Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
THE HOPF BIFURCATION AND ITS APPLICATIONS 27 SECTION 2 THE CENTER MANIFOLD THEOREM In this section we will start to carry out the program outlined in Section 1 by proving the center manifold theorem. The general invariant manifold theorem is given in HirschPugh-Shub [1]. Most of the essential ideas are also in Kelley [1] and a treatment with additional references is contained in Hartman [1]. However, we shall follow a proof given by Lanford [1] which is adapted to the case at hand, and is direct and complete. We thank Professor Lanford for allowing us to reproduce his proof. The key job of the center manifold theorem is to enable one to reduce to a finite dimensional problem. In the case of the Hopf theorem, it enables a reduction to two dimensions without losing any information concerning stability. The outline of how this is done was presented in Section 1 and the details are given in Sections 3 and 4. In order to begin, the reader should recall some results about basic spectral theory of bounded linear operators by consulting Section 2A. The proofs of Theorems 1.3 THE HOPF BIFURCATION AND ITS APPLICATIONS 28 and 1.4 are also found there. Statement and Proof of the Center Manifold Theorem We are now ready for a proof of the center manifold theorem. It will be given in terms of an invariant manifold ~, for a map not necessarily a local diffeomorphism. Later we shall use it to get an invariant manifold theorem for flows. Remarks on generalizations are given at the end of the proof. (2.1) Theorem. Center Manifold Theorem. Let a mapping of a neighborhood of zero in a Banach space Z. ~ We assume that is D~(O) further assume that D~(O) the spectrum of ck + l , k > 1 ~ be Z into ~(O)= O. and that has spectral radius 1 We and that splits into a part on the unit circle and the remainder which is at a non-zero distance from the unit circle.* D~(O) Y Let denote the generalized eigenspace of belonging to the part of the spectrum on the unit circle; assume that Y has dimension d < 00. V Then there exists a neighborhood submanifold o and tangent to a) of at Y. V 0, of dimension ~. If and d, passing through x EM and~(x) E V, (x) E M (Local Attractivity): n = 0, 1, 2, ••• , from Z in such that (Local Invariance): then b) M 0 of ~n(x) to ~n(x) E V If then, as M n ~ 00, for all the distance goes to zero. We begin by reformulating (in a slightly more general way) the theorem we want to prove. We have a mapping neighborhood of zero in a Banach space Z into Z, ~ with *This holds automatically if Z is finite dimensional or, more generally, if D~(O) is compact. of a THE HOPF BIFURCATION AND ITS APPLICATIONS = ~(o) O. We assume that the spectrum of D~(O) 29 splits into a part on the unit circle and the remainder, which is contained in a circle of radius strictly less than one, about the origin. The basic spectral theory discussed in Section 2A guarantees the existence of a spectral projection P of Z belonging to the part of the spectrum on the unit circle with the following properties: i) and P (I-P)Z PZ D~(O). The spectrum of the restriction of D~(O) to lies on the unit circle, and iii) to so the subspaces D~(O), are mapped into themselves by ii) PZ commutes with (I-P)Z We let The spectral radius of the restriction of is strictly less than one. X denote striction of D (0) to D~(O) (I-P)Z, D~(O) Y. Then ~(x,y) to Z X = Y denote and X e Y B PZ, A denote the res- denote the restriction of and (Ax+X*(x,y), By + Y*(x,y», where A is bounded linear operator on X with spectral radius strictly less than one. B is a bounded operator on Y with spectrum on the unit circle. (All we actually need is that the spectral radius of B- 1 is no larger than one.) k X* is a c + l mapping of a neighborhood of the origin in X e Y into zero at the origin, i.e. DX(O,O) Y Ck + l is a in = X 0, X with a second-order X(O,O) = 0 and and mapping of a neighborhood of the origin e Y into Y with a second-order zero at the 30 THE HOPF BIFURCATION AND ITS APPLICATIONS origin. ~ We want to find an invariant manifold for Y to at the origin. mapping into u X, which is tangent Such a manifold will be the graph of a which maps a neighborhood of the origin in with = u(O) 0 and Du(O) Y = o. In the version of the theorem we stated in 2.1, we assumed that Y was finite-dimensional. We can weaken this assumption, but not eliminate it entirely. (2.2) Assumption. ¢ on function Y 1 which is I Iyl origin and zero for k l c + There exists a on a neighborhood of the I > 1. Perhaps surprisingly, this assumption is actually rather restrictive. if Y real-valued is finite-dimensional or if Y It holds trivially is a Hilbert space; for a more detailed discussion of when it holds, see Bonic and Frampton [1]. We can now state the precise theorem we are going to prove. (2.3) Theorem. Let the notation and assumptions be as above. Then there exist {y E Y: Ilyll < d e: > 0 into X, ck-mapping and a u* from with a second-order zero at zero, such that a) IIYII ~ <d c X Ell Y, i.e. the graph of in the sense that, i f (xl'Yl) II ylll with b) r u* = ((x,y) The