two new approaches to computing hopf bifurcation problems

13
Chaos, So/mm & Frtzcrals Vol. 4. No. 12. PP. 2203-2215, 1994 Copyright 0 1994 &via Science Ltd Printed in Great Britam. All riehts reserved 0960-0779/9%7.00 + .@I 0960-0779(94)E0125-9 Two New Approaches to Computing Hopf Bifurcation Problems WU BAISHENG Department of Mathematics, Jilin University, Changchun 130023, P.R. China and TASSILO KfjPPER Institute of Mathematics. University of Cologne, Weyertal 86-90, D-50931 Kdln, Germany (Received 23 March 1994) Abstract-We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter F; practical experience shows that one gets convergence for F sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures. 1. INTRODUCTION We consider the autonomous system of ordinary differential equations dY _ = F(y, 4 dr (1) where F : R” x R + R” (n 2 2) is a smooth map and A is a bifurcation parameter. The determination of periodic solutions bifurcating off steady-states along the lines of Hopf bifurcation is by now a well-established subject [6]. Nevertheless, we propose two new approaches which are robust, avoid the typical drawbacks, are easy to implement and can be realized by standard routines. Moreover, we expect our approach to be accessible to more general bifurcation situations. We assume that there is a family of periodic solutions of (1) bifurcating from a branch y”(A)(Al < A < &) of solutions of the stationary equations F(y, a> = 0 (2) at A = A,, E (A,, AZ), y = y”(&) = yo. This is guaranteed by the standard hypotheses on Hopf bifurcation: (Hl) F is twice continuously differentiable in a neighbourhood of (y,,, A”). (H2) The Frechet derivative [(y”(A), A)(A, < A < &) has a pair of simple complex

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Chaos, So/mm & Frtzcrals Vol. 4. No. 12. PP. 2203-2215, 1994 Copyright 0 1994 &via Science Ltd

Printed in Great Britam. All riehts reserved 0960-0779/9%7.00 + .@I

0960-0779(94)E0125-9

Two New Approaches to Computing Hopf Bifurcation Problems

WU BAISHENG

Department of Mathematics, Jilin University, Changchun 130023, P.R. China

and

TASSILO KfjPPER

Institute of Mathematics. University of Cologne, Weyertal 86-90, D-50931 Kdln, Germany

(Received 23 March 1994)

Abstract-We consider the computation of Hopf bifurcation for ordinary differential equations. Two new extended systems are given for the calculation of Hopf bifurcation problems: the first is composed of differential-algebraic equations with index 1, the other consists of differential equations by using a symmetry inherited from the autonomous system of ordinary differential equations. Both methods are especially suitable for calculating bifurcating periodic solutions since they transform the Hopf bifurcation problem into regular nonlinear boundary value problems which are very easy to implement. The bifurcation solutions become isolated solutions of the extended system so that our methods work both in the subcritical and supercritical case. The extended systems are based on an additional parameter F; practical experience shows that one gets convergence for F sufficiently large so that a substantial part of the bifurcating branch can be computed. The two methods are illustrated by numerical examples and compared with other procedures.

1. INTRODUCTION

We consider the autonomous system of ordinary differential equations

dY _ = F(y, 4 dr

(1)

where F : R” x R + R” (n 2 2) is a smooth map and A is a bifurcation parameter. The determination of periodic solutions bifurcating off steady-states along the lines of Hopf bifurcation is by now a well-established subject [6]. Nevertheless, we propose two new approaches which are robust, avoid the typical drawbacks, are easy to implement and can be realized by standard routines. Moreover, we expect our approach to be accessible to more general bifurcation situations.

