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Existence of periodic solutions for the Belousov-Zaikin-Zhabotinskiĭ reaction by a theorem of Hopf
JOURNAL
OF DIFFERENTIAL
EQUATIONS
20, 399-403
(1976)
Existence of Periodic Solutions for the Belousov-Zaikin-Zhabotinskii Reaction by a Theorem IN-DING Department
of Hopf
Hsij
of Mathematics, State University of New York at Buffalo, Amherst, New York 14226 Received May 25, 1974
I. INTRODUCTION The purpose of this paper is to illustrate the simplicity of applying a theorem of Hopf [4] in establishing the existence of periodic solutions to the equations of Field and Noyes [l] for the Belousov-Zaikin-Zhabotinskii reaction, and thereby to publicize Hopf’s theorem. This theorem is not as well known and available as it should be. It is a powerful, direct tool, a general theorem of ordinary differential equations that directly establishes existence of periodic solutions to nonlinear systems of orders exceeding 2. It has the disadvantage, compared to applications of fixed point theorems, that is a local theorem, local with respect to a parameter. It has the advantage of possibly leading to a successful analysis of orbital stability. The Belousov-Zaikin-Zhabotinskii reaction is an oscillating oxidation reaction that has attracted much attention recently because of several of its aspects [l-8, 10-121. Here we are concerned with the model proposed by Field, KorBs, and Noyes [l, 21 for the kinetics of this reaction. The basic model is: ff = s(y - xy + x - qxp, j = s-y fx - y - xy),
(1)
5i = w(x - x), wheref, s, q, and w are positive real parameters and (x, y, z) are concentrations, and hence nonnegative. The behavior of the resulting system of differential equations has been examined by Field and Noyes [l], who showed numerically that this system has a stable, closed, periodic orbit in (x, y, z) space. Murray [8] and Hastings and Murray [3] have proved various results for system (1). They have used a theorem of Pliss [9] to show the existence of periodic solutions to (1) for a wide range of the parameters and have made a careful study of the trajectories, showing that all solutions of (1) are oscillatory. Copyright 0 1916 by Academic Press, Inc. All rights of reproduction in any form reserved.
399
400
IN-DING
HSii
We use a different approach, the theorem of Hopf. This approach has the advantage that it may lead to a proof of the orbital stability of the periodic solutions. It has the disadvantage that we cannot show existence of periodic solutions for as wide as a range of the parameters as Murray and Hastings. THEOREM (HoPF). Let G be an open connecteddomain in W, c > 0, and let F be a real analytic function defined on G x [-c, c]. Consider the dafferatial system: ff = F(x, CL), where XEG, 1~~1
Suppose there is an analytic, real, vector function g defined on [-c, c] such that F(g(p), ,u) = 0. Thus OM can expandF(x, II) about g(p) in the form F(x, CL)= L,i + F*(% p),
* = x - g(p),
where L, is an n x ti real matrix which dependsonly on CL,and F*(%, p) is the nonlinear part of F. Supposethere exist exactly two complex conjugate eigenvalues ol(tL),E(p) of L, with the properties Re(or(0)) = 0
and
Re(40)) # 0
(’ = d/dp).
Then there exists a periodic solution P(t, c) with period T(E) of (2) with /.L = p(e), such that ~(0) = 0, P(t, 0) = g(0) and P(t, e) #g@(c)) for all sz.&-zktZy small l # 0. Moreover P(E), P(t, B), and T(e) are analytic at E = 0, and T(0) = 27r// Im LX(O)].Th ese “small” periodic solutions exist for exactly one of three cases:either only for p > 0, or only for p < 0, or only for p = 0.
II. PERIODIC SOLUTIONS The critical points for the system (1) are y = (0, 0,O) and ~a = (x0 , y. , so), where 20 = XOYYO =fxo/U +x0) = QW -tf) - !Pol, al-% = (1 - f - 9) + [(I - f - 412+ 4q(l + f F’“.
(3)
(4)
The eigenvalues h corresponding to the critical point y. satisfy the equation X3+ Ah2 + Bh + C = 0,
(5)
where A=w+ol, B = C2qxo2 + x&q - 1) + f I + OIW, c = W~oP7Xo+
(4 - 1)+fl, fx= SYO + [(l/s) + 2pl x0 +
[(l/s) - s].
(6)
BELOUSOV-ZAIKIN-ZHABOTINSKIi
401
REACTION
By using (3) and (4) and a little algebra, we conclude that A, C, and (Yare positive for any positive f, q, w and s. Suppose 2qxo2 + x,(q - 1) + f < 0, then there exists a unique LEMMA 1. w, > 0 such that at w = w0 corresponding to the critical point y0 of the system (1) are two pure imaginary eigenvahes, fi(B(w,))lla, and one negative real ezgenvalue, -A(w,). Moreover, 2m, = -[a” + f (1 - %)I + ib.” + f (1 - 40?[2qx,2 + xO(q - 1) + f]}‘l”.