manifold < e: The manifold in the sense that, i f (~'Yn) ~n(x,y) for all n > 0, Ilxll then < e: = x u*(y) and is invariant for u* ~ and if (u* (y) ,y) Xl = u* (Yl) . is locally attracting for r u* are such that then < Ilyll I e: Ilyll II xnll < < e: e: , and if IIYnl1 < e: ~ THE HOPF BIFURCATION AND ITS APPLICATIONS lim n+ oo I IXn - u*(Yn) II 31 o. = Proceeding with the proof, it will be convenient to assume that than 1. IIA II < 1 and that IIB-lil is not much greater This is not necessarily true but we can always make it true by replacing the norms on (See Lemma 2A.4). X, Y by equivalent norms. We shall assume that we have made this change of norm. It is unfortunately a little awkward to exl plicitly set down exactly how close to one I IB- , I should be taken. We therefore carry out the proof as if 1 IB-li I were an adjustable parameter; in the course of the argument, I IB- l , I. we shall find a finite number of conditions on In principle, one should collect all these conditions and impose them at the outset. The theorem guarantees the existence of a function u* defined on what is perhaps a very small neighborhood of zero. Rather than work with very small values of x,y, scale the system by introducing new variables calling the new variables again does not change make < X*, Y*, k + 1, tiplying A,B, x and but, by taking y). s we shall xis, y/s (and This scaling very small, we can together with their derivatives of order as small as we like on the unit ball. X*(x,y), Y*(x,y) by the function Then by mul- ¢(y) whose existence is asserted in the assumption preceding the statement of the theorem, we can also assume that are zero when I Iyl I > 1. sup II x 11::1 y unrestricted x*(x,y), Y*(x,y) Thus, if we introduce sup jl,j2 jl+j2::k+l THE HOPF BIFURCATION AND ITS APPLICATIONS 32 we can make A as small as we like by choosing E The only use we make of our technical assumption on very small. Y is to A arrange things so that the supremum in the definition of may be taken over all y and not just over a bounded set. ¢, Once we have done the scaling and cutting off by we can prove a global center manifold theorem. That is, we shall prove the following. (2.4) Lemma. Keep the notation and assumptions of the A is SUfficiently small (and if center manifold theorem. If liB-II I is close enough to one), there exists a function defined and k u*, Y, times continuously differentiable on all of with a second-order zero at the origin, such that a) invariant for b) lim n+ oo I Ix n ru* The manifold "xii If As with n u*(y),y E Y} {(x,y) Ix is in the strict sense. 0/ - u * (y = ) II = I IB-li < 0 I, 1, and y is arbitrary then (where (xn'Yn) = we shall treat 0/ n (x,y)). A as an adjustable parameter and impose the necessary restrictions on its size as they appear. It may be worth noting that A depends on the choice of norm; hence, one must first choose the norm to make I /B- l , I close to one, then do the scaling and cutting off to A small. make To simplify the task of the reader who wants. to check that all the required conditions on I IB- l " can be satisfied simultaneously, we shall note these conditions * with a as with (2.3)* on p. 34. The strategy of proof is very simple. manifold of u); M of the form we let {x = u(y)} (this stands for the graph M denote the image of some mild restrictions on u, We start with a M under 0/. With we first show that the manifold THE HOPF BIFURCATION AND ITS APPLICATIONS ~M 33 again has the form {x = u(y)} A for a new function u. If we write ffu for u we get a (non- linear) mapping u 1+'9'"u from functions to functions. and only if u = ffu, The manifold M is invariant if so we must find a fixed point of do this by proving that ~ We ff is a contraction on a suitable A is small enough). function space (assuming that More explicitly, the proof will be divided into the following steps: I) II) Derive heuristi~ally a "formula" for Show that the formula obtained in Jv. I) yields a well-defined mapping of an appropriate function space U into itself. III)t Prove that ff is a contraction on has a unique fixed point IV) U and hence u*. Prove that b) of Lemma (2.4) holds for u*. We begin by considering Step I). I) i) y. = This means that ii) ~(u(y),y), we should proceed as follows Solve the equation y for u(y), To construct Let By + Y*(u(y) ,y) y u(y) is the be the Y-component of of functions ~(u(y),y). X-component of Le., u(y) = Au(y) +X*(u(y),y). II) (2.1) (2.2) We shall somewhat arbitrarily choose the space u we want to consider to be tone could use the implicit function theorem at this step. For this approach, see Irwin [1]. THE HOPF BIFURCATION AND ITS APPLICATIONS 34 . {u: Y .... X I Dk+l u U j 0,1, ••• ,k+l, continuous; all y; U{O) for = = Du{O) o}. We must carry out two steps: Prove that, for any given i) y a unique solution y E Y. for each that~, Prove ii) u E U, equation (1) has And defined by (2.2) is in U. To accomplish (i), we rewrite (2.1) as a fixed-point problem: It suffices, therefore, to prove that the mapping _ Y 1+ B-1 Y - is a contraction on Y. -) B-1 Y * (u (-) Y ,y We do this by estimating its deriva- tive: I IDy[B-1Y-B-ly*(U(Y),y)] I I ::11B-lll by the definitions of A and l + D2 Y*(U(y) ,y) II :: 2AIIB- ll U. If we require l 2AIIB- ll < 1, Y is a function of y, By the inverse function theorem, of y. Next we establish (ii). and < 1 for all = 0, D.