We assume that there is a family of periodic solutions of (1) bifurcating from a branch y”(A)(Al < A < &) of solutions of the stationary equations

F(y, a> = 0 (2)

at A = A,, E (A,, AZ), y = y”(&) = yo. This is guaranteed by the standard hypotheses on Hopf bifurcation:

(Hl) F is twice continuously differentiable in a neighbourhood of (y,,, A”). (H2) The Frechet derivative [(y”(A), A)(A, < A < &) has a pair of simple complex

2204 WU BAISHENG and T. KiiPPER

conjugated eigenvalues p(A), &A) such that

44 = 44 + iP(A)

with a(&) = 0, /3(h) = on > 0, a’(&) f 0. (H3) F,,(y”, &) has no eigenvalues of the form kiw,(k = 0, +2, +3, . . .). As usual we normalize the system (1) such that the periodic solutions have a fixed period

21r: i.e. we set s = cot and obtain

qy, A, w) := us - F(y. A) = 0,

(3b)

where CZn(jW, iW”)rsp. C:,([w, rW’> denote the Banach spaces of continuous rsp. continuously differentiable 2a-periodic functions y : IF!, + R” employed with the usual norms.

The numerical computation of Hopf bifurcation consists mainly of two steps: the detection of the bifurcation points and the determination of the emanating limit cycles.

The methods presented by Langford [12] and Weber [22] to compute periodic orbits bifurcating from a trivial stationary solution are based on expansions and need derivatives of higher order of the right-hand side which are expensive to evaluate. The region of convergence of Seydel’s method [16] strictly depends on the choice of two parameters, and in addition, the right-hand side of the resulting two-point boundary value problem is not continuous. Jepson and Keller [7] derived a continuation method using pseudo-arclength normalization to compute the bifurcating branches. Due to the high singularity of their system at the Hopf bifurcation point the method converges only within a cone neighbour- hood of the bifurcation point.

The usual extended systems determining the periodic solutions do not exclude the stationary solutions. Hence, especially in cases when an unstable orbit close to a stable equilibrium is to be computed special care is required [ 11, 191. Recently, Dellnitz [l] proposed a numerical method based on Fourier expansions of 2~ periodic functions. However, their approach has the same drawbacks as the previous one. Although there is already a vast amount of methods to treat Hopf bifurcation (see [19] for a review) we propose two new approaches.

Both are constructed along the following lines: first we set up an augmented system which is then embedded into a regular family of problems with a uniquely determined branch of solutions. For the numerical realization these systems are converted into regular boundary-value problems which can be treated by standard routines. In this way we obtain two methods by which the bifurcating periodic solutions are calculated directly. In both cases the right-hand sides are easy to implement.

The first approach leads to a DAE-system, and it is mainly designed to determine the periodic solutions if the branch of steady-state solutions is easily accessible. The resulting boundary-value problem in this case consists of n + 3 equations. The second one exploits the symmetry of Hopf bifurcation and it leads to a system of differential equations. Here we obtain a system of 2n + 3 equations, but with the additional advantage due to symmetry that we only have to solve the BVP on [0, rr] instead of [0, 2n] so that the amount of work is considerably reduced.

2. PRELIMINARIES

To formulate the methods we collect some standard facts of Hopf bifurcation. To simplify the notation we denote Z$(y,,, A,,) by F? etc. throughout the paper.

Hopf bifurcation problems 2205

Let Fit denote the transpose of Fe. Condition (H2) implies that there are vectors

p = a + ib, q = c +id in C”/(O) such that (cf. [4], VIII, 02)

Fy,p = iwOp, F:‘q = -ia+q, (44

p-lp = 2, q-‘p = 2, qfp = 0. (4b)

Let L denote the linear differential operator

LX := & - F” ” ds

?.x, x E &Iw, IFP).

Then

L”z := ‘o,,$ + F;‘z, z E &([w, 1w”)

is the adjoint operator of L with respect to the inner product

(x, z) = $;z’(s)x(r)dS.

