h s,

13 4. INVARIANT MANIFOLDS OF HYPERBOLIC SYSTEMS 83 x(t 0 ) 2 W s implies x(t) 2 W s for all t 0 x(t 0 ) /2 W s implies there exists >0, t 1 t 0 for which x(t) > for all t t 1. There also exists a C k manifold W u and a C k function h u : P u (U)! E s, where P u (U) is the project of U onto E u such that h u (0) = u (0) = 0 and W u is the graph of h u. x(t 0 ) 2 W u implies x(t) 2 W u for all t apple t 0 x(t 0 ) /2 W u implies there exists >0 and t 1 apple t 0 such that x(t) > for all t apple t 1. Proof: We need to prove the existence of functions h s and h u whose graphs are manifolds W s and W u.we then need to establish that these manifolds are invariant sets of f. So let E s denote the stable eigensubspace of the linearized system ẏ = Ay. We know this subspace is A- invariant so that AE s E s. Moreover, we also know that E s is also invariant with respect to the flow e At, so that e At E s E s. Finally, for any y s 2 E s, we know that there exists >0and C s > 0 such that e At y s apple C s e t y s, t 0 The same remarks will apply to the unstable eigensubspace E u, so that e At E u E u and for any y u 2 E u there exist constants >0 and C u > 0 such that e At y u applec u e t y u, t apple 0 We now express the solution to the ODE in integral form, Z t x(t) =e A(t t0) x(t 0 )+ e A(t t 0 ) g(x( ))d Since R n is the direct sum of the stable and unstable eigensubspaces (i.e. R n = E s E u ), one can split the state into the sum of two parts x(t) =x s (t)+x u (t) and write an integral equation for x s and x u separately. (49) (50) where g s = P s g and g u = P u g. Z t x s (t) = e A(t t0) x s (t 0 )+ e A(t t 0 x u (t) = e A(t t0) x u (t 0 )+ Z t t 0 e A(t ) g s (x( ))d ) g u (x( ))d We now look for a solution that lies in W s and that therefore remains in a neighborhood of the origin for all t 0. If we look at equation (50) for x u (t), we see note that x u is only impacted by the positive eigenvalues of A and so the only way the solution stays in a neighborhood of the origin is if we let t 0!1. In other words, we can assert lim e At0 x u (t 0 ) apple lim C ue t 0!1 t 0!1 t 0 =0

14 84 4. LINEARIZATION AND INVARIANT MANIFOLDS So we can conclude from equation (50) that with t 0 = 1 for finite t x u (t) = Z t e A(t 1 ) g u (x( ))d will remain close to the origin and adding this to the equation for x s with t 0 =0yields, x(t) =e At x s (0) + Z t e (t 0 ) g s (x( ))d Z 1 e A(t t ) g u (x( ))d Now let x s (0) = x 0 s 2 E s and consider the space C 0 (R 0, R n ) of continuous functions from the half line R 0 =[0, 1) into R n. This space is not a Banach space under the L 1 norm but it is a Banach space under the norm kxk µ = sup e µt x(t) t2[0,1) So we consider the space C 0 µ(r 0, R n ) of continuous functions with bounded µ norm where µ =min(, ). This space confines its attention to functions that decay exponentially at a rate consistent with the eigenvalues of the linearized system matrix A. So let us define an operator F : C µ 0! Cµ 0 F [x](t) =e At x s (0) + Z t e A(t 0 such that ) g s (x( ))d Z 1 e A(t t ) g u (x( ))d Using the same ideas we had in proving the local uniqueness theorem 30 of chapter 3, we can show that F is a contraction on C 0 µ for x 0 s that are sufficiently close to the equilibrium point. So this implies that the integral equation has a unique solution x(t) but we also assert it gives us a function h s : E s! E u. Namely we are stating that x(t) =x s (t)+x u (t) where and x s (t) =e A(t t0) x s (t 0 )+ x u (t) = When t = t 0, we get that x s (t 0 )=x s and which is the desired function. h s (x s )=x u (t 0 )= Z 1 e A(t t Z t Z 1 t 0 e A(t ) g u (x( ))d ) g s (x( ))d t 0 e A(t ) g u (x( ))d We now prove that h s has the desired properties. To prove the first part consider the integral equation x s (t) =e At x s (0) + Z t e A(t 0 ) g s (x( ))d where we ve chosen t 0 =0for convenience. Use the triangle inequality and take the limit as t!1 Z t lim x s(t) apple lim C s e t x s (0) +limc s e µt t!1 t!1 t!1 apple C s lim t!1 e t x s (0) + C s ( µ) lim t!1 e µt kxk µ =0 0 e ( µ)(t ) e µt x( ) d