%.(O) First take j = 0: -%.(0) IliVu(y)11 y. Note y is a Ck + l function By what we have just proved, Thus to show ~ E U, I I DjiVu(y) II for each * depending also on the function u. 5'u E c k + l • (2.3) y equation (2.1) has a unique solution that IID1Y*(u(y),y)Du(y) y, what we must check is that j = 0,1,2, ..• ,k+l (2. 4) (2.5) = O. ~ IIAII·llu(Y)11 + Ilx*(u(Y),Y)11 < IIAII + A, so if we require IIAII + A < 1, then I Wu{y) II < 1 for all y. (2.6) * THE HOPF BIFURCATION AND ITS APPLICATIONS To estimate D9U we must first estimate 35 Dy(y). By differentiating (2.1), we get (2.7) yU: Y where + Y is defined by = yU(y) Y*(u(y) ,y). By a computation we have already done, I IDYu(y)!! Now B + DY u ~ 2A B[I+B-1DY u ) (2.3)*), B + DY u for all and since y. 2AI IB-11! < 1 (by is invertible and The quantity on the right-hand side of this inequality will play an important role in our estimates, so we give it a name: (2.8) 1IB- l , I Note that, by first making by making A y small, we can make very close to one and then as close to one as we like. We have just shown that II Dy (y) II < y for all Differentiating the expression (2.2) for D5U(y) = y. YU(y) (2.9) yields [A DU(y) + DXu(y») Dy(y); } and (2.10) (xu (y) = X*(u(y) ,y». Thus 11D9U(y) II < (IIAII + 2A) .y, (2.11) 36 THE HOPF BIFURCATION AND ITS APPLICATIONS so if we require (IIAII + 2A) Y (2.12) < 1, we get 11D%(y) II < 1 for all y. We shall carry the estimates just one step further. Differentiating (2.7) yields By a straightforward computation, so Now, by differentiating the formula (2.10) for D~, we get 2 D :§ilu(y) so If we require (2.13) we have IID2~(y)11 <1 for all y. At this point it should be plausible by imposing a sequence of stronger and stronger conditions on I I Djy(y) II < 1 for all y, Y,A, that we can arrange j = 3,4, •.. ,k+1. * * THE HOPF BIFURCATION AND ITS APPLICATIONS 37 The verification that this is in fact possible is left to the reader. ~u To check (2.5), i.e. u = 0, Du yeo) o 0) IL%. (0) o D% (assuming we note that since 0 is a solution of AU(O) + X(u(O) ,0) = 0 ~u(O) 0, By + Y(u(y) ,y) 0 and [A Du(O) + D1X(0,0)Du(0) + D2X(0,0)] ·Dy(O) [A· 0 + 0 + 0]· Dy (0) = O. This completes step II). Now we turn to III) III) We show that ~ is a contraction and apply the contraction mapping principle. What we actually do is slightly more complicated. i) norm. Since We show that U ~ is a contraction in the supremum is not complete in the supremum norm, the con~ traction mapping principle does not imply that point in U, but it does imply that the completion of ii) U has a fixed ~ has a fixed point in with respect to the supremum norm. We show that the completion of U with respect to the supremum norm is contained in the set of functions from to y X th k with Lipschitz-continuous and with a second-order zero at the origin. point u* ~ of derivatives Thus, the fixed has the differentiability asserted in the theorem. We proceed by proving i) I I u l -u 2 I 10 = Consider i). u l ' u 2 E U, supl lu l (y)-u 2 (y) I I· y Let and let Yl(y), Y2(y) the solution of y u 1,2. denote 38 THE HOPF BIFURCATION AND ITS APPLICATIONS We shall estimate successively 11:i\-Y211 0' and Subtracting the defining equations for Yl'Y2' II9"u 1 ~211 O. we get so that (2.14) Since I I Du 1 1 10 2 1, we can write (2.15) Inserting (2.15) in (2.14) and rearranging, yields 1 (l-n·IIB- 11) IIY1-Y211 I 1Y1 -Y 2 1I 0 i. e. 2. >.·IIB-lll·llu2-ulII0' ~ >.. yo I I u 2 - u 1 1I • I J (2.16) Now insert estimates (2.15) and (2.16) in to get If we now require a 3t = II All (l+y>.) + >. (l+2y>.) < 1, will be a contraction in the supremum norm. (2.17)* 39 THE HOPF BIFURCATION ANO ITS APPLICATIONS ii) The assertions we want all follow directly from the following general result. (2.5) Lemma. a Banach space Y that, for all Let (un) be a sequence of functions on with values on a Banach space nand Assume y E Y, j O,1,2, ..• ,k, is Lipschitz continuous with Lipschitz and that each constant one. (un(y)) X. Assume also that for each y, the sequence converges weakly (i.e., in the weak topology on to a unit vector a) u u(y). X) Then k th derivative has a Lipschitz continuous with Lipschitz constant one. b) y and ojun(y) converges weakly to oju(y)* for all j = 1, 2 , ••• , k • If X, Yare finite dimensional, all the Banach space technicalities in the statement of the proposition disappear, and the proposition becomes a straightforward consequence of the Arzela-Ascoli Theorem. We postpone the proof for a moment, and instead turn to step IV). IV) Ilxll < 1 We shall prove the following: and let y E Y be arbitrary. Let Let (xl'Yl) * This statement may require some interpretation. n,y,ojun(y) x E X with 'I'(x,y) . For each j y is a bounded symmetric j-linear map from to X. What we are asserting is that, for each Y'Yl' ..• 'Yj' the sequence (ojun(y) (Yl' •.