(6)

(7)

(5)

Now set

u 1= Re(e’“p) = acoss - bsins, u2 = Im(eisp) = asins + bcoss, (8)

(1: = Re(e’“q) = ccoss - dsins, u,* = Im(e”q) = csins + dcoss. _ (9)

Then ([4], VII, $1.4; VIII, $2)

ker L = span{u,, o,}, ker L” = span {UT, u:}, (10)

(Ui, u,> = a,, (U,* U,) = 6, (i, i = 1, 2), (11)

range L = (ker L”)‘. (12)

Let 1 denote a constant vector with nonzero projection on span {a, b}. There is (except for a sign) a uniquely determined vector c = (cl, cZ)’ such that

c: + c; = 1, .!‘(~,a + c,b) = 0. (13)

We use c to specify a solution

of the linear system

h,(s) := C,Q(S) + C&(S) E &(Iw, IfY) (14)

w dk - F”,h = 0 ‘Ids J

satisfying (ho, ho) = 1 and f’h,,(O) = 0.

3. THE DAE-APPROACH

(15)

The reduction of a branching system of differential equations for the detection of Hopf-points to a DAE-system has already been pointed out by Seydel [18] but, to our knowledge, the regularity of the corresponding DAE-system has not been proved so far. Here, we are going to close that gap and, in addition, we propose a numerical method for calculating periodic orbits that bifurcate from a Hopf-bifurcation point based on the

2206 WU BAISHENG and T. KijPPER

DAE-system. Besides the theoretical aspects collected in Theorem 3.1 its main advant- age is given by the regular augmented system (30) whose properties are described in

Theorem 3.2. Define the map

G1 : R” x C;,(R, Rn) x R x R --$ 52” x C&R, [w”)

bY

F(Y > 4

G'(y, h, A, (0) := wdh/ds - F,Jy, A)h 1

(h, h) - 1

l’h(0) I

and set up the DAE-system

G’(y, h, A, o) = 0.

XRXR

(16)

(17)

Theorem3.1. Let (Hl), (H2) and (H3) hold at (yO,&,) E KY x R. Then U” :=

(y,,, ho, &, q) solves the DAE-system (17) and the linearization of G’ in u’) is non- singular.

Proof. We verify that the linearization G!O: R” X C:,(R, W) x 53 x R+ R” x C,,(R, IL!“) x R’ XR is one-to-one and onto.

Given w = (wt, w2, w3, w4)’ E R” x C2,(R, R”) x R xR, we wish to find u1 = (x, u, r, k)’ E R” X C:,(R, R”) X R X IR satisfying

Gf,o~’ = w. (18)

This equation has the form

Fpx + rFy = wi, (19a)

o do _ Fou + k dhO 0

“ds ’ ds FyLvhox - rFJhhO = w2 (19b)

2( ho, 0) = w3, (19c)

I’u(0) = wq. (19d)

Since F: is nonsingular by condition (H3), there exist unique solutions g and XI in [w”

for each of the following equations F;g = -F;, GOa)

Fjtx, = wl. (2Ob)

From equation (19a) we get

x = rg + x1.

Substituting (21) into (19b) leads to

wdu _ FUu + k dho - - ds ’ ds

r(F$g + F~,Jh~~ = w2 + Fy,hoxl.

Define (see (13))

fif = c,uT + c&, a; = c& - c,v;.

(21)

(22)

(23)

Hopf bifurcation problems 2207

Then by (10) and (13) we see ker L* = span { fif, a,*>.

Using the Fredholm alternative (12) and formula (23), we derive from (22)

k dho

1 ) ds’ a; - r&hog + &ho, 6:) = (fir, w2 + F;Yhoxl), (244

k dho

1 1 ds’ 6; - r(FO,,hog + F;mAho, 6;) = (U;, w2 + FO,,hoxl). (24b)

By utilizing (ll), (13), (14) and (23), and the Appendix of [15] one concludes that the

coefficient matrix of the left-hand side of (24) becomes

[: ;:::;I

which is invertible by condition (H2). Then (k, r) is uniquely determined from (24), we

also determine JC by (21). Let u. be the unique solution of

%$) - F& = -kdho + r( FFAg + F;,)ho + w2 + F;Yh,,x, (25) ds

where u. E X1 = {x E C:,(R, R”)l( x, fi,*) = 0, i = 1,2}. Then from (22) we get

u = /3,u, + p2u2 + ug.

