- Email: [email protected]
Dissipative structures in a soluble non-linear reaction-diffusion system
Volume 60A, number 5
PHYSICS LETTERS
21 March 1977
DISSIPATIVE STRUCTURES IN A SOLUBLE NON-LINEAR REACTION-DIFFUSION SYSTEM R. LEFEVER, M. HERSCHKOWITZ-KAUFMAN and J.W. TURNER Université Libre de Bruxelles, Faculté des Sciences, CampusPlaine U.L.B., Code Postal 231, 1050 Bruxelles, Belgium Received 2 February 1977 The steady state solutions of a non-linear reaction diffusion system are evaluated exactly. This bifurcation diagram as well as their stability is discussed.
We present a chemical model in which seif-organization processes may take place and lead to coherent non-equilibrium behaviour (dissipative structures [1]). It corresponds to an isothermal reaction-diffusion systern whose simplicity, or rather minimality of ingredients permits an exact determination of the steady state branches of solutions. The analysis of the mathematical and physical properties of this model can be pushed beyond the domain accessible to usual perturbation methods such as bifurcation theory [2—4]or singular perturbation theory [5—7].In some cases it is possible to carry out a stability analysis which deter-
variables w = x ~and r’ = r/D~2,the steady state solutions are given by —
(~~2
=
K
—
\ di / with 2 F(w) =
— ~-
at ar2 (1) ~=f(x,y)+D1~, 2 ~X=cf(x,y)+D2~, at ar where D 1 and D2 are the diffusion coefficients, c is a constant andf(x, y) stands for the chemical 2y terms. Bx (B Here, c = —1 and we shall take f(x, y) = x is a constant parameter), which corresponds to a set of two irreversible reactions: 2x+y 3x andB+x -÷y + D We consider fixed and equal boundary conditions: x(O)=x(l)=~y(O)=y(1)’=B/~.Intermsofnew
1
~o\ ~
~
2 +
—
(~ ~ )
(3)
—
and p = BD 2/D1
mines exactly the spectrum of eigenvalues associated to particular inhomogeneous branches of solutions. Points corresponding to an exchange of stability of the inhomogeneous solutions can in general be calculated. The model describes in a one-dimensional medium of length 1, the diffusion and reaction of two cornpounds x, y according to kinetic laws of the form [8]:
(2)
F(w)
(4)
.
Fixing ~ and varying p, three domains are encountered, as illustrated in fig. 1.
40
F(w)
18
/
20
/
\ •~..
~r3
N
/
—
0
—
~. ~
—
— -—
—
—~
~.
Y
I .
~ These reactions are a simplitication of the trimolecular scheme (Brusselator) originally used to introduce the concept of dissipative structure (see ref. [1]).
\
0
-20 -~
I
-1.5
I
0
~I
1.5
3
W
Fig. 1. Plot of F(w) versus w for various values of p and ~ fixed.
389
Volume 60A, numberS
PHYSICS LETTERS
(1) For p <~2/2,the only stationary solution possible is the homogeneous state where the concentrations are constant and equal to the value at the boundaries, (II) For ~2/2
/
21 March 1977
where F(p~\e)is the incomplete elliptic integral of the first kind with modular angle c. The slope at the ongin of the half-period solution being positive or negative, its amplitude is given by: ~
_(a--y)~
sin~
— ~——~-—
2 or sin p
___
7
—
______
w (K)
L(K)
=
2
dw’
respectively.
Only when the two maxima of F(w) are equal (i.e.
for p
=
22) are the two half-period solutions for a
w(K) is the value of w at which the integral of(S) becomes singular. Note that for any o
given K completely symmetrical. For a system ol length!, the only adniissible solutions are those which correspond to a half-period L~(K)= / or such that: n L~ (K) + n2L2(K) = I. with In1 U7 = 0 or I -
first decreases, passes through a minimal value at r 1/2 and then increases again up to the value w = 0 in r = 1. The profile whose concentration at r = 1/2 is the smallest and the homogeneous state w(r) = 0 are stable; the other one is unstable. (III) For p > ~2, F(w) presents two maxima above the w-axis. They are located at w’ = —~ and w” = -----
is a subcritical region in the upper or lower branch depending on the relative height of the peaks in fig. I. The asypiptotes K’ and K” of these branches correspond to half-period solutions of infinite length. When p <~2 the asymptote K’ merges with the w-axis:
where
=
(5)
where t~-~ . n2 are positive integers. It follows that K can only assume a discrete set of values and that there is a inaximuni to the number of periods which can be contained in I . In other words, in a finite system there is a quantization of the acceptable values ofK and a multiplicity of solutions which increases rapidly with the size of the system. Fig. 2 is a bifurcation diagram of the fundamental 112 is steady state solutions three values ofp(±K There plotted versus L(K) asfor bifurcation parameter).