15 4. INVARIANT MANIFOLDS OF HYPERBOLIC SYSTEMS 85 which means lim t!1 x s (t) =0and consequently h s (0) = 0 because Z 1 h s (0) = lim x u (t) apple lim C u e µt t!1 t!1 Then we differentiate the equation for f to get D xs h s = apple Z 1 e A(t t t e ( +µ)(t C u ( + µ) lim t!1 e µt kxk µ =0 ) D x g u (x( ))D xs x( )d This integral converges uniformly since D xs h s is bounded and so we conclude by the estimate D xs h s (0) = lim t!1 D xs x u (t) =0 Z 1 D xs h s (0) = lim D xs x u (t) apple lim C u e µt t!1 t!1 apple t e ( +µ)(t C u ( + µ) lim t!1 e µt kxk µ =0 To prove the second assertion, consider the integral equation. If x(t 0 ) 2 W s, then x u (t 0 )= Z 1 t 0 e A(t ) g u (x( ))d and x s (t 0 )=x s. Solving equations for x u (t) and x ( t) with this initial data we get x s (t) = e A(t t0) x s + x u (t) = Z 1 e A(t t Z t t 0 e A(t ) g u (x( ))d ) g s (x( ))d ) e µ x( ) d ) e µ x( ) D xs x( ) d Adding these two equations and setting t 0 =0gives the equation that characterizes W s, and so x(t) 2 W s for all t 0. Finally, to prove the final property, if we write the original ODE in integral form then x(t) =e A(t t0) x u (t 0 )+ Z t t 0 e A(t ) g u (x( ))d + e A(t t0) x s (t 0 )+ Z t t 0 e A(t ) g s (x( ))d Let 0 be the smallest positive real part of A s eigenvalues and let be the largest positive real part. By the triangle inequality, we can then write x(t) C u e 0 (t t 0) x u (t 0 ) C u e µt Z t for t C s e (t t0) x s (t 0 ) C s e µt Z t t 0 e ( +µ)(t C u e 0 (t t 0) x u (t 0 ) e µt + µ kxk µ ) e µ x( ) d e ( µ)(t ) e µt x( ) d t 0 C s e (t t0) x s (t 0 ) + µ e µt kxk µ t 0 sufficiently large. This shows that if x u (t 0 ) 6= 0, so that x(t 0 ) /2 S s, then x(t) must eventually leave a neighborhood of the origin. }

16 86 4. LINEARIZATION AND INVARIANT MANIFOLDS How do we compute the stable and unstable manifolds? We derive an equation for h s or h u and approximate these functions by higher order Taylor polynomials. These equations can be written as ẋ = Ax + f(x, y), x 2 E s ẏ = By + g(x, y), y 2 E u We now substitute y = h s and by the chain rule obtain D x h s ẋ = By + g(x, h s ) or D x h s (Ax + f(x, h s )) = Bh s (x)+g(x, h s ) where all functions are only functions of x. This equation is satisfied by h s (x). In a similar way h u (y) satisfies D x h u (By + g(h u,y)) = Ah u + f(h u,y) Example: " d x1 dt x 2 # = " #" x1 x 2 # + " x 2 2 x 2 1 # So we want h s (for stable manifold) that satisfies Dh s (Ax + f(x, h s )) = Bh s + g(x, h s ) where A = 1, B =1, f(x, h s )= h 2 s, and g(x, h s )=x 2. This is We consider a power series solution of with Substituting into the above equation yields, or rather equating coefficients gives dh s dx ( x h2 s(x)) = h s (x)+x 2 h s (x) =ax 2 + bx 3 + dh s dx =2ax +3bx2 + (2ax +3bx 2 + )( x (ax 2 + bx 3 + ) 2 )=ax 2 + bx x 2 2ax 2 3bx 3 + =(a + 1)x 2 + bx 3 + 2a = a +1 ) a =1/3 3b = b ) b =0