• 'Yj)) of elements of X converges in the weak topology on X to oju(y) (Yl' ... 'Yj). THE HOPF BIFURCATION AND ITS APPLICATIONS 40 Then II xIII ~ 1 and ~ CY.·llx-u*(y) II, Ilxl-u*(Yl) II where CY. is as defined in (2.17). [ I x n -u* (Y n ) II < cy'nll x-u* (Y) (2.18) By induction, II as 0 -+ n -+ 00, as asserted. To prove I Ixll I < 1, we first write Xl = Ax + X(x,y), /lxlll ~ IIAlj·llxll + It ~ so that "All + It < 1 by (2.6) To prove (2.18), we essentially have to repeat the estimates made in proving that j7 is a contraction. Let Yl the solution of On the other hand, by the definition of Y l = Y l we have By + Y(x,y). Subtracting these equations and proceeding exactly as in the derivation of (2.16), we get Next, we write xl = Ax + X(x,y). Subtracting and making the same estimates as before, we get be 41 THE HOPF BIFURCATION AND ITS APPLICATIONS as desired. This completes step IV). Let us finish the argument by supplying the details for Lemma (2.5). Proof of Lemma (2.5) We shall give the argument only for ization to arbitrary k k = 1; the general- is a straightforward induction argu- ment. Yl'Y2 E Y We start by choosing and ¢ E X* and con- sider the sequence of real-valued functions of a real variable From the assumptions we have made about the sequence (un)' it follows that lim ~n(t) = ¢(u(Yl+tY2)) n->-QO for all t, ~n(t) that I~~ (t)1 < = ~(t) is differentiable, that II ¢ 1/ •II y 1// for all n, t and that for all n, t l , t . 2 By this last inequality and the Arzela- Ascoli Theorem, there exists a subsequence verges uniformly on every bounded interval. rily denote the limit of this subsequence by hence, passing to the limit j ->- 00, we get ~' n j (t) which con- We shall temporaX(t). We have THE HOPF BIFURCATION AND ITS APPLICATIONS 42 1jJ(t) =1jJ(O) + which implies that 1jJ (t) f: X(T) dT, is continuously differentiable and that 1jJ I (t) X(t) • To see that lim1jJ~(t) = 1jJ'(t) nT OO (i.e., that it is not necessary to pass to a subsequence), we note that the argument we have just given shows that any subsequence of (1jJ' (t» n has a subsequence converging to 1jJ I (t) ; this implies that the original sequence must converge to this limit. Since we conclude that the sequence converges in the weak topology on shall denote by only suggestive. DU(Yl) (Y2); we have to a limit, which we this notation is at this point By passage to a limit from the correspond- is a bounded linear mapping of norm for each X** < 1 from We denote this linear operator by yto X ** THE HOPF BIFURCATION AND ITS APPLICATIONS i.e., the mapping to ~ y Du(y) 43 is Lipschitz continuous from Y L(Y,X ** ). The next step is to prove that this equation together with the norm-continuity of will imply that u is (Frechet)-differentiable. Du(y) y~ The integral in (2.19) may be understood as a vector-valued Riemann integral. By the first part of our argument, for all ¢ E X** and taking Riemann integrals commutes with continuous linear mappings, so that for all ¢ * EX. Therefore (2.19) is proved. The situation is now as follows: if we regard u as a mapping into X** We have shown that, which contains then it is Frechet differentiable with derivative other hand, we know that want to conclude that x. X u actually takes values in t+O DU(Yl) (Y2) and belongs to u(Yl+ tY 2)-u(Yl) t the difference quotients on the right all belong to is norm closed in X But ~~l~ X On the it is differentiable as a mapping into This is equivalent to proving that for all Du. X, X** the proof is complete. CJ X, and 44 THE HOPF BIFURCATION AND ITS APPLICATIONS (2.6) Remarks on the Center Manifold Theorem It may be noted that we seem to have lost some 1. differentiability in passing from that ~ is ck + l to and only concluded that fact, however, the k th ~ u* u* , since we assumed u* ck . is In we obtain has a Lipschitz continuous derivative, and our argument works just as well if we ~ only assume that k th has a Lipschitz continuous deriva- tive, so in this class of maps, no loss of differentiability occurs. k th Moreover, if we make the weaker assumption that the derivative of ~ is uniformly continuous on some neigh- borhood of zero, we can show that the same is true of (Of course, if X and u*. Yare finite dimensional, continuity on a neighborhood of zero implies uniform continuity on a neighborhood of zero, but this is no longer true if X or Y is infinite dimensional). 2. As C. Pugh has pointed out, if ~ is infinite- ly differentiable, the center manifold cannot, in general, be taken to be infinitely differentiable. that, if ~ It is also not true is analytic there is an analytic center manifold. We shall give a counterexample in the context of equilibrium points of differential equations rather than fixed points of maps; cf. Theorem 2.7 below. This example, due to Lanford, also shows that the center manifold is not unique; cf. Exercise 2.8. Consider the system of equations: ~ 0, = - .where zero. h (2.20 ) is analytic near zero and has a second-order zero at We claim that, if h is not analytic in the whole com- THE HOPF BIFURCATION AND ITS APPLICATIONS p1ex plane, there is no function neighborhood of (0,0) u(Y1'Y2)' 45 analytic in a and vanishing to second order at (0,0), such that the manifold is locally invariant under the flow induced by the differential equation near (0,0). To see this, we assume that we have an invariant manifold with Straightforward computation shows that the expansion coefficients c, , are uniquely determined by the requirement of J 1 ,J2 invariance and that (j1+ j 2) : (j1): h j1 + j2 , where h(Y1) If the series for series for h u(o'Y2) = L j.:. 