%)I” (7)
Proof. One easily shows that there is a unique w,, > 0 such that ws satisfies the equation AB = C, which is cmJ2+ [a” + f (1 - x0)] w + a(2qxfJ2+ x&z -
1) + f) = 0.
(8)
Therefore Eq. (5) can be rewritten in the form (A + A(w,)) (A2 + B(woo,))= 0. Thus the three eigenvalues are -A(w,,) follows directly from (8). i
and -&i(B(w,,))1/2. Conclusion (7)
Now let w = ws + E. Since the coefficients of (5) are continuous in W, the eigenvalues are continuous in E. Hence there exists an E,,> 0 such that the eigenvalue -A( B) is real and negative for any 1E 1 < c,, , while the other two are complex conjugates; E(c) -J$(B(c))llz, where J(O) = A(O), B(O) = B(O), and A(0) and B(0) stand for the coefficients in (6) evaluated at E = 0. LEMMA 2. Proof.
Under the hypothesis of Lemma 1, E’(0) < 0.
Differentiating
(5) with respect to E, we find that
xl(,) =
-A’(E) X2(,) - B’(E) X(E) - C’(,) 3X2(c) + 2A(c) A(E) + B(e) - ’
Note that if c = 0, the three roots of (5) are distinct; hence the implicit function theorem applies so that x’(e) exists for sufficiently small B. But X’(C) = E’(E) j--lW2(~)
B’(<)i.
Thus E’(0) = Re ]
A’(0) B(0) - C’(0) - B’(0)(fB1la(O)i) -2B(O) + 2A(0)(&iB118(O))
’
= 2BON--A’(O)B(O)+ c’(O)- B’(O)A(O)1 4B2(0) + 4A2(0) B(0)
’
402
IN-DING
HSii
where B(0) > 0, and --A’(O) B(O) + C’(0) - B’(0) A(O)
= -[24xoa+x,(p-1)+fl--orw,+2qx,2+x,(q-l)+fx,--(w,+ol) = -2 = -{[a”
- 2oreu,- f(1 - x0) + f(l
- x0)]” - 4a2[2qx,,2+ x0(4 - 1) + f]}‘j2
< 0. Thus E’(0) < 0.
[
Lemmas 1 and 2 show that the hypotheses for Hopf’s theorem are fully satisfied. We therefore have proved the following theorem. THEOREM 3. If 2qxe2 + x& - 1) + f < 0, then there exists a w,, > 0 such that (1) has a periodic solution for exactly one of three cases: either only for each w E (w,, - q, , wo), or only for each w E (wO , w,, + q,), for some positive e0 , or only for w = w, . The value of w, is given by (7).
ACKNOWLEDGMENT The writing
author would this paper.
like
to thank
Professor
N. D. Kazarinoff
for guidance
in
REFERENCES 1. R. J. FIELD AND R. M. NOYES, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Gem. P&s., (1974). 2. R. J. FIELD, E. K~R&, AND R. M. NOYES, Oscillation in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic-acid system, 1. Amer. Chenz. Sot. 99 (1972), 8649. 3. S. P. HASTINGS AND J. D. MURRAY, The existence of oscillatory solutions in the Field-Noyes model for the Belousov-Zhabotinskii reaction, Preprint, 1974. 4. E. HOPF, Abzweigung einer periodischen Lijsung von einer station&en Liisung eines differential Systems, Ber. Verb. Sacks. Akad. Wiss. Leipsig. Math.-Nut. 94 (1942), 3-22. 5. L. N. HOWARD AND N. KOPELL, Spatial structure in the Belousov reaction. I. External gradients; II. Diffusion and target patterns. Preprints. 6. N. KOPELL AND L. N. HOWARD, Horizontal bands in the Belousov reaction, Science 180 (1973), 1171. 7. N. KOPELL AND L. N. HOWARD, Plane wave solutions to reaction diffusion equations, Preprint. 8. J. D. MURRAY, On a model for the temporal oscillations in the BelousovZhabotinskii reaction, in press.
BELOUSOV-ZAIKIN-ZHABOTINSKIi
REACTION
403
9. V. A. PLISS, “Nonlocal Problem of the Theory of Oscillations,” Academic Press, New York, 1966. 10. A. T. WINFREE, Spiral waves of chemical activity, Science 175 (1972), 634-636. wave of chemical activity in three dimensions, 11. A. T. WINFREE, Scroll-shaped Science 181 (1973), 937-939. “Oscillatory Processes in Biological and Chemical Systems,” 12. A. M. ZHABOTINSKI~, p. 149, Science Publ., Moscow, 1967.