+ B/s, while w
= 0 has become a minimum. Depending on the value of ~ the highest peak is either at w’(p
<2~2)or at w”(p > 2~).In either case there exists a range of values for K, respectively 0
~
sL
~-
/
i
390
=
_________________________ ~ )(~ —6)
F(~~\e)(i
=
1, 2),
(6)
- -
I
/
2~
0~
--~~
~
/
-
/4-5
I
I -4 -_______
L i (K)
-
~
~ 4.
0.85 0 87 .05 12 £ I’ig. 2. Bifurcation diagram of K172 versus!1.4for three values of the ratio of diffusion coefficients.
Volume 60A, number 5
PHYSICS LETTERS
therefore the situation described in II can be viewed as a particular case of III in which the bifurcation point moves to infinity. We should like to indicate some results [9, 10] concerning the stability of the steady state solutions considered here. It is known that at a bifurcation point on a stable branch, the subcritical part of a new branch is unstable whereas its supercritical part is stable. Furthermore it is also known that solutions which appear as successive bifurcations from the homogeneous state are at first unstable. These results are restricted to the immediate vicinity of the bifurcation point. In the system described here, putting B = 2~2and q = D2/D1 = 1, the spectrum of eigenvalues around a new branch can be calculated exactly for any value of the length of system taken as bifurcation parameter. When q ‘~‘ 1, the eigenvalues can in general no longer be calculated; nevertheless the points of exchange of stability, if they exist, can be determined [10]. As an example, when q = 1, w
Csn(Xr,k)
with 2 k — —
(9)
This is no longer necessarily the case when the ratio of diffusion coefficients q becomes larger than 1. It is also interesting to note that it is for q> 1 that the formation of patterns can be viewed as a symmetry breaking instability due to the presence of diffusion and not as in some one-component systems [11, 12] as a result of the effect of diffusion on a first order type of phase transition. We thank Professor I. Prigogine for his stimulating interest and numerous discussions. The research reported in this paper is related to a programme sponsored by the Solvay Company Brussels.
References [1] G.
Nicolis and 1. Prigogine, Selforganization in nonequi-
librium systems (Wiley lnterscience, New York, 1977).
[2] J.F.G. Auchmuty and G. Nicolis, Bull. Math. Biol. 37 (1975) 323. [31 M. Herschkowitz-Kaufman, Bull. Math. Biol. 37 (1975) 589.
—
— ~
2
~2
~
— — —
2~’~2~ ~“
—ç
~ < ~
~-
~“-
After linearisation of the equations of motion (1) around the steady state solution (9), that part of the spectrum which can lead to an exchange of stability is related to the eigenvalues i~of
a2u = [6k2 sn2 (r, k) ~]U, ar
—
21 March 1977
—
(10)
i.e. a Lame equation whose general solution is known. In this particular case it is found that the second branch remains unstable for all values of the parameter of bifurcation 1.
[41 T.J. Mahar and B.J. Matkowsky, A model biochemical
reaction exhibiting secondary bifurcation, SIAM J. Appi. Math. (1977), to be published. [5] P. Ovtoleva and J. Ross, J. Chem. Phys. 63 (1975) 3398. [6] P.C. Fife, J. Chem. Phys. 64(1976)554. [7] J.A. Boa, Studies in Appl. Math. LIV (1975) 9. [8] Systems obeying equations of this type are commonly found in artificial membrane reactors, see for example:
D. Thomas and J.P. Kernevez, eds., Analysis and control
[9] [10] [11] [12]
of immobilized enzyme systems (North-Holland, Amsterdam, 1976). M. Herschkowitz-Kaufman and R. Lefever, in preparation. J.Wm. Turner, in preparation. F. Schldgl, Z. Physik 253 (1972) 147. D. Bedaux, P. Mazur and R.A. Pasmanter, The ballast resistor; an electro-thermal instability in a conducting wire I; The nature of the stationary states, Physica (1977), to be published.