2 j h 'Y1 J has a finite radius of convergence, the diverges for all non-zero Y2' The system of differential equations has nevertheless many infinitely differentiable center manifolds. one, let h(Y1) agreeing with To construct be a bounded infinitely differentiable function h on a neighborhood of zero. Then the manifold defined by (2.21) is easily verified to be globally invariant for the system THE HOPF BIFURCATION AND ITS APPLICATIONS 46 0, dx dt (2.22) and hence locally invariant at zero for the original system. (To make the expression for sketch its derivation. volve x u less mysterious, we The equations for Yl'Y2 and are trivial to solve explicitly. do not inA function u defining an invariant manifold for the modified system (2.22) must satisfy for all t, Yl' Y2. The formula (2.21) for u is obtained by solving this ordinary differential equation with a suitable boundary condition at 3. ~ t = _00.) Often one wishes to replace the fixed point by an invariant manifold on a normal bundle of 9. V. V 0 of and make spectral hypotheses We shall need to do this in section This general case follows the same procedure; details are found in Hirsch-Pugh-Shub [lJ. The Center Manifold Theorem for Flows The center manifold theorem for maps can be used to prove a center manifold theorem for flows. time t We work with the maps of the flow rather than with the vector fields themselves because, in preparation for the Navier Stokes equations, we want to allow the vector field generating the flow to be only densely defined, but since we can often prove that the time t-maps are 00 C this is a reasonable hypothesis for many partial differential equations (see tails) • Section 8A for de- 47 THE HOPF BIFURCATION AND ITS APPLICATIONS (2.7) Let z Theorem. Center Manifold Theorem for Flows. and let Ft o < t < zero for Ft(X) cO be a ck + l is C 0 semi flow defined in a neighborhood of Assume T. norm * away from 00 be a Banach space admitting a jointly in Ft(O) t and and that for 0, = x. t > O. Assume that the spectrum of the linear semi group DF (0): Z -+ Z is of the t tal form e t (01U0 2 ) where lies on the unit circle (Le. e ta 2 lies inside the °1 lies on the imaginary axis) and e unit circle a nonzero distance from it, for is in the left half plane. Let Y t> 0; i.e. 02 be the generalized eigen- space corresponding to the part of the spectrum on the unit circle. a o Assume dim Y d < 00. Then there exists a neighborhood V of submanifold d passing through and tangent to M CV Y at of dimension 0 0 in Z and such that (a) If x EM, t > 0 (b) If t > 0 and then Ft(x) E V, Ft(X) EM V for all n 0,1,2, .•• , and F~(X) then remains defined and in F~(X) -+ M as n -+ This way of formulating the result is the most convenient for it applies to ordinary as well as to partial differential equations, the reason is that we do not need to worry about "unboundedness" of the generator of the flow. Instead we have used a smoothness assumption on the flow. The center manifold theorem for maps, Theorem 2.1, applies to V and M F , t > O. However, we are claiming that t can be chosen independent of t. The basic reason e~ch 48 THE HOPF BIFURCATION AND ITS APPLICATIONS for this is that the maps~t} commute: Fs F t -- F t+s- 0 F 0 F ' where defined. However, this is somewhat overs t simplified. In the proof of the center manifold theorem we would require the F to remain globally commuting after they t $. have been cut off by the function That is, we need to ensure that in the course of proving lemma 2.4, A can be chosen small (independent of are globally t) and the Ft'S defined and commute. The way to ensure this is to first cut off the Z outside a ball B in such a way that the turbed in a small ball about tity outside of of F t 0 00 function C and is 0 outside where T = it is easy to see that Z G t which coincides with f B. remains a Ft , 0 semiflow. t and and are the iden- x which is one on a neighThen defining (2.23) extends to a global semiflow t on ~ t ~ T on a neighborhood of B. Moreover, Gt (For this to be true we required the smoothness of the norm on smoothness in T, Z and that for jointly * ) t > 0 . F t Now we can rescale and chop off simultaneously the outside F t in are not dis- It f(F (x))ds, o s zero, and which is the identity outside Ck + l -< t t This may be achieved by joint continuity B. and use of a borhood of 0, 0 < t F F B as in the above proof. has G t Since this does not affect on a small neighborhood of zero, we get our result. *In linear semi group theory this corresponds to analyticity of the semigroup; it holds for the heat equation for instance. For the Navier Stokes eqations, see Sections 8,9. t