OF DIFFERENTIAL
EQUATIONS
20, 399-403
(1976)
Existence of Periodic Solutions for the Belousov-Zaikin-Zhabotinskii Reaction by a Theorem IN-DING Department
of Hopf
Hsij
of Mathematics, State University of New York at Buffalo, Amherst, New York 14226 Received May 25, 1974
I. INTRODUCTION The purpose of this paper is to illustrate the simplicity of applying a theorem of Hopf [4] in establishing the existence of periodic solutions to the equations of Field and Noyes [l] for the Belousov-Zaikin-Zhabotinskii reaction, and thereby to publicize Hopf’s theorem. This theorem is not as well known and available as it should be. It is a powerful, direct tool, a general theorem of ordinary differential equations that directly establishes existence of periodic solutions to nonlinear systems of orders exceeding 2. It has the disadvantage, compared to applications of fixed point theorems, that is a local theorem, local with respect to a parameter. It has the advantage of possibly leading to a successful analysis of orbital stability. The Belousov-Zaikin-Zhabotinskii reaction is an oscillating oxidation reaction that has attracted much attention recently because of several of its aspects [l-8, 10-121. Here we are concerned with the model proposed by Field, KorBs, and Noyes [l, 21 for the kinetics of this reaction. The basic model is: ff = s(y - xy + x - qxp, j = s-y fx - y - xy),
(1)
5i = w(x - x), wheref, s, q, and w are positive real parameters and (x, y, z) are concentrations, and hence nonnegative. The behavior of the resulting system of differential equations has been examined by Field and Noyes [l], who showed numerically that this system has a stable, closed, periodic orbit in (x, y, z) space. Murray [8] and Hastings and Murray [3] have proved various results for system (1). They have used a theorem of Pliss [9] to show the existence of periodic solutions to (1) for a wide range of the parameters and have made a careful study of the trajectories, showing that all solutions of (1) are oscillatory. Copyright 0 1916 by Academic Press, Inc. All rights of reproduction in any form reserved.
399
400
IN-DING
HSii
We use a different approach, the theorem of Hopf. This approach has the advantage that it may lead to a proof of the orbital stability of the periodic solutions. It has the disadvantage that we cannot show existence of periodic solutions for as wide as a range of the parameters as Murray and Hastings. THEOREM (HoPF). Let G be an open connecteddomain in W, c > 0, and let F be a real analytic function defined on G x [-c, c]. Consider the dafferatial system: ff = F(x, CL), where XEG, 1~~1
Suppose there is an analytic, real, vector function g defined on [-c, c] such that F(g(p), ,u) = 0. Thus OM can expandF(x, II) about g(p) in the form F(x, CL)= L,i + F*(% p),
* = x - g(p),
where L, is an n x ti real matrix which dependsonly on CL,and F*(%, p) is the nonlinear part of F. Supposethere exist exactly two complex conjugate eigenvalues ol(tL),E(p) of L, with the properties Re(or(0)) = 0
and
Re(40)) # 0
(’ = d/dp).
Then there exists a periodic solution P(t, c) with period T(E) of (2) with /.L = p(e), such that ~(0) = 0, P(t, 0) = g(0) and P(t, e) #g@(c)) for all sz.&-zktZy small l # 0. Moreover P(E), P(t, B), and T(e) are analytic at E = 0, and T(0) = 27r// Im LX(O)].Th ese “small” periodic solutions exist for exactly one of three cases:either only for p > 0, or only for p < 0, or only for p = 0.
II. PERIODIC SOLUTIONS The critical points for the system (1) are y = (0, 0,O) and ~a = (x0 , y. , so), where 20 = XOYYO =fxo/U +x0) = QW -tf) - !Pol, al-% = (1 - f - 9) + [(I - f - 412+ 4q(l + f F’“.
(3)
(4)
The eigenvalues h corresponding to the critical point y. satisfy the equation X3+ Ah2 + Bh + C = 0,
(5)
where A=w+ol, B = C2qxo2 + x&q - 1) + f I + OIW, c = W~oP7Xo+
(4 - 1)+fl, fx= SYO + [(l/s) + 2pl x0 +
[(l/s) - s].
(6)
BELOUSOV-ZAIKIN-ZHABOTINSKIi
401
REACTION
By using (3) and (4) and a little algebra, we conclude that A, C, and (Yare positive for any positive f, q, w and s. Suppose 2qxo2 + x,(q - 1) + f < 0, then there exists a unique LEMMA 1. w, > 0 such that at w = w0 corresponding to the critical point y0 of the system (1) are two pure imaginary eigenvahes, fi(B(w,))lla, and one negative real ezgenvalue, -A(w,). Moreover, 2m, = -[a” + f (1 - %)I + ib.” + f (1 - 40?[2qx,2 + xO(q - 1) + f]}‘l”.