Finally, substituting (26) into (19c,d) one obtains

(26)

(27)

The determinant of the coefficient matrix of (27) is nonzero, since I is orthogonal to c,a + czb with nonzero projection on span(a) b} . Hence Br, &, u and thus u1 are uniquely determined. It is easily verified that u1 = 0 if w = 0. 0

The regularity of G f,o permits us to embed the DAE-systems (17) into a family of DAE-systems such that there is a uniquely defined branch of isolated solutions passing through u”. This family is constructed by approximating F,(y , A)h by a difference quotient.

Since the solutions are isolated they are easy to determine, and in that way they will provide a robust way of calculating the bifurcating periodic solutions of the original system.

Define

F(y, h, A, E) := [F(Y + Eh, 4 - F(Y 9 ~I/E (E # 0)

F,(Y t n>h (E = (0, (28)

Kl(y, h, A, w, E) := wdh/ds - F(y, h, A, E)

(29)

Then K’ is a Cl-map from [w” X C:,([w, I?) X [w X [w into R” X C2@, If??) X [w X 5% and a direct application of the implicit function theorem gives:

Theorem 3.2. Let (Hl), (H2), (H3) hold at (yo, &,) E IF!” X R. Then the DAE-system

K’(y, h, A, w. E) = 0 (30)

2208 WU BAISHENG and T. KijPPER

has a continuous family of isolated solutions (Y’(E), /Z’(E), A’(E), O’(E)) for /F] less than some ~~~ > 0 such that (y’(O), h’(O), L’(O), ~~(0)) = (yo. ho, 4, IX~).

Corollary 3.3. The family (Y’(E) + E/Z’(E), jll(&))(l / E -C co) forms a continuous branch of nontrivial periodic solutions of (1) with period T(E) = 2rr/t0’(&) bifurcating at (y,,, 4,) from the branch of stationary solutions.

Remark 3.4. Through theoretical estimates it is usually difficult to specify the size of c,, or. more interestingly, of A(F). In our numerical calculations E,) was sufficiently large to compute a substantial part of the bifurcating branch such that standard procedures could be employed for further continuation of the branch.

Remark 3.5. For E # 0 our approach is theoretically equivalent to Seydel’s branch switching to periodic orbits [17]. We note, however, that Seydel’s method leads to a singular system if applied in the bifurcation point. This difficulty (which might arise in the case of narrow areas of suitable initial data) is removed by our approach which leads to a regular system even for E = 0.

Since F,.(yo, 4,) is nonsingular, the DAE-system is of index 1. If the (locally) unique smooth branch of stationary solutions ~“(1 - 6, < A < 2 + 6,) through y’(&) = y,, is expli- citly known (as happens in case of a trivial solution) we can reduce it to a system of n + 3 equations

dh -1 LE(y”(Q h. A, E), ds w

(31a)

db = h’h

ds ’

dA -0 ds ’

do 0 -= , ds

(3lb)

(3lc)

(31d)

together with the boundary conditions

h(0) - h(2a) = 0,

b(O) = 0,

6(2r) - 21T = 0,

/‘h(O) = 0.

(32a)

(32b)

(32~)

(32d)

We obtain a regular boundary value problem over [0,2?r] which can be solved by standard routines.

4. THE SYMMETRY INDUCED APPROACH

While the DAE-approach is particularly useful if the primary branch of steady-state solutions is already known we now present a procedure which gives the periodic solutions without any knowledge of the steady solutions. The augmented system is given by

differential equations; it heavily exploits the underlying symmetry of Hopf bifurcation

[4,191.

