5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

138 (iii). If q(0+) /s

138 (iii). If q(0+) /s the adjoint eigenvector, i.e. q(0+)A(~o, Po) = 0, normalized such that (q(O+ ), D~A(,~o,/to)p(0)) = 1, then D.i(/to) = -(q(0+), First we recall the tIopf bifurcation theorem in finite dimensions, see for instance [Go1:85]. Consider the system of ODE = f(z,/t), (7.3) where x E R n and/t E R, i.e. p = 1. We assume that (Hfl) f(O,/t) = 0, f E C k, k > 2. If we let L(/t) = D~f(0,/t), then we assume that (Hf2) L(/to) has simple eigenvalues at :l:iwo and no other eigenvalue equals kiwo, k E I, (af3) TZe(D~,a(/to)) # O, where a(/t) is the branch of eigenvalues of L(p) through/wo at # =Po- Theorem 7.2 (ttopf bifurcation for a system of ODE) Let the above hypotheses be sat- isfied and let p be the eigenvector of L(/to) at iwo. Then there exist C k-1 functions/t*(e), w*(E) and x*(e), defined for e sufficiently small, such that at/t = /t*(e), z*(e) is a periodic solution of (7.3). Moreover/t* and w* are even in e, /t(0) = ~to, w(0) = wo and x'(E)(t) = ,TZe(e~'p) + o(E). In addition, if z is a small periodic solution of this equation with/t close to ~to and period close to 2_.~ then modulo a phase shift,/t =/t*(E) and z = x*(e). ol 0 , We recall the equation on the center manifold ( note that we do not assume that this equation is two dimensional) fl = Z(/to)y + P00*(/to) r°*N,noa(C(y, ~), v). (7.4) With respect to a basis in X(/to), this is an equation in fln. It is a consequence of (H(1) and the assumption on g that (Hfl) is satisfied. At/t =/to, i.e. v = 0, the eigenvalues of the linearization are given by the purely imaginary roots of the equation det(A(A,po)) = 0. To satisfy (Hf2) we assume (H(2) At p =/to, the equation det(A(,\,/t0)) = 0 has simple roots at ~ = +iwo and no other root equals )~ = kiwo, k E l. The eigenfunction of A(#o) at eigenvalue iwo is given by p(e) = p(0)e (7.5) where p(0) is a nontrivial solution of the equation A(iwo, po)p(0) = 0. Let q(0+) ~ 0 satisfy q(O+)A(-iwo,/to) = 0. If q(t) = q(0+) + ft g(r) dr (7.6) a(t) = f h then q(t) is an eigenfunction of A*(po) at the eigenvatue -iwo: A'(/to)q = -iwoq, (7.7)

and 139 (q,p) = foh d-"~)p(-r) (7.8) = q(O+)DxA(iwo, Po)p(O). We let P be the projection operator on the two dimensional subspace of Xo(Po) given by and we write PC = (q, ¢)P + (q, ¢)/T, (7.9) ¢ = u + v, u e 7~(P), v e Af(P). We let z = (q, ¢), ~, = (~, ¢). If w are coordinates in (I - P)X0(P0) then with respect to the coordinates z, ~., w the linear part of (7.4) is given by the matrix M(#) = M(p0) + D,M(po)(P-po)+o(p-po),where i~ 0 0 ) (7.10) M(#0) = o -i~ M(#) has a branch of eigenvalues, say a(p), through/tOo, and DI, a(po) = D,Mll(I~o) 0 ~o h = (q, Po°'r O* dD~,C(O, po)p(-O)) = (q, r ®" L h dD,C(O, po)p(-O)) -- (q(o+), dD,.C(O,.o)p(O)e So the condition that guarantees the transversality is (He3) TCe(q(O+)D~,A(iwo, Po)p(O)) ~ O. = We now state the Hopf bifurcation theorem for a system of FDE. Theorem 7.3 (Hopf bifurcation for a system of FDE) Assume (H~I-H~3) and let g be as in (7.1). Then there exist C k-1 functions p*(e), ¢*(e) and w*(e), with values in R, Xo(Po) and R respectively, defined for e sufficiently small, such that the solution of (7.1) with initial condition ¢ = C(¢*(,).p*(e)- Po) is ~ periodic. Moreover, #'(e) and w*(,) are even in, and if x is any small periodic solution of this equation with p close to Po and period close to ~--~ then modulo a translation zo = C(¢*(e),#*(e) - #o) and p = p*(e). tao' Proof. The assumptions guarantee that Theorem 7.2 applies to (7.4). So on the center manifold (7.1) has a periodic orbit. Conversely, any small periodic solution of (7.1) lies on the center manifold and hence is a small periodic solution of (7.4). [] Remark. The hypothesis (H(1) is somewhat restrictive. For instance, the problem = - + g(=,,#)

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