391
PHYSICS LETTERS
21 March 1977
DISSIPATIVE STRUCTURES IN A SOLUBLE NON-LINEAR REACTION-DIFFUSION SYSTEM R. LEFEVER, M. HERSCHKOWITZ-KAUFMAN and J.W. TURNER Université Libre de Bruxelles, Faculté des Sciences, CampusPlaine U.L.B., Code Postal 231, 1050 Bruxelles, Belgium Received 2 February 1977 The steady state solutions of a non-linear reaction diffusion system are evaluated exactly. This bifurcation diagram as well as their stability is discussed.
We present a chemical model in which seif-organization processes may take place and lead to coherent non-equilibrium behaviour (dissipative structures [1]). It corresponds to an isothermal reaction-diffusion systern whose simplicity, or rather minimality of ingredients permits an exact determination of the steady state branches of solutions. The analysis of the mathematical and physical properties of this model can be pushed beyond the domain accessible to usual perturbation methods such as bifurcation theory [2—4]or singular perturbation theory [5—7].In some cases it is possible to carry out a stability analysis which deter-
variables w = x ~and r’ = r/D~2,the steady state solutions are given by —
(~~2
=
K
—
\ di / with 2 F(w) =
— ~-
at ar2 (1) ~=f(x,y)+D1~, 2 ~X=cf(x,y)+D2~, at ar where D 1 and D2 are the diffusion coefficients, c is a constant andf(x, y) stands for the chemical 2y terms. Bx (B Here, c = —1 and we shall take f(x, y) = x is a constant parameter), which corresponds to a set of two irreversible reactions: 2x+y 3x andB+x -÷y + D We consider fixed and equal boundary conditions: x(O)=x(l)=~y(O)=y(1)’=B/~.Intermsofnew
1
~o\ ~
~
2 +
—
(~ ~ )
(3)
—
and p = BD 2/D1
mines exactly the spectrum of eigenvalues associated to particular inhomogeneous branches of solutions. Points corresponding to an exchange of stability of the inhomogeneous solutions can in general be calculated. The model describes in a one-dimensional medium of length 1, the diffusion and reaction of two cornpounds x, y according to kinetic laws of the form [8]:
(2)
F(w)
(4)
.
Fixing ~ and varying p, three domains are encountered, as illustrated in fig. 1.
40
F(w)
18
/
20
/
\ •~..
~r3
N
/
—
0
—
~. ~
—
— -—
—
—~
~.
Y
I .
~ These reactions are a simplitication of the trimolecular scheme (Brusselator) originally used to introduce the concept of dissipative structure (see ref. [1]).
\
0
-20 -~
I
-1.5
I
0
~I
1.5
3
W
Fig. 1. Plot of F(w) versus w for various values of p and ~ fixed.
389
Volume 60A, numberS
PHYSICS LETTERS
(1) For p <~2/2,the only stationary solution possible is the homogeneous state where the concentrations are constant and equal to the value at the boundaries, (II) For ~2/2
/
21 March 1977
where F(p~\e)is the incomplete elliptic integral of the first kind with modular angle c. The slope at the ongin of the half-period solution being positive or negative, its amplitude is given by: ~
_(a--y)~
sin~
— ~——~-—
2 or sin p
___
7
—
______
w (K)
L(K)
=
2
dw’
respectively.
Only when the two maxima of F(w) are equal (i.e.
for p
=
22) are the two half-period solutions for a
w(K) is the value of w at which the integral of(S) becomes singular. Note that for any o
given K completely symmetrical. For a system ol length!, the only adniissible solutions are those which correspond to a half-period L~(K)= / or such that: n L~ (K) + n2L2(K) = I. with In1 U7 = 0 or I -
first decreases, passes through a minimal value at r 1/2 and then increases again up to the value w = 0 in r = 1. The profile whose concentration at r = 1/2 is the smallest and the homogeneous state w(r) = 0 are stable; the other one is unstable. (III) For p > ~2, F(w) presents two maxima above the w-axis. They are located at w’ = —~ and w” = -----
is a subcritical region in the upper or lower branch depending on the relative height of the peaks in fig. I. The asypiptotes K’ and K” of these branches correspond to half-period solutions of infinite length. When p <~2 the asymptote K’ merges with the w-axis:
where
=
(5)
where t~-~ . n2 are positive integers. It follows that K can only assume a discrete set of values and that there is a inaximuni to the number of periods which can be contained in I . In other words, in a finite system there is a quantization of the acceptable values ofK and a multiplicity of solutions which increases rapidly with the size of the system. Fig. 2 is a bifurcation diagram of the fundamental 112 is steady state solutions three values ofp(±K There plotted versus L(K) asfor bifurcation parameter).