Hopf bifurcation problems 2209

Define the bounded linear operator S : CZn([W, Rn) + C&R, [w”) by

(SY)(S) := Y(S + n) (S E [w). (33)

Of course, S maps C:,([w, UV) as well as C2,,(R, R’) into itself. The invariance of (3) under the translation is expressed by

@(Sy,Aw) = sqy, A 4. (34)

Further S satisfies

s f I, s2 = I, SVj = -Vj, sv,*= -VI* (i = 1,2) (35)

which leads to the decomposition

C*,(R, IWn> = c, 0 c_, c;,([w, [w”) = c: 0 c! (36)

where

c+ := {x E C2&R, rw”)lSX = +x>, c: := {x E c:,([w, rwfl)lSx q +x> (37)

consist of symmetric and antisymmetric elements of C2,@, R’), C:,(Iw, R”), respectively. By (34), @ maps C: x R! x R into C,, and <D,(y,,, ho, ma) = w. d/ds - Fe := L maps C: into C+, respectively, and it is well known that

L : C: + C, bijective.

Define the map G2 by

G2(y, h, A, o) := [~~~;-~‘;)h~.

Clearly, u” = (yo, ho, &, wo) is a solution of

G2(y, h,A, w) = 0. (39)

If we consider G* as a map from C:,([w, R”) x C:,([w, [w”) x R x R! into C,,([w, Rn) x &,([w, Rn) x Rx!&’ then the linearization in u” can be seen to be singular, and then additional care has to be taken into account if (39) is considered as an augmented system locating Hopf bifurcation points (see Griewank-Reddien [5]). We avoid these difficulties by considering the restriction of G* to C: x C! X R X R which we denote by G*.

Theorem 4.1. Let (Hl), (H2) and (H3) hold at (yo,&) E 5%” x R. Then u” = (yo, ho, wo) E C!+ x C! x [w x IR solves the system

@(U) = 0; (40)

moreover, the Frechet derivative z’$ is nonsingular.

Proof We prove that GiO : C’: x C! x R x R + C, x C_ x IF! x R is one-to-one and onto. For given z = (zi, z2, z3, z4)’ E C, x C_ x R x R, we want to find u2 = (w, v, m, n)’ E C: x C! x If2 x R such that

@zl2 = z. (41)

This equation may be unfolded by

2210 WU BAISHENG and T. KiiPPER

(42b)

2( ho, u) = zi (42~)

PO(O) = ZJ. (42d)

Since L: C: -+ C, is bijective, from (42a) one solves by using (20a)

w = mg -t- M’, (43)

where w1 E C’: is the unique solution of the equation @r, dw,/ds - Ftwl = z!. Substituting (43) into (42b) leads to

(r) do - F’Jo + n dh,J ” ds ’

__ - ds

m(Fj:,.g + F’:,Jh,, = z2 + F’:.,how,. (44)

Similarly to the proof of the Theorem of 3.1, there is by (H3) a unique vector (m, n) such that y1 := -ndh,/ds + rn(F’l,g + FyA)hn + z2 + Ft,,houil E range L, we also obtain w from (43).

By using the fact that rl E C and letting u”,, E X2 be the unique solution of the following equation

d&l 0 5 w,- - F>u,, = r

ds (45)

where X2 = {x E C! ) ( x. 6:) = 0, j = 1,2}, from (42) we obtain

0 = /$u, + /&11? + c?,,. (46)

Similarly as in the proof of Theorem 3.1, &, pz, u and thus u2 are uniquely determined. It is easily verified that U’ = 0 if w = 0. 0

Again the regularity of c2 permits the embedding of (40) into a family of differential equations in a regular way. To preserve symmetry we approximate FJ (y , h)h by a central difference quotient and F(y, A) by a symmetric mean value [22]. Set

F(y, h, A, E) := [[F(y + t-h, A) - F(y - Eh, A)]/& (F# 0)

I F,.(Y 1 A)h (P = 0) ’

I co% - ;[ F(y + oh, A) + F(y - Eh, d)]

n, A, (0, E) := odh/ds - $(y, h, A. E)