17 5. RECTIFICATION AND LINEARIZATION (HARTMAN-GROSSMAN) THEOREM 87 and so h s (x) = x2 3 + o(x4 ) 5. Rectification and Linearization (Hartman-Großman) Theorem The rectification and linearization theorems are two fundamental results regarding the conjugacy of a nonlinear systems flows. In particular, the rectification theorem shows that the flows about a non-critical point (also called a regular point) of the vector field are topologically conjugate to a constant velocity field. In this regard, one can say that the rectification theorem shows that flows around these regular points are uninteresting. The linearization theorem is more commonly referred to as the Hartman-Großman theorem. It shows that the flows around a hyperbolic equilibrium are locally topologically conjugate to the system s linearization. y rectified flow x original flow z t FIGURE 4. Rectification Theorem (right) original flow (left) rectified flow The following form of the rectification theorem applies to a time-varying equation of the form ẋ = f(t, x). The theorem extends the state space of this system by treating time as another state variable,, that satisfies =1. The theorem then asserts that this is topologically conjugate to a dynamical system whose state, y, is constant for all time, z. This relationship between the two flow fields is, perhaps, more apparent in Fig. 4. The right panel shows the flows of the original system in the (x, t) plane. The rectification theorem asserts that these flows can always be straightened out (i.e. rectified) into the straight flows shown on the left panel of the Fig. 4. THEOREM 43. (Rectification Theorem) Let (x 0, 0 ) be a regular point of the vector field ẋ = f(, x) = 1 then there exists a neighborhood U R n+1 of (x 0, 0 ) and a diffeomorphism g : U! V R n+1 such that (y, z) =g(x, ) satisfies the differential equation, ẏ =0 ż =1 Proof: So we define a map h : R n+1! R n+1 that takes points on the line {(y, z) 2 R n+1 : y constant} onto the orbits {( (x 0 ), ) 2 R n+1 : x 0 constant}. We need to show that h is a diffeomorphism, then g = h 1 is the desired map. So define h to take the values ( t (x 0 ), (t)) = h(y(t),z(t))

18 88 4. LINEARIZATION AND INVARIANT MANIFOLDS where y(0) = x 0 is a constant and z(0) = 0. Because is differentiable, then clearly h will also be differentiable with respect to both y and z. So the Jacobian of h is " # I t D (y,z) h = 0 1 at x = t0 and = 0. Since the determinant of D (y,z) h 6= 0, we can use the inverse function theorem to infer that h is a diffeomorphism and the proof is complete. } The rectification theorem essentially states that the behavior of flows in the neighborhood of a regular point is uninteresting. In the vicinity of a hyperbolic equilibrium, however, the flows becomes much more interesting because they are conjugate to the flows of the system s linearization about that equilibrium. This result is known as the Hartman-Großman theorem. Is is an important result for it suggests that one can use methods from linear systems theory to study the local behavior of a nonlinear system. Its restriction to local behavior of course is very restrictive, but in the context of the regulation problem for feedback control, it is still very useful. The proof of this theorem is done by focusing on the 1-map for the system. In other words, consider the original system ẋ = f(x) with a hyperbolic equilibrium, x 0, and let ẋ = A(x x 0 ) be its linearization about x 0. Let be the flow for the original system and define the one map as 1(x) = (1,x), in other words the flow for a single unit of time. We then focus on establishing the conjugacy of this map 1 to the 1-map for the linearized system, e A. Establishing conjugacy means there is a diffeomorphism H such that (51) H( 1 (x)) = e A H(x). Because we are focusing on the behavior of the flows about a hyperbolic equilibrium, we can express the 1-map, 1(x), as an additive perturbation of the linearized flow e A x. In a small neighborhood of the equilibrium, that perturbing function will be an appropriately bounded function and for such well behaved perturbations it becomes possible to split equation (51) in a manner that constructs a contraction mapping whose fixed point, H, is the diffeomorphism whose existence is needed for the conjugacy of 1 and e A. The preceding discussion, of course, only investigates the conjugacy of the 1-map. But once that is done, it is relatively straightforward to establish proof. t (x) is conjugate to e At for 0 apple t apple 1, and that would complete the The remainder of this section proves the Hartman-Großman theorem using the method outlined above. In our study, it will be convenient to introduce two norms for the linear space of continuously differentiable functions C 1 (R n, R n ). The 0-norm will be defined as kuk 0 = sup x2r n u(x) When C 1 (R n, R n ) is associated with this norm we obtain the Banach space, C 0, consisting of all C 1 functions of bounded amplitude. We will need to consider a more restrictive set of C 1 functions whose derivatives are also bounded. This gives rise to the 1-norm kuk 1 = kuk 0 + kd x uk 0