+ B/s, while w
= 0 has become a minimum. Depending on the value of ~ the highest peak is either at w’(p
<2~2)or at w”(p > 2~).In either case there exists a range of values for K, respectively 0
~
sL
~-
/
i
390
=
_________________________ ~ )(~ —6)
F(~~\e)(i
=
1, 2),
(6)
- -
I
/
2~
0~
--~~
~
/
-
/4-5
I
I -4 -_______
L i (K)
-
~
~ 4.
0.85 0 87 .05 12 £ I’ig. 2. Bifurcation diagram of K172 versus!1.4for three values of the ratio of diffusion coefficients.
Volume 60A, number 5
PHYSICS LETTERS
therefore the situation described in II can be viewed as a particular case of III in which the bifurcation point moves to infinity. We should like to indicate some results [9, 10] concerning the stability of the steady state solutions considered here. It is known that at a bifurcation point on a stable branch, the subcritical part of a new branch is unstable whereas its supercritical part is stable. Furthermore it is also known that solutions which appear as successive bifurcations from the homogeneous state are at first unstable. These results are restricted to the immediate vicinity of the bifurcation point. In the system described here, putting B = 2~2and q = D2/D1 = 1, the spectrum of eigenvalues around a new branch can be calculated exactly for any value of the length of system taken as bifurcation parameter. When q ‘~‘ 1, the eigenvalues can in general no longer be calculated; nevertheless the points of exchange of stability, if they exist, can be determined [10]. As an example, when q = 1, w
Csn(Xr,k)
with 2 k — —
(9)
This is no longer necessarily the case when the ratio of diffusion coefficients q becomes larger than 1. It is also interesting to note that it is for q> 1 that the formation of patterns can be viewed as a symmetry breaking instability due to the presence of diffusion and not as in some one-component systems [11, 12] as a result of the effect of diffusion on a first order type of phase transition. We thank Professor I. Prigogine for his stimulating interest and numerous discussions. The research reported in this paper is related to a programme sponsored by the Solvay Company Brussels.
References [1] G.
Nicolis and 1. Prigogine, Selforganization in nonequi-
librium systems (Wiley lnterscience, New York, 1977).
[2] J.F.G. Auchmuty and G. Nicolis, Bull. Math. Biol. 37 (1975) 323. [31 M. Herschkowitz-Kaufman, Bull. Math. Biol. 37 (1975) 589.
—
— ~
2
~2
~
— — —
2~’~2~ ~“
—ç
~ < ~
~-
~“-
After linearisation of the equations of motion (1) around the steady state solution (9), that part of the spectrum which can lead to an exchange of stability is related to the eigenvalues i~of
a2u = [6k2 sn2 (r, k) ~]U, ar
—
21 March 1977
—
(10)
i.e. a Lame equation whose general solution is known. In this particular case it is found that the second branch remains unstable for all values of the parameter of bifurcation 1.
[41 T.J. Mahar and B.J. Matkowsky, A model biochemical
reaction exhibiting secondary bifurcation, SIAM J. Appi. Math. (1977), to be published. [5] P. Ovtoleva and J. Ross, J. Chem. Phys. 63 (1975) 3398. [6] P.C. Fife, J. Chem. Phys. 64(1976)554. [7] J.A. Boa, Studies in Appl. Math. LIV (1975) 9. [8] Systems obeying equations of this type are commonly found in artificial membrane reactors, see for example:
D. Thomas and J.P. Kernevez, eds., Analysis and control
[9] [10] [11] [12]
of immobilized enzyme systems (North-Holland, Amsterdam, 1976). M. Herschkowitz-Kaufman and R. Lefever, in preparation. J.Wm. Turner, in preparation. F. Schldgl, Z. Physik 253 (1972) 147. D. Bedaux, P. Mazur and R.A. Pasmanter, The ballast resistor; an electro-thermal instability in a conducting wire I; The nature of the stationary states, Physica (1977), to be published.
391