see Renz [1] for further details. THE HOPF BIFURCATION AND ITS APPLICATIONS Exercise. (2.8) for flows). d(x,y) dt = (nonuniqueness of the center manifold X (x,y) = (-x,y 2 ). Let X(x,y) 49 Solve the equation and draw the integral curves. Show that the flow of Theorem 2.7 with X satisfies the conditions of the y-axis. y center manifold for the flow. Show that the y-axis is a Show that each integral curve in the lower half plane goes toward the origin as that the curve becomes parallel to the y-axis as t t ~ and ->- -00. Show that the curve which is the union of the positive y-axis with any integral curve in the lower half plane is a center manifold for the flow of X. (see Kelley [1], p. 149). Exercise. (2.9) (Assumes a knowledge of linear semi- group theory). Consider a Hilbert space space) and let A Let be a K: H H ->- H (or a "smooth" Banach be the generator of an analytic semigroup. Ck + l mapping and consider the evolution equations dx dt (a) on H Ax + K(x), x(O) = X (2.24) o Show that these define a local semiflow c k +l which is in (t,x) Duhamel integral equation for t > O. x(t) = eAtx o + Ft(x) (Hint: Solve the t- f 0 e ( t-s ) AK(x(s))ds by iteration or the implicit function theorem on a suitable space of maps (references: Segal [1], Marsden [1], Robbin [1]). (b) of i~variant Assume K(O) = 0, DK(O) = O. Show the existence manifolds for (2.24) under suitable spectral hypotheses on A by using Theorem 2.7. 50 THE HOPF BIFURCATION AND ITS APPLICATIONS SECTION 2A SOME SPECTRAL THEORY In this section we recall quickly some relevant results in spectral theory. Schwartz [1,2]. For details, see Rudin [1] or Dunford- Then we go on and use these to prove Theorems 1. 3 and 1.4. Let space E T: E and let {A E ~[T - AI Then I ITI a(T) I· + Let E be a bounded linear operator on a Banach a(T) denote its spectrum, a(T) = is not invertible on the complexification of is non-empty, is compact, and for r(T) r(T) = SUp{IAI A E a(T),IAI El < denote its spectral radius defined by IA E a(T)}. r(t) is also given by the spectral radius formula: r(T) limit I ITnl I lin • n+oo (The proof is analogous to the formula for the radius of convergence of a power series and can be supplied without difficulty; cf. Rudin [1, p.355].) The following two lemmas are also not difficult and are THE HOPF BIFURCATION AND ITS APPLICATIONS 51 proven in the references given: (2A.l) Lemma. tion and define (2A.2) Lemma. d(01,02) > 0, E l and o (T ) i E2 o .• 1. f(T) Let = La L a n zn f(z) Tn then On' Suppose otT) 0 (f (T» = f (0 (T» • E i E = El e E2 0 1 U 02 = then there are unique such that where T-invariant subspaces and if = Ti TIE.' then 1. is called the generalized eigenspace of Let Lemma 2A.2 is done as follows: curve with be an entire func- o in its interior and O 2 0i' be a closed in its exterior, then A is not al- Note that the eigenspace of an eigenvalue ways the same as the generalized eigenspace of A. In the finite dimensional case, the generalized eigenspace of A is the subspace corresponding to all the Jordan blocks containing A in the Jordan canonical form. (2A. 3) Lemma. Let T, 0 1 , and O 2 be as in Lemma °1 °2 2A.2 and assume that d(e , e ) > O. Then for the operator tT the generalized eigenspace of e i e tT is Ei • Proof. Write, according to Lemma 2A.2, E Thus, From this the result follows easily. (2A.4) Lemma. Let T: E + E [] be a bounded, linear THE HOPF BIFURCATION AND ITS APPLICATIONS 52 operator on a Banach space E. Let r be any number greater than r(T), the spectral radius of I on equivalent to the original norm such that ITI I E We know that Proof. II Tnll sup Therefore, n~a we see that (~~~ r sup n~a I E r n I :s and let Let r > cr(A) I = sUb A: E + Ie of I tAl I E i.e. n~a r II Tn(x) II n r n be a bounded operator A E cr(A) , Re on E A< r). such that for Then t > a, < e rt. Note that (see Lemma 2A.l) e etA; sup I}/n. II Tn+l(x) II n~ (i.e. if there is an equivalent norm Proof. = r. 0 rl xl . Lemma. I xl < Ilxll.2lxl.2 I T(x) Hence I l'7nl r(T) = lim I f we define < "'. r II Tn+l(X) II n+l r Then there is a norm is given by is a norm and that I I Tnn " ) I I x I I . (2A.5) on r(t) T. rt e rt > lim I I entAl I lin. is > spectral radius Set n+'" Ixl = sup n;::a II entA(x) II e rnt t~a and proceed as in Lemma 2A.4. There is an analogous lemma if CJ r < cr (A) , giving e rt With this machinery we now turn to Theorems 1.3 and 1.4 of Section 1: (1.3) operator. Let Theorem. Let T: E cr(T) C {zlRe z < a} + E be a bounded linear (resp. cr(T) ~ {zlRe z > a}), then the origin is an attracting (resp. repelling) fixed point for the flow ¢t = e tT of T. '!'HE POPF BIFURCA,TION AND ITS APPLICATIONS Proof. {Z This is immediate from Lemma 2A.5 for if IRe z < O}, is compact. 53 there is an r < 0 with 0 (T) < r, tA Thus le I.:5 e rt .... 