19 5. RECTIFICATION AND LINEARIZATION (HARTMAN-GROSSMAN) THEOREM 89 With these definitions, we can now state and prove a preliminary theorem that represents the 1-map as an additive perturbation of e A. THEOREM 44. (Lemma for Hartman-Großman Theorem) Let E R n be open, x 0 2 E, N (x 0 ) E for some >0. Let f 2 C 1 (E,R n ) such that f(x 0 )=0and A = D x f(x 0 ). Let : R n+1! R n denote the flow generated by the initial value problem, ẏ = f(y), y(0) = x For any 2 (0, 1) there exists d>0 such that p 2 C 1 (R n, R n ) with p(x 0 )=0, D x p(x 0 )=0, kpk 1 <, and for all x 2 N d (x 0 ) the 1-map satisfies 1(x) =e A (x x 0 )+p(x) Proof: Without loss of generality we assume x 0 = 0 and we let B = e A = D x 1 (0) for notational convenience. From our earlier local existence and uniqueness theorems, we know that there exists an >0 such that the IVP has a solution for all t 2 [ 1, 1] for any initial conditon in N (0). For any d>0, we are going to introduce a C 1 cutoff function ( 1 if 0 apple x appled d(x) = 0 if x 2d d : R n! R such that with D x d (x) apple 2 d. This function is essentially 1 within the neighborhood of radius d and quickly goes to zero outside of that radius. It is used to cutoff a function in a way that it remains C 1 even after the cutoff. We use the cutoff function d to cut off the nonlinear part of the flow. This yields the perturbation term, p(x) = d ( 1 (x) Bx) Clearly p(0) = 1 (0) = 0 and since d =1in a neighborhood of 0, it follows that D x p(0) = D x 1 (0) B = 0. Since is C 1 in x we can choose d small enough to ensure So the norm of p(x) will therefore be sup D x 1 (x) B < = x2n d (0) 2 +6 p(x) = d(x) 1(x) since the cutoff function ensures p(x) =0for x perturbation term, p(x). Bx apple x = 2 d 2d. This gives us a upper bound on the amplitude of the We also need to obtain a bound on the derivative of p(x). The cutoff function was chosen in such a way that cutting off 1 was done in a manner that kept the derivatives of p bounded. In particular, we see D x p(x) apple D x d (x) 1(x) Bx + d(x) D x 1 (x) B apple 2 x + apple 5 d and so the 1-norm of the perturbation function is bounded as kpk 1 applekpk 0 + kd x pk 0 apple 2 d +5 = 2 d +5 2 d +6 <

20 90 4. LINEARIZATION AND INVARIANT MANIFOLDS The 1-norm of the perturbation is bounded by the arbitrarily chosen 2 (0, 1) provided we choose d small enough } With the preceding preliminary theorem, we are now able to formally state and prove the Hartman-Großman theorem. THEOREM 45. (Hartman-Großman Theorem) Let E R n be open, x 0 2 E and N d (x 0 ) E for some d>0. Let f 2 C 1 (E,R n ) such that f(x 0 )=0and A = D x f(x 0 ) is hyperbolic. Then there are open neighborhoods of x 0 2 U and 0 2 V and a homeomorphism H : U! V such that the flow (t, x) of equation ẏ = f(y), y(0) = x is topologically flow equivalent to the flow of its linearization, e At x. In other words, for all x 2 U and t apple1. H( (t, x)) = e At H(x) Proof: Without loss of generality we assume that x 0 =0. We also assume that a similarity transform has been applied so that the 1-map for the linearized system is " # e B = e A A s 0 = 0 e Au where A s is an s s matrix whose eigenvalues are stable and A u is a u u matrix whose eigenvalues are unstable. Let us now consider the function (52) T (x) =Bx + p(x) where p(x) is the perturbation obtained by passing 1 (x) B(x) through a cutoff function d. We know that T (x) = 1 (x) for x 2 N d (0), but outside of this neighborhood they will differ, though that difference is bounded. We now introduce a candidate for the homeomorphism H. Let X denote the Banach space of C 0 (R n, R n ) functions with bounded 0-norm and let us define (53) H(x) =x + h(x) where h 2 X. Now consider x 2 N (0) where H is a homeomorphism satisfying 1 = T and note that the condition for local conjugacy is that (54) H(T (x)) = BH(x) substituting from equations (53) for H and (52) for T yields, T (x)+h(t (x)) = Bx + Bh(x)

