J. Austral. Math. Soc. Ser. B 35(1993), 103-113 NUMERICAL COMPUTATION OF SYMMETRY-BREAKING BIFURCATION POINTS B. S. ATTILI1 (Received 14 February 1989; revised 26 February 1991) Abstract We consider symmetry-breaking bifurcation points which arise in parameterdependent nonlinear equations of the form /(JC, X) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundaryvalue problems to obtain a finite-dimensional problem. 1. Introduction Many physical problems can be formulated as a parameter-dependent nonlinear equation of the form /Oc,A.) = 0, (1.1) where / is a C2-function which maps Rn x R into R". The point (x0, Xo) is called a regular point of f(x, X) = 0 if fx(x, X) is nonsingular. It is called a simple singular point of f{x, X) = Oif fx(x, X) is singular; / j 0 = fx(x0, Xo) and f°J each has one-dimensional null space spanned by 0o and Vo respectively. Whether (or not) bifurcation occurs at a singular point depends on the type of singularity, which is described by (in)equalities involving the derivatives of f(x,X) at (x0, Xo). Typically, turning points, and transcritical and pitchfork bifurcation occur. 'K.F.U.P.M., P.O. Box 1927, Dhahran 31261, Saudi Arabia © Australian Mathematical Society, 1993, Serial-fee code 0334-2700/93 103 Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 104 B. S. Attili [2] To compute such points, we use the extended system proposed by Griewank and Reddien [4, 5]; that is, where g(x, k) = uJ fx(x, k) • v and u and v are the left and right null vectors of /JJ0 respectively. More on this system will be given in Section 3 (see also Attili [1,2]). The outline of this paper will be as follows. Some assumptions and definitions are given in Section 2. In Section 3, we present the relation between symmetrybreaking bifurcation points and pitchfork bifurcation, and details on the extended system will also be presented. In this section we also present the main theorem of this paper, which shows that a symmetry-breaking bifurcation point is an isolated solution of the extended system. Numerical details are given in the final section. It is important to note that Werner and Spence [9] have considered computation of such points using a different extended system; also some other investigators considered systems which do not utilise the symmetry, see Moore [6] and Werner [10]. But the system considered here is more direct, and reduces the number of equations and unknowns considerably, in comparison with the system considered by Werner and Spence [9]. 2. Assumptions and definitions Before defining what is meant by a symmetry-breaking bifurcation point, we shall recall some standard theory on bifurcation. The point (*0> k0) is a simple singular point of f(x, X) — 0 if f(x0, k0) = 0, N(f°) =span{</>0} and *(/?) = iy € R" : fh = 0}. DEFINITION 2.1. A simple bifurcation point (xo, k0) is called a pitchfork bifurcation point of f(x, k) = Oif V0TA° = 0, ^o T />o0o = Oand ^ ( / A + Z i ^ o V o ) # 0 where f°v0 = — / x °. If Vro"/tox0o0o # 0, the bifurcation point is called transcritical. We shall assume that (Al): There exists 5 e L(R"), the bounded linear operators mapping R" into R", with 5 # / and S2 = I and f(Sx, A.) = Sf(x, k) for all JC e Rnandk e Rn. Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 [3] Numerical computation of symmetry-breaking bifurcation points 105 Here (Al) will give rise to the natural decomposition R" = R"s 0 R"a, (2.1) /?? = [x e R" : Sx = x] (2.2) where is the set of symmetric elements, and Rna = {x e R" : Sx = -x} (2.3) is the set of antisymmetric elements, see Sattinger [7]. The decomposition arises because x e R" can be written as 2x = (/ + S)x + (/ - S);t,where Sx e R" and (/ - S)x e R"a. To see this relation, note that S(I + S)x = Sx + S2x = Sx + Ix = (/ + 5)JC which implies that (/ + S)x e Rns and also (/ - S)x e Rna since S(/ - S)* = (5 - S2)x = (5 - 7)jc = - ( / - 5)JC. DEFINITION 2.2. If (x0, Xo) is a singular point of f(x, X) = 0, then (x0, Xo) is called a symmetry- breaking bifurcation point of f(x, k) ifx0 e ^?" and0 o € /?^ where span{0o} = 3. Symmetry breaking and pitchfork bifurcation points The singular points in definition (2.2) will be shown to be pitchfork bifurcation points of f{x, X) = 0 under certain generic conditions. Before doing that, we state some results which can be obtained from assumption (Al). LEMMA 3.1. Assume that (Al) holds. Then for all X e R andx, w,v 6 Rn,we have MSx,X) fASx,X)Sv fkxiSx,X)Sv fxxiSx,X)SvSw = SfdSx.k), = SfASx,X)v, = SfkxiSx,X)v, = P - ; SfxxiSx,X)vw. All results in (3.1) can be obtained by differentiating fiSx, X) = Sfix, X) with respect to X or x. Using (3.1), one can show that for X e R and x e /?" fix,X),Mx,X)eR"s. (3.2) This is true if x e /?; then Sx = x and so fix, X) = fiSx, X) = Sfix, X), which implies that fix,X) e R". The same can be said about /x(x, X). Note Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 106 B. S. Attili that R" and /?£ are invariant with regard to fx(x, k) and fkx(x, since if v € R" then Sv = v and so fAx,k)v = Sfx(x,k)v. [4] k). This follows (3.3) Also, for v, w 6 R" or v, w e fl", fxx(x, k)vw e /?". The reason is that if v, w are in R", then Sv = v and 5io = w, which yields / „ ( * , A)vu> = fxx(x, X)Svw = Sfxx(x, X)vw, and thus fxx(x, k)vw e /?;. If v e R", w e R" thenfxx(x,k)vw e Rna. LEMMA 3.2. For every x € /?", we have ^x = 0. PROOF. Since R" is invariant with respect to f° as in (3.3), then R" is the range of ./j 0 . The conclusion now follows from the Fredholm Alternative. Standard theory of bifurcation shows that at a symmetry-breaking bifurcation point, the solution set of (1.1) will consist of two smooth transversally intersecting branches (see Brezzi, Rappaz and Raviart [3]); that is, C, = {(x, k):k = ko and Ca = {(JC, k) : k = k0+oU2), where Cs and Ca refer to the symmetric and antisymmetric branches respectively. The Implicit Function Theorem will guarantee that Cs C R" since f£ is restricted to R" is an isomorphism of /?". Hence Ca bifurcates from a symmetric branch of solution Cs at a symmetry-breaking bifurcation point. Such branching is given by 0o € /?a- To compute symmetry-breaking bifurcation points, let us consider the extended system -ulfx(x,k)v where u = \jr0 and v = <pQ, the left and right null vectors of f° respectively, g{x, k) is obtained from the following lemma. LEMMA 3.3. The systems Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 [5] Numerical computation of symmetry-breaking bifurcation points 107 and where v and u are n x 1 vectors, vectors, are both uniquely solvable g is a scalar function, and T and R are n x l 8 = ~uTfxV, 3 ?) f/ie prime denotes differentiation with respect to x or A.. Here 7"T and /? are normalisations for v and u respectively, which forces fx to be singular, see Attili [1,2] and Griewank and Reddien [4, 5]. Also v and R are chosen to be in Rna. It follows from (3.2) and (3.3) that F(x, A.) maps Rf x R into R" x R. Hence a symmetry-breaking singular point (*o, A-o) corresponds to a solution (JC0, A.o) of (3.4). Moreover, the next theorem shows the relevance of (Al) for pitchfork bifurcation. See Brezzi, Rappaz and Raviart [3]. THEOREM 3.1. Assume that (Al) it holds and let (x0, A.o) be a symmetry breaking simple singular point. Let F(x, A.), as in (3.4), be a mapping from R" x R into R" x R then (x0, A.o) is an isolated solution of (3.4) if and only if(xo, A.o) is a pitchfork bifurcation point. PROOF. The Jacobian of (3.4) is At (*<), A.o), consider the system r /? A° l r^i-o L ^ o / ^ tfUfo J ' L <* \ which when expanded gives (i) (») ^ Since -/ A ° = / ? • u0 and /X°0O = 0, then from (3.9-i), we have (3.10) Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 108 B. S. Attili [6] Substituting (3.10) in (3.9-ii) we get & T lf > = 0. (3.11) Moreover t/^J fxx(f>o<po = 0 since (x0, k0) is a pitchfork bifurcation point, and hence (3.11) becomes «[^ 0 T (/A + f?x4>ovo)] = 0. But ^ofxX(f>o + fxxfovo # 0 since (x0, Ao) is a pitchfork bifurcation point, this yields a = 0. Thus / = £0 O which implies t e /?", but ? e /?", then ^ e Rns PI /?^ = {0} and hence (3.10) has only the trivial solution. The solution {t, a)J is nontrivial if and only if there exist ana e R and t e R" such that A° + ft = o, and or equivalently, the null space of (3.8) is trivial if and only if and t satisfies A° + ft = o. Thus (xo, ^-o) is a pitchfork bifurcation point. The above theorem shows that the symmetry-breaking bifurcation point is an isolated solution of the extended system (3.4), hence it will be used to compute such points. Moreover, the same system can be used to compute certain double singular points (x0, kQ) of f{x, k) = 0, as we shall show here that (x0, ^-o) is again an isolated solution to (3.4), where F(x, k) = 0 (the extended system) maps R" x R into R" x R. Now assume that N(f?) = span{0o0i}, <£o e K - {0} and </>, e Rns - {0}, fl(/°) = {yeRn: fry = 0, i = 0, 1}, (3.12) (3.13) and tfx =0 forx e R", xfrjx = 0 for* e Rna. (3.14) With these assumptions, we will have the following theorem. Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 [7] Numerical computation of symmetry-breaking bifurcation points 109 THEOREM 3.2. Assume that (Al) holds and let (x0, k0) be a double singular point with x0 € R". Assume also that (3.12), (3.13) and (3.14) hold. With F(x,k) given by (3.4) considered as a mapping on R" x R, then (x0, XQ) is an isolated solution of (3.4) if and only if f® g range(/J°) and t/fo/t°J</>o0i # 0. PROOF. Consider ^!</>oJ• U _ r 0 > (3l5) where the matrix in (3.15) above is the Jacobian of (3.4). Equation (3.15) implies (i) f!'t+af° = 0, (ii) ^o T />or+a^ 0 T /c>o = 0. ^O) Now since / ° £ range(/;c0), then rfrjf° ^ 0 and so (3.16-i) implies that f] f?t+ax//] f° = 0. Further since ifrjf? = 0, t h e n a ^ A 0 = O . b u t ^ / " # 0, this shows that a = 0. Now (3.16-i) becomes f°t — 0, and thus t e N(f°), or equivalently t = a</>0 + b<f>\. Substituting this in (3.16-ii) we obtain «^oT/,>o0o + bflf^fa = 0. (3.17) From (3.12), </>0 € /?„ and so f°x(pO(f>o 6 /?" as shown earlier in this section, also irjf°x4>o4>o = 0 since \frjx = 0 for every x e R" (Lemma 3.2), (3.17) becomes b\//Q fxx(po<f)i = 0, which means b = 0 since irj f°x4>o<l>i # 0, and so / = a</>0Here f = a</>0 implies t e Rna since 0 O € ^ J . but t € R" since F ( x , A.) maps /?; x R into /?; x /?, which shows t e R? D Rna - {0} and so a = 0. Thus (3.15) has only the trivial solution. The other side of the "if and only i f statement follows directly. A result similar to the above is stated without proof in Werner and Spence [9], where they considered a different extended system than the one