0 as t .... +00. D 0(T)~ as 0 (T) Next we prove the first part of Theorem 1.4 from Section 1. (1.4) Theorem. Banach manifold P Let X cl be a and be such that o (dX (p 0)) C {z E <c IRe z < O}, vector field on a = X(PO) then O. If is asymptotically stable. Proof. Po = O. We can assume that P is a Banach space E As above, renorm I letA I I < e - d , where A E and find € > 0 and that such that dX (0) • = From the local existence theory of ordinary differen- . * , there is a tial equat10ns r-ball about for which the 0 time of existence is uniform if the initial condition lies in this ball. such that where o Let R(x) Find X X(x) - DX(O) ·x. = Ilx II 2 r 2 implies IIR(x)11 <a.llxll a. = €/2. Let show that if remains in B be the open X E B, o Band .... r /2 ball about O. We shall 2 then the integral curve starting at X 0 exponentially as t .... +00. o This will prove the result. Let Suppose x (t) x(t) be an integral curve of remains in B for 0 <t X < T. starting at Then noting that *See for instance, Hale [1], Hartman [1], Lang [1], etc. x • o THE HOPF BIFURCATION AND ITS APPLICATIONS 54 x x(t) = x I: I: + o + o x (x (s)) ds A(x(s)) + R(x(s)) ds gives (the Duhamel formula; Exercise 2.9), x(t) = e tAx o + It 0 e (t-s)A R(x(s)) ds I: and so Ilx(t)11 Thus, letting ~e-tExo f (t) = e tE Ct Ijx(t) ~xO f(t) + + Ct e-(t-s)Ellx(s)llds. II, I: f(s) ds and so (This elementary inequality is usually called Gronwall's inequality cf. Hartman [1].) Thus Hence x(t) E B, extended in x(t) + 0 t as 0 < t < T so x(t) may be indefinitely and the above estimate holds. t + +00. It shows ~ The instability part of Theorem 1.4 requires use of the invariant manifold theorems, splitting the spectrum into two parts in the left and right half planes. The above analysis shows that on the space corresponding to the spectrum in the right half plane, (2A.6) PO is repelling, so is unstable. Remarks. Theorem 1.3 is also true by the same THE HOPF BIFURCATION AND ITS APPLICATIONS 55 proof if T is an unbounded operator, provided we know o (e tT ) etO(T) which usually holds for decent T, (e.g.: if the spectrum is discrete) but nee,d not always be true (See Hille-Phillips [1]). This remark is important for applications to partial differential equations. require which is unrealistic for partial R(x) = 0(1 Ixl I) differential equations. Assume F in t However, the following holds: O(DFt(O» = etO(DX(O» and we have a cO flow cl with derivative strongly continuous (See Section 8A), then the conclusion of 1.4 is true. and each t Theorem 1.4's proof would F t is in the notation above, This can be proved as follows: we have, by Taylor's theorem: where R(t,x) as long as is the remainder. x remains in a small neighborhood of -2Et < e and hence because some E > 0 R(t,x) = J: Now we will have and x in some neighborhood of {DFt(sx),x - DFt(O) ·x} ds. x and 00 + 0 Exercise. (2A.7) F: E + E a exponentially as cl map with inside the unit circle. U of 0 Let such that if is a stable point of E t + for remember remains close to 0 ~ be a Banach space and F(O) = 0 and the spectrum of DF(O) Show that there is a neighborhood x EU, F(n) (x) F. -Et < e Therefore, arguing as in Theorem 1.4, we can conclude that Ft(x) 0 '" this is 0; + 0 as n + 00; Le. 0 56 THE HOPF BIFURCATION AND ITS APPLICATIONS SECTION 2B THE POINCAR~ MAP We begin by recalling the definition of the Poincare map. In doing this, one has to prove that the mapping exists and is differentiable. cO of flows Ft(X) In fact, one can do this in the context such that for each t, F t ck , is as was the case for the center manifold theorem for flows, but here with the additional assumption that t t > 0. as well for Ft(X) is smooth in Again, this is the appropriate hypo- thesis needed so that the results will be applicable to partial differential equations. However, let us stick with the ordinary differential equation case where vector field X F t is the flow of a at first. First of all we recall that a closed orbit flow on a manifold Yet) X L > y O. ck such that The least such M of a is a non-constant integral curve y(t+L) L y Yet) for all is the period of y. t ER and some The image of is clearly diffeomorphic to a circle. (2B.l) Definition. Let y be a closed orbit, let THE HOPF BIFURCATION AND ITS APPLICATIONS m E Y, say m Y(O) = and let S 57 be a local transversal sec- tion; i.e. a submanifold of codimension one transverse to , Y(Le. Y (0) is not tangent to S). '21 eM Let x IR be the domain (assumed open) on which the flow is defined. A Poincare map of (PM 1) (PM 2) there is a function p(x) (PM 3) if = o ->- W l where: are open neighborhoods of k diffeomorphism; is a C P for all P: W S WO'W l e and is a mapping Y 0: W o ->- IR m E S, such that x E WO' (X,T-O(X)) E'21 and F(X,T-O(X)); and finally, t E (O,T-O(X)), then ~ F(x,t) W o (see Figure 2B.l). (2B.2) Theorem (Existence and uniqueness of Poincare maps). (i) If X is a closed orbit of is a ck vector field on M, and Y then there exists a Poincare map of X, Y. (in ->- Wl a local transversal section S at m E and pI are locally conjugate. (ii) S' at P: W o If m ' E Y), then P is a Poincare map That is, there are open neighborhoods m' E S I, W e Wo 2 and a n Wl ' k C and ~ I: 0 pI Y also (in , of W of m E S, W 2 2 , H: W ->- W ' such that 2 2 diffeomorphism I W2 e Wo and the diagram , 1 0- (W,) ; j .W W2----1.8....'