I

(h,h)-1

I.

l’h(0). I

It is obvious that K2 maps C\ x C! x RX iw into C, x C_ x R x R and that

K2(y, h, A. (0, -E) = K’(y, h, A, w, E). Applying the implicit function theorem to the augmented system

K2(y, h, A, cc). E) = 0, (47)

we get:

Theorem 4.2. Let (Hl), (H2), (H3) hold at (y,,, 4,) E IR” x R. The system (47) with the restriction K’: C: x C! x R x R+ C, x C_ x R x R has a continuous family of isolated solutions U(F) = (Y’(F), h’(E), A2(&), a?(&)) E C!+ x C! x R x R for 1~1 less than some F,, > 0 such that (y’(O), h’(O), A”(O), o*(O)) = ( y,,, ho, 4). q); moreover u(--E) = U(F).

Hopf bifurcation problems 2211

Corollary 4.3. The family (Y’(E) + eh2(&), A2(&))(l 1 E < E,,) forms a continuous branch of

nontrivial periodic solutions (both solutions differ only by a phase difference r~) of (1) with

period T(E) = 2n/e?(.s). The system (42) is easily converted into a regular boundary value problem of dimension

2n + 3:

d y -= ds

&[F(y + Eh, 4 + F(.Y - eh, A>],

dh -= btv. h, A, E), ds 20.~ li ’

dA 0 -= ds ’

?!!t = h’h ds ’

dw 0 -= 3 ds

with boundary conditions

Y(0) Y(T) = 0,

(48b)

(48~)

(4Sd)

(4Sd)

(494

h(0) + h(n) = 0,

6(O) = 0,

&a-) - 7r = 0,

(49b)

(49c)

(49d)

th(0) = 0. (4W

Remark4.4. Because of the additional symmetry y E C: and h E CL integrations of the system need only be carried out on the semi-interval [0, ~1. Although the dimension is nearly doubled the overall effort is reduced at the same time through the shortening of the interval.

5. NUMERICAL REALIZATION AND EXAMPLES

For ]E] less than some co > 0 we have derived two regular boundary value problems of dimension n + 3 on [0,27r] and 2n + 3 on [0, a]. The choice E = 0 determines the bifurcation point; for E # 0 we obtain two systems for the periodic solutions which exclude the stationary solution completely and for that reason apply likewise to super- and subcritical bifurcation.

Both systems are very easy to implement and can be treated by standard routines to solve boundary value problems; in our examples we used the subroutine ‘DBVFD’ of the IMSL-package for SUN-workstations.

The convergence is guaranteed since the solutions are isolated by Theorem 3.2 and 4.2 [lo].

For E = 0 both systems yield the Hopf bifurcation point. It is obvious of course to calculate the stationary solutions first. Garratt-Moore-Spence [2] have proposed an efficient method to detect Hopf bifurcation points which seems very favourable in connections with our approaches. Although their method only provides an approximation y’, Al, or, u1 and b’ for the exact values y , ’ A0 coo a and b they can supply starting values , ,

2212 WU BAISHENG and T. KIPPER

for the boundary value problems (31), (32) and (48), (49); for example an approximation for ho can be taken as

&, = Fl(al toss - 6’ sins) + Fz(a sins + b’ toss)

where c”,, T? satisfy

z:+z;= 1, l’a’c”, + I’b’? 2 = 0

and the vector I is chosen as in Griewank-Reddien [5]. In our numerical examples we followed this approach.

The Hopf bifurcation points can of course also be computed by direct methods [5,7, 161. Griewank-Reddien [5] directly supply the quantities y”, il”, co”, a and 6; in the approach by Roose-HlavaEek [16] the vectors a and b are obtained by inverse iteration.

If only the Hopf bifurcation points need to be computed the direct approach [5,7, 161 seems to be more efficient but if the bifurcating periodic solutions have to be determined as well our method appears to be favourable.

We illustrate our methods by several examples which have been established in the literature as test examples.

