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Birhythmicity and a route to turbulence through limit cycle fusion in a simple autocatalytic system
Volume 143, number 8
PHYSICS LETTERS A
29 January 1990
BIRHYTHMICITY AND A ROUTE TO TURBULENCE THROUGH LIMIT CYCLE FUSION IN A SIMPLE AUTOCATALYTIC SYSTEM Simone MORI Dipartimento di Fisica, Università di Parma, 43100 Parma, Italy
and Enrico DI CERA’ Istituto di Fisica, Università Cattolica, Largo F. Vito 1, 00168 Rome, Italy Received 27 July 1989; revised manuscript received 13 October 1989; accepted for publication 20 November 1989 Communicated by A.P. Fordy
A simple system of autocatalytic reactions shows bistability, birhythmicity, and a route to turbulence through “fusion” of two independent limit cycles.
Coexistence of multiple steady states is a dynamical feature often encountered in nonlinear systems [1]. Dynamical phenomena such as bistability, birhythmicity and chaos are commonly obtained as a result of multiple bifurcations ofthe variables in discrete evolution when dealing with difference equations [2]. On the other hand, these phenomena are less frequently observed in the case of differential equations ofbiological and biochemical interest [3]. For example, the presence of multiple periodic regimes in a set of differential equations modelling an enzymatic reaction has been reported only recently [4]. The connection between this dynamical behaviour and the occurrence of chaos through a cascade of period-doubling bifurcations has also been discussed [5]. We have recently drawn attention to a class of autocatalytic reaction networks subject to mass constraints [6] that may be of relevance in the description of “imperfect” catalytic activity, as seen in the case of self-splicing RNAs [7]. The properties of these networks can be described with relatively simple differential equations, such as those derived for the the hypercycle Underimperfect suitable conditions reactiontheory system[81. involving cat-
alysts undergoes a cascade of period-doubling bifurcations leading to chaos [91.We have now found that a simple extension of the reaction scheme leads to the occurrence of bistability, birhythmicity, as well as a very peculiar route to turbulence via limit cycle fusion, which is the purpose of this paper to report. The reaction scheme of interest here includes four “imperfect catalysts”, X’s, and two “cofactors”, Y’s, as follows: (Ia) X2+X3—2X3+Y,
(Ib)
,
Y23+X4~2X4,
(lc)
X4 +X~-+2X~+Y2.
(ld)
.
.
The underlying differential equations are derived by application of mass action kinetics, i.e., .
.
dx, /dt = x1 (k4x4
.
—
k1 Yi x2)
,
dx2idt=x2(i~1v~x1—i~2x3)
(2a) (2b)
dx3/dt=x3(k2x2 —k3v2x4) (2c) 1d k k ~ ~ t=x~( 3~v2x3 4x1) (2d) where the k’s are kinetic constants, and x’s and v’s refer to the concentration of catalysts and the cofac,
—
To whom correspondence should be addressed,
369
Volume 143, number 8
PHYSICS LETTERS A
tor respectively. The reaction scheme (1) is driven far from equilibrium by a constant energy source included in the kinetic constants [9]. Since the total concentration of each cofactor is taken to be constant, we must have the following conservation conditions,
29 January 1990
x? = (c—rn 1 )/x2, ..v~=[rn+l—~J(rn—3)(rn+l)]/2,
(6a) (6b)
x~=(c—rn—l)/x~,
(6c)
x~=[rn+ 1 +~J(rn_3)(m+1 )]/2,
(6d)
—
and
y1+x2=rn1=const,
(3a)
y2+x4=rn2=const,
(3b)
x?=(c—rn—l)/x2, where rn1 and rn2 are the total amounts of cofactors, and x2 and x4 correspond to the concentrations ofYi andY2 bound to X1 and X3 respectively. One sees from eqs. (2a)—(2d) that only three variables are truly independent due to conservation of the catalysts, i.e., ~ dx,/dt=O, and hence x~+x2+x3+x4= c=const. In a previous study [9] we have shown that in the presence of only one “unbuffered” cofactor, i.e., for rni~cand rn2>>c, the system of equations period-doubling bifurcations. Here we are interested (2a)—(2d) gives rise to chaos through a cascade of in the case of two unbuffered cofactors, i.e., for rni, rn2 and c of comparable magnitude. The salient dynamical properties ofthe system can be shown for a simplified case in which all the k’s are assumed to be equal and thus included in the time variable. A further assumption is made that rn1=rn2=rn, so that the system of equations (2a)— (2d) becomes a function of only two parameters, rn and c, and the reaction scheme becomes invariant under the transformations x1 x~and x2 x4, i.e., ~
(4a)
,
(4b)
dx3/dl=x3[x2—(rn—x4)x4]
,
(4c)
dx4/dt=x4[(rn—x4)x3—x1]
.
(4d)
2/dtx2[(mx2)xi~3]
x~=[rn+l+..,/(m—3)(rn+l)]/2,
(7b)
(c—rn 1) /x9, x~=[rn+l—~/(m—3)(rn+l)]/2.
(7c) (7d)
X~=
—
s
a
~
+
0
6 x4
s
b
x3
There are three sets of non-trivial solutions of eqs. (4a)—(4d) in the physically meaningful range O~x~c. The first one, x?=x~=(c—2m+2)/2, x9=x~=rn—l,
(5a) (5b)
is a saddle point. The other4,two are equidisandpoints are given by tant from the first one in P 370
I
. .
~
,
dx1/dt=x1[x4—(rn—x2)x2]
(7a)
___________
+
0
6
x4
Fig. I. Numerical analysis ofeqs. (4a)—(4d) forc = 10: (a) two stable foci for the m=5; (b)state two values stable given limit by cycles m=5.5. Crosses depict steady eqs. for (5)—(7) in thetext.
Volume 143, number 8
PHYSICS LETTERS A
5
x3
x3
:
29 January 1990
5
~
~
6
x3
6
x3
:
o
6
:
6
Fig. 2. Numerical analysis of eqs. (4a)—(4d) for c= 10 and m=5.556085 showing a homoclinic orbit in the x 3—x4 plane (a) and in the x2—x4 plane (b) with a direct involvement of the sad die point Crosses depict the steady state values given by eqs (5)— (7)inthetext x3
+
They are responsible for bistability (see fig la) and give rise to birhythmicity (see fig ib) for values of m above the Hopf bifurcation m= [c+2+~J(c_2)2+32]/4
(8)
which can be shown by normal mode analysis to be exactly the same for both fixed points. By further increasing the value of~nthe two independent periodic orbits coalesce into a homoclinic orbit (see figs. 2a and 2b) involving the saddle point. This orbit then evolves toward a unique, stable orbit (see fig. 3a) which, beyond a critical value of m, is destabilized toward an aperiodic regime whereby formation of a
_