--», commutes. y) of n 0W, THE HOPF BIFURCATION AND ITS APPLICATIONS 58 P(x)= F(x,t - 8{x» Figure 2B.l Here is the idea of the proof of existence of further details see Abraham-Marsden [1). trarily. By continuity, borhood U o assumption, of m. of at x U 2 t-differentiable at = m P P tive of FT-O(x) (x) E S. This is P(x), will be as differentiable as at m FT arbi- of m. t = 0 By and is and hence also in a neighborhood Therefore, there is a unique number such that tion S is S (for is a homeomorphism of a neigh- to another neighborhood FT+t(X) transverse to of m FT Choose P F o(x) near zero and by construcis. The derivative is seen to be just the projection of the derivaon a diffeomorphism, T S m P (this is done below). will be as well. particular, this means Ft t > 0 FT is CJ For partial differential equations, a semi-flow i.e. defined for Hence if F is often just t (See Section 8A). In is not generally a diffeomorphism. THE HOPF BIFURCATION AND ITS APPLICATIONS (For instance if 59 is the flow of the heat equation on L , 2 t is not surjective). For the Poincare mapping this means F F t that we will have a for t > 0), morphism. but P P (if Ft(X) is differentiable in t, x is just a map, not necessarily a diffeo- This is one of the technical reasons why it is im- portant to have the center manifold theorem for maps and not just diffeomorphisms. As we outlined in Section 1, there is a Hopf theorem for diffeomorphisms and this is to be eventually applied to the Poincare map circle for P P to deduce the existence of an invariant and hence an invariant torus for Ft. For partial differential equations, this seems like a dilemma since P need not be a diffeomorphism. can be overcome by a trick: However, this first apply the center manifold theorem to reduce everything to finite dimensions--this does not require diffeomorphisms; then, as proved in Section BA, (see BA.9) in finite dimensions F and hence P will autot matically become (local) diffeomorphisms. Thus the dilemma is only apparent. Let us now prove the fundamental results concerning the derivative of P of Ft. E, and we can let so we can relate the spectrum of P to that It suffices to do the computation in a Banach space m = 0, the origin of We begin by calculating (2B.3) Lemma. Banach space. Let dP(O) Let in terms of F : E + E be a t be on a closed orbit 0 E. Ft. Cl flow on a y with period dF f O. Let ated by V T E = F e V t ~ (0) It=o = V. and let and F Let V be the subspace gener- be a complementary subspace, so that 1 2 (Ft(X,y), Ft(X,y)). Let P: F + F Ft{x,y) 60 THE HOPF BIFURCATION AND ITS APPLICATIONS be the Poincare map associated with y at the point O. Then 2 dP (0) = d1F 'I (0) . Proof. Let 'I(X) the above notation). be the time at Then by definition of P(x) so aF dP(x) Letting P(x) 1 (x) in P, 1 F 'I (x) (x, 0) 1 (X,O)d'I(X) + d1F'I(x) (X,O). at (x,O) = 0, ('I-O (x) we get a F1 1 (O)d'I(O) + d1F (0) . 'I 2 aF ) (O,V) by construc(0), at'I (0) at dP (0) = 1 aF 'I However d t (0) = (a F a F1 'I tion,so a t (0) = o. Thus, -IT- ~ dF (2B.4) Lemma. Proof. ~ (0,0) (0,0) ds = dF 'I So, dF of 'I s ds Is=O dF t = dt (0,0) It='I v (O,O)ls=o (d1F'I(0,0)'0 + d 2 F 'I (0,0)V) (0,0) (O,V) d F (0,0)V 2 'I ( 2B• 5 ) 0 d F (0,0)V = V. 2 'I dF 'I + S Is=O 1 d1F'I(0). dP (0) V. Lemma. 0 CJ (dF'I (0 , 0) ) CJ (P (0» U {l}. (This is true of the point spectrum, too). Proof. * where Let A The matrix of dF (0,0) 'I is indicates some unspecified matrix entry. E~. Then A E CJ(dF'I(O,O» iff dF'I(O,O) - AI is not THE HOPF BIFURCATION AND ITS APPLICATIONS I - l o r is not onto. But (dF T (0, 0) - AI) (~) = (dP ( (dP (0) 0: - Ao:, Clearly, °)0:, 1 Ea(dFT(O,O». (0:,6) we can choose 6 Since The former implies A 'I 1, for any 0:, There- 1'1 A Ea(dP(O}}. On the other hand, let is not onto, then clearly neither is dFT(O,O} - AI. 0. A Ea(dFT(O,O». so that the second component is onto. A Eo(dP(O)}. dP(O) - AI and such that the expression Assume the latter. = A '11 is zero or the map is not onto. fore, + 6) + (- A0: , - A6 ) (2B. 1) Assume A E a(dP(O». If, * (0:) * (0:) + 6 (1 - A». Then either, there exists (2B.l) 61 Suppose there exists Then choosing 6 = *(o:) '-1' 0: such that we see that dP(O}o: - Ao: A E o:(dF (0,0». 0 T A Consult Abraham-Marsden [1] Abraham-Robbin [1] or Hartman [1, Section IV. 6, IX.IO] for additional details on this and and the associated Floquet theory. One of the most basic uses of the Poincare map is in the proof of the following. (2B.6) with period Theorem. T. Let Let m E y be a closed orbit for y and suppose dFT(m) inside the unit circle except for one point circle. Then y Proof. 1 F t has spectrum on the unit is an attracting (stable) closed orbit. By Demma 2B.5, the condition on the spectrum means that the spectrum of the derivative of the Poincare map P at m is inside the unit circle. an attracting fixed pojnt for struction of P that y P. Hence from 2A.7 m is It fo110ws from the con- is attracting for Ft. o 62 THE HOPF BIFURCATION AND ITS APPLICATIONS (2B.7) Exercise. (a) Give the details in the last step of the above (b) If, in 2B.6, proof. P has an attracting invariant circle, give the details of the proof that this yields an attracting invariant 2 torus for the flow.