%)I” (7)
Proof. One easily shows that there is a unique w,, > 0 such that ws satisfies the equation AB = C, which is cmJ2+ [a” + f (1 - x0)] w + a(2qxfJ2+ x&z -
1) + f) = 0.
(8)
Therefore Eq. (5) can be rewritten in the form (A + A(w,)) (A2 + B(woo,))= 0. Thus the three eigenvalues are -A(w,,) follows directly from (8). i
and -&i(B(w,,))1/2. Conclusion (7)
Now let w = ws + E. Since the coefficients of (5) are continuous in W, the eigenvalues are continuous in E. Hence there exists an E,,> 0 such that the eigenvalue -A( B) is real and negative for any 1E 1 < c,, , while the other two are complex conjugates; E(c) -J$(B(c))llz, where J(O) = A(O), B(O) = B(O), and A(0) and B(0) stand for the coefficients in (6) evaluated at E = 0. LEMMA 2. Proof.
Under the hypothesis of Lemma 1, E’(0) < 0.
Differentiating
(5) with respect to E, we find that
xl(,) =
-A’(E) X2(,) - B’(E) X(E) - C’(,) 3X2(c) + 2A(c) A(E) + B(e) - ’
Note that if c = 0, the three roots of (5) are distinct; hence the implicit function theorem applies so that x’(e) exists for sufficiently small B. But X’(C) = E’(E) j--lW2(~)
B’(<)i.
Thus E’(0) = Re ]
A’(0) B(0) - C’(0) - B’(0)(fB1la(O)i) -2B(O) + 2A(0)(&iB118(O))
’
= 2BON--A’(O)B(O)+ c’(O)- B’(O)A(O)1 4B2(0) + 4A2(0) B(0)
’
402
IN-DING
HSii
where B(0) > 0, and --A’(O) B(O) + C’(0) - B’(0) A(O)
= -[24xoa+x,(p-1)+fl--orw,+2qx,2+x,(q-l)+fx,--(w,+ol) = -2 = -{[a”
- 2oreu,- f(1 - x0) + f(l
- x0)]” - 4a2[2qx,,2+ x0(4 - 1) + f]}‘j2
< 0. Thus E’(0) < 0.
[
Lemmas 1 and 2 show that the hypotheses for Hopf’s theorem are fully satisfied. We therefore have proved the following theorem. THEOREM 3. If 2qxe2 + x& - 1) + f < 0, then there exists a w,, > 0 such that (1) has a periodic solution for exactly one of three cases: either only for each w E (w,, - q, , wo), or only for each w E (wO , w,, + q,), for some positive e0 , or only for w = w, . The value of w, is given by (7).
ACKNOWLEDGMENT The writing
author would this paper.
like
to thank
Professor
N. D. Kazarinoff
for guidance
in
REFERENCES 1. R. J. FIELD AND R. M. NOYES, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Gem. P&s., (1974). 2. R. J. FIELD, E. K~R&, AND R. M. NOYES, Oscillation in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic-acid system, 1. Amer. Chenz. Sot. 99 (1972), 8649. 3. S. P. HASTINGS AND J. D. MURRAY, The existence of oscillatory solutions in the Field-Noyes model for the Belousov-Zhabotinskii reaction, Preprint, 1974. 4. E. HOPF, Abzweigung einer periodischen Lijsung von einer station&en Liisung eines differential Systems, Ber. Verb. Sacks. Akad. Wiss. Leipsig. Math.-Nut. 94 (1942), 3-22. 5. L. N. HOWARD AND N. KOPELL, Spatial structure in the Belousov reaction. I. External gradients; II. Diffusion and target patterns. Preprints. 6. N. KOPELL AND L. N. HOWARD, Horizontal bands in the Belousov reaction, Science 180 (1973), 1171. 7. N. KOPELL AND L. N. HOWARD, Plane wave solutions to reaction diffusion equations, Preprint. 8. J. D. MURRAY, On a model for the temporal oscillations in the BelousovZhabotinskii reaction, in press.
BELOUSOV-ZAIKIN-ZHABOTINSKIi
REACTION
403
9. V. A. PLISS, “Nonlocal Problem of the Theory of Oscillations,” Academic Press, New York, 1966. 10. A. T. WINFREE, Spiral waves of chemical activity, Science 175 (1972), 634-636. wave of chemical activity in three dimensions, 11. A. T. WINFREE, Scroll-shaped Science 181 (1973), 937-939. “Oscillatory Processes in Biological and Chemical Systems,” 12. A. M. ZHABOTINSKI~, p. 149, Science Publ., Moscow, 1967.