21 6. CENTER MANIFOLD THEOREM 91 Using T (x) = Bx + p(x) yields which we re-arrange as Bx + p(x)+h(t (x)) = Bx + Bh(x) (55) h(t (x)) = Bh(x) p(x) Now recall that we ve already transformed the states so they are aligned with the stable and unstable eigensubspaces of A. So we can also split equation (55) in this manner by simply projecting onto the eigensubspaces. In particular, we consider the h s (x) as the elements of h(x) associated with the stable eigensubspace and h u (x) as the elements of h u (x) associated with the unstable eigensubspace. A similar projection can be applied to p(x) as well. We may therefore decompose equation (55) into two equations (56) (57) h s (T (x)) = e As h s (x) p s (x) h u (T (x) = e Au h u (x) p u (x) Note that the the equation (56) will be a contraction on E s and equation (57) is an expansion on E u. We can turn equation (57) into a contraction by reversing the direction of time and rewriting it as (58) h u (x) = e Au h u (T (x)) + e Au p u (x) We then use the inverse function theorem 39 to prove that T 1 exists and is C 1 so we can use the transformation y = T 1 (x) to rewrite equation (56) as (59) h s (x) = e As h s (T 1 (x)) p s (T 1 (x)) These two equations are both contractions defined on the Banach space X, so we can infer the existence of a unique function h : R n! R n that satisfies equation (54). This map H constructed can be shown to be a homeomorphism. This is done by constructing a continuous inverse H 1, using methods similar to those employed in constructing H. The preceding arguments establish the topological conjugacy between the time-one flows. To prove that this conjugacy can be extended to 1 (t, x) and e At where t < 1, we introduce the map H(x) = Z 1 0 e sa H( (s, x))ds and show that this is a homeomorphism for all (t, x) 2 [ 1, 1] N d (0). This is done through a computation to verify H( (t, x)) = e At H(x). } 6. Center Manifold Theorem Theorem 42 established the existence of local invariant manifolds, W s and W u, that are tangent to the stable and unstable eigensubspaces of a nonlinear system s linearization. In theorem 45, we went one step further and demonstrated that the flows on these invariant manifolds were topologically equivalent to the flows on the linearization s eigensubspaces. These particular theorems, however, only apply about the hyperbolic

22 92 4. LINEARIZATION AND INVARIANT MANIFOLDS equilibria of the system. If the linearization has a center eigensubspace, then the behavior of the system on a so-called center manifold can no longer be predicted by the linearization. This section reviews results regarding the system s behavior on a center manifold. As we will see, the center manifold plays an important role in studying the stability and local bifurcations of nonlinear systems. The following discussion is drawn closely from Carr s monograph [Car82] on the center manifold theorem. So consider the system ẋ = f(x) where f : D! R n is locally Lipschitz in D R n with an equilibrium point at x =0. We confine our attention to system s whose linearization only has eigenvalues with non-positive real parts. For such systems the Hartman-Großman theorem (45) allows us to rewrite this system as (60) ż = Az + f(z,y) ẏ = By + g(z,y) where A has c eigenvalues with zero real parts, B has s eigenvalues with negative real parts, and c + s = n. The functions f : R c R s! R c and g : R c R s! R s are C 2 with f(0, 0) = 0, f 0 (0, 0) = 0, g(0, 0) = 0, and g 0 (0, 0) = 0. Note that if f and g are identically zero (i.e. the system is linear), then there are two invariant manifolds (subspaces) E c = {x =[z,y] T : y =0} and E s = {x =[z,y] T : z =0}. The invariant manifold E c is called the center manifold. We d like to extend this idea from the linear system to the nonlinear system. The invariant manifold theorem 42 established the existence of stable and unstable invariant manifolds for a hyperbolic system. For equation (60), however, we will not try to establish the existence of the invariant set. Rather we will simply define a center manifold to be a C 2 function h : R c! R s associated with equation (60) such that y = h(z), h(0) = 0 and h 0 (0) = 0. In other words, we will define what we mean by a center manifold by the function h. Our task will then be to prove 1) that such a function exists locally for the system in equation (60), 2) to establish that the asymptotic behavior of z and y is determined solely by the behavior on z, and 3) to establish methods for constructing h. We start first with the following theorem regarding the existence of h. THEOREM 46. (Center Manifold - Existence) Consider the system ż = Az + f(z,y) ẏ = By + g(z,y) where f : R c R s! R c and g : R c R s! R s such that f(0, 0) = 0, f 0 (0, 0) = 0, g(0, 0) = 0 and g 0 (0, 0) = 0. We assume the eigenvalues of A have zero real parts and the eigenvalues of B have negative " # T z real parts. Then there exists >0 such that for all x = where x <, there is a C 2 function y h : R c! R s where h(0) = 0 and h 0 (0) = 0.

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