We start with two examples where the stationary solution is known explicitly and apply equations (31,32).

Example 1 (Lorenz equations [ 131)

dyl - = P(Y2 - Y,). dt

dy-2 __ = (-YlY3 + RY, - yz),

dt

dy3 __ = (~1~2 - by,).

dt

We choose A = R, b = 4, p = 15, then (Seydel [18], p. 67) one branch of steady state solutions is given by (yl, y2, y3) = (S, S, R - 1) for R 2 1 where S = (bR - b)ln. Subcritical Hopf bifurcation occurs at 4, = R. = 33, y,, = (82/2,8d2,32)’ with o,, = 8j/3. We take I = (l,O, 0)‘.

The corresponding boundary value problem (31), (32) has been solved by the IMSL- subroutine ‘DBVFD’ for E ranging between 0 and 17 in steps of AE = 0.5 resp. AF = 0.1 for the first step. Convergence was rapid until E = 16.5; for F = 16.5; for F = 17 convergence failed. The bifurcating periodic solutions are unstable. The corresponding values for A(E) : = A’(c) and the period T(E) are given in Table 1.

Example 2 (A feedback inhibition (Glass [3]))

d y _ = Ay + G(y, ,I) dt

where

&, A) = 1 (1 + 2YZ -1

1 2 (1 + 2y,)” + 1 1 - 2Yr

Hopf bifurcation problems 2213

Table 1. Bifurcation parameter and period for Lorenz equations

0 33.00000 0.4534499 0.1 32.99975 0.4534548 0.5 32.99352 0.4535749 1.0 32.97391 0.4539519 2.5 32.83531 0.4566322 5.0 32.32381 0.4667919 7.5 31.49809 0.4861454

10.0 29.95654 0.5203347 13.0 27.03628 0.6068964 16.5 20.41063 1.1108953

The stationary solution is y = (0, 0,O)'. Hopf bifurcation occurs at &, = 4 with T(0) := T,, = 3.627599. We take I = (1, 0,O)'. Continuation in E worked without difficulties until E = 0.3 in steps of AE = 0.1. To capture the steep ascent of h(e) we continued by a smaller stepsize of AE = 0.01 until E = 0.33. We recall that E = 0.33 corresponds to ,I= 8.62; hence by this approach the covered range of A is larger than elsewhere, for example, in Langford’s method [12] convergence fails at A = 5.16. All of the periodic solutions are stable. For values of A( E) : = AI(E) and T(E) see Table 2.

Since the stationary solution is not explicitly available in the next 2 examples we use these to illustrate the second method (48), (49).

Example 3 (CSTR [20])

dyl ~ = -yl + DA1 - yl)exp(y2),

dt

dy2 - = -y2 + BD,(l - ydexp(y2) - Lb.

dt

As bifurcation parameter we choose the Damkohler number D, = A and we fix p = 3, B = 16.2. Further we take 1 = (1.0)‘. At A = & = 0.2464796 there is subcritical Hopf bifurcation. Using equations (48, 49) we followed the bifurcating branch of periodic solutions (0 G E < 1.5) with a stepsize of AE = 0.1; at E = 1.6 convergence eventually failed. The subcritical periodic solutions branch is stable.

The corresponding values of A(E) : = A2(~), T( E as well as the initial values ~~(0) + ) &h,(O) := y:(O) + phi and ~~(0) + .sh2(0) := y:(O) + c/z:(O) are given in Table 3.

Table 2. Bifurcation parameter and period for the feedback inhibition

F A(&) T(E)

0 4.000000 3.627599 0.01 4.000990 3.627635 0.05 4.025034 3.628466 0.10 4.104104 3.630434 0.15 4.215164 3.631331 0.20 4.503697 3.626264 0.25 4.960033 3.603373 0.30 6.032106 3.522151 0.32 7.171688 3.431834 0.33 8.614472 3.332485

2214 WU BAISHENG and T. KIPPER

Table 3. Bifurcation parameter, period and initial values for the CSTR

E k(e) T(e) Y,(O) + Fh,(O) L’*(O) + F/I:(O) _

0 0.2464796 1.186331 0.9063992 3.670917 0.1 0.2460758 1.189359 0.9056697 3.604795 0.2 0.2448568 1.198709 0.9034647 3.531676 0.3 0.2428036 1.215205 0.8997361 3.451704 0.5 0.2361646 1.276177 0.8874083 3.272680 0.7 0.2267856 1.390541 0.8683230 3.071931 0.9 0.2171668 1.566337 0.8434322 2.853118 1.1 0.2090265 1.786776 0.8129564 2.611359 1.3 0.2016349 2.044413 0.7742273 2.333250 1.5 0.1935918 2.344450 0.7221019 1.996580

Example 4 (Lorenz’ fourth-order system [ 141)

dyl __ = - Y1-t 2/I - y: + (y: + A/2,

dt

dy2 ~ = -Y2 + (YlY2 - Y3Y4) - <y: - y:p,

dt

dY3 ~ = -Y3 + (Y2 - YlKY3 + YJA

dt

dy4 __ = -Y4 + (Y2 + Yl>(Y3 - YJ2.

dt

A rigorous bifurcation analysis is given in [9]. Choosing I = (1, 0, 0,O)' we continue the solution of the resulting boundary value problem (48, 49) in steps of AE = 2.0 over the range [0,72], with a corresponding range for h given by [3.8,459.9]; at E = 74 convergence eventually fails. The precise numbers similar to Example 3 are given in Table 4. When E G 4, the bifurcating periodic solutions are stable; but for F 3 6, they become unstable.

The examples show that both methods work in such a way that they provide both the Hopf bifurcation point and a substantial part of the bifurcating branch of periodic solutions both in the subcritical and supercritical case. We want to point out that our method does not depend on the stability of bifurcating solutions. Of course, the range of .s for which we obtain a solution of the corresponding boundary-value problem by the iteration method in the DAE resp. symmetry induced approach depends on the problem. In all cases we were

Table 4. Bifurcation parameter, period and initial values for Lorenz fourth-order system

4 El T(F) ? Y204 + Eh2@) Y3(0) + h(O)

0 3.85313 1.50756 1.49771 2.69046 1.18864 0.80447 2 5.35846 1.37177 1.76541 4.83637 0.68556 0.58322 4 9.27961 1.13725 2.22805 7.88404 0.45286 0.44648 6 14.8782 0.95150 2.64903 11.2995 0.36129 0.38888

10 29.6572 0.72257 3.38429 18.4160 0.32247 0.37580 20 71.6897 0.43110 6.59104 33.8899 0.85105 1.14921 30 114.207 0.29244 10.7043 47.1631 1.47217 2.34139 40 161.531 0.22174 14.4729 60.5268 1.98270 3.55113 50 215.385 0.17923 17.8533 74.2277 2.40698 4.76313 60 281.077 0.15116 20.6350 88.6845 2.71856 5.94780 70 394.097 0.13243 21.4261 106.430 2.71208 6.95299 72 459.874 0.13030 20.1238 112.900 2.50262 6.98677

able to follow the branches further computation of the cedures.

As a central advantage of

Hopf bifurcation problems 2215

sufficiently far away from the bifurcation points so that the branches can be carried out by standard continuation pro-

our methods we notice that they are very easy to implement and that they lead to robust boundary-value problems.

Acknowledgement-WWu Baisheng was supported by a grant of the Volkswagen-Stiftung.

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2. T. J. Garratt, G. Moore and A. Spence, Two methods for the numerical detection of Hopf bifurcations, in Bifurcation and Chaos: Analysis, Algorithms, Applications, edited by R. Seydel, F. W. Schneider, T. Kiipper and H. Troger, ISNM 97. no. 129-133. Birkhauser. Boston. MA (1991).

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