presented here. 4. Numerical examples 4.1. Coupled cell reaction. Consider the nonlinear operator Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 110 B. S. Attili [8] where x = (x\, x2) € ./? x R, which represent a two-cell exothermic reaction (see Werner and Spence [9]), where h(xi,x2, A.) = xi +e(x, -x2)-Xexp[xi/(l +Sx,)}. Here e is some coupling coefficient and the exponential term is the Arrhenius reaction rate term. Solving the extended system (3.4) with e = .01 and £ = .23 we obtained (Xi,x2,k) = (7.7773552,7.7773261,-47827289) = (2.430591,2.4306056,-51124817), which are two simple and (xux2,k) symmetry-breaking bifurcation points. With e = 0, the simple symmetrybreaking bifurcation coalesces with the simple turning point. This means that assumption (Al) will split the double turning point referred to in Theorem 3.2 into a simple turning point and a simple symmetry-breaking bifurcation point. 4.2. The Duffing equation. The system (3.4) in the previous section will be a basis for the computation of symmetry breaking; that is, pitchfork bifurcation points. Such points will be isolated solutions of (3.4). For more on the experimental aspects, consider the Duffing equation which exhibits secondary bifurcation, see Seydel [8]; that is, Jc + 2;c3=A.cosf, x(0)=x(2n). (4.1) We are seeking 2n-periodic solutions. The assumption (Al) of Section 2 is satisfied directly for (4.1) because of the symmetry properties of the cosine for S € (Sa, Sfi, Sy), where (Sax)(t) = x(-t), (Spx)(t) = -x{n-t), (4.2) (Syx)(t) = -x(t - 7t). Equation (4.1) has a branch of 2n-periodic functions which are symmetric with respect to all symmetries. On this branch there are several symmetrybreaking bifurcation points at which two symmetries are broken and one symmetry is preserved. See Werner and Spence [9]. We shall discretise (4.1) using the shooting method to obtain a finite-dimensional problem. We start by dividing the interval [0, 2n] into four equal subintervals. Let or (0) = S\, x(0) = s2, x(n) = s3 and JC(TT) = *4 be initial guesses at 0 and n. With s\ and s2 as initial guesses, solve the two initial-value problems on [0, n/2] and backward on [37r/2, 2n] to obtain xx, x\ at n/2 and x4, x4 at 37r/2 respectively. Also with s^ and s4 as initial guesses, solve the two initial-value Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 [9] Numerical computation of symmetry-breaking bifurcation points 111 problems backward on [n/2, n] and forward on [n, 3n/2] to obtain x2, x2 at n/2 and JC3, i 3 at 3n/2 respectively. Now " xx -x2 F(s, k) = X\ -x2 (4.3) X3 -Xi, -x4 _ For the finite-dimensional problem, (Sax)(t) = —x(t) will be equivalent to saying s2 = 0 and s4 = 0, and so an analogous symmetry to Sa is 5,= 1 0 0 0 0 - 1 0 0 0 0 1 0 0 0 0 - 1 (4.4) The second symmetry (Spx)(t) = —x(n — t) will be analogous to 0 0 - 1 0 0 0 0 1 - 1 0 0 0 0 1 0 0 (4.5) Also, (SYx)(t) = —x(t — n) is analogous to 0 0 - 1 0 0 0 0 - 1 - 1 0 0 0 0 - 1 0 0 (4.6) To show that assumption (Al) holds for the finite-dimensional problem, we shall deal with F(s, k), where F(s, k) = QF(s, k) and Q= l l l - l 1 1 l l i i - l l l l - 1 1 (4.7) Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 112 B. S. Attili [10] Note that solving F(s, k) — 0 is equivalent to solving F(s,k) = 0 , since Q is invertible. The function F is given by F(s, k) = X\ — Xi + *3 — X4 — X\ + X2 + XT, — X4 X\ — X2 + X3 — X4 + X\ — X2 — JC3 + X4 —X\ + X2 + X-} — Xj + X\ — X2 + X3 — X4 Xl — JC2 — *3 + X4 + X\ — X2 (4.8) Now, under the 5i symmetry, we have si (4.9) S\s = Si \ s4 J \ —s4 Solving the initial-value problems with S\, —s2 and s3, —s4, we obtain x4, —x4 on [0, n/2], JC3, — x3 on [n, n/2], x2, —x2 on [n, 3n/2] and xu —xt on [In, 3n/2]. Using these values we find F(Sys, k) is the same as F(s, k) with the second component and the fourth component reversed in sign, which is SiF(s, k), and so Similarly, one can prove the same result for the other symmetries, which shows that assumption (Al) is satisfied. For the computations, we can find the symmetric solution by solving jci +2x\ = kcost, (4.10) on [0, n/2]. In order to find the antisymmetric solution, we solve jc, +2x\ = kcost, 5i, (4.11) on [0, n/2]. There will be no need to solve the other initial-value problems on the other subintervals. The solutions may be extended based on the symmetries. We transform (4.10) and (4.11) into a first-order differential equation to obtain X\ = X2 , x2 = — 2x\ + kcost, (4.12) together with the initial conditions Xi(0) = Si and ^2(0) = 0 in the symmetric case and ^ ( 0 ) = 0 and x2(0) = st in the antisymmetric case. In the symmetric case, we need to satisfy x, (n/2) = 0 and so Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293 [11] Numerical computation of symmetry-breaking bifurcation points To find g we used R = 1 and 7 T = 1. With this choice, g = BF/dsx. function dF/dsi can be obtained by solving the initial-value problem u, = v2, v2 = — 6u]vi, V](0) = 0, 113 The v2(0) = 1 and 3 F / 3 A. is obtained by solving u>i = w2, w2 = cost — 6u\w\, tUi(0) = 0, u;2(0) = 0. Using Newton's method, (£) = 0 was solved with ^ = 2.88 and A, = 11 as initial guesses. We find s{ = 2.97410 and A. = 12.0776. The value of g = —.111768 - 10. The same system was solved using other initial guesses, s\ — 1.2 and X = 1.35. We find 5, = 1.33150 and A. = 1.57408. Note that this problem was solved by Werner and Spence [9] using finite differences, and similar results were obtained. Acknowledgement The author would like to thank the referees for their valuable comments. References [1] B. Attili, "Multiple shooting and the calculation of some types of singularities in B.V.P's", Int. J. Comp. Math. 32 (1990) 97-111. [2] B. Attili, "A direct method for the characterization and computation of bifurcation points with corank 2", Computing 48 (1992) 149-159. [3] F. Brezzi, J. Rappaz and P. Raviart, "Finite dimensional approximation of nonlinear problems, Part III, Simple bifurcation points", Numer. Math. 38 (1981) 1-30. [4] A. Griewank and G. Reddien, 'The approximation of generalized turning points by projection methods with super convergence to the critical parameter", Num. Math. 14 (1975) 354-366. [5] A. Griewank and G. Reddien, "Characterization and computation of generalized turning points", SIAMJ. Numer. Anal. 21 (1984) 186-196. [6] G. Moore, "The numerical treatment of nontrivial bifurcation points", Num. Fund. Anal. Optim. 2(1980)441^72. [7] D. Sattinger, "Group theoretic methods in bifurcation theory", in Lecture Notes in Math. 762,(1979). [8] R. Seydel, "Branch switching in bifurcation problems for ordinary differential equation", Num. Math. 41 (1983) 93-116. [9] B. Werner and A. Spence, "The computation of symmetry breaking bifurcation points", SIAMJ. Num. Anal. 21 (1984) 388-399. [10] H. Werner, "On the numerical approximation of secondary bifurcation problems", in Lecture Notes in Math. 878, (1981), 407^425. Downloaded from https://www.cambridge.org/core. IP address: 3.235.21.12, on 27 May 2020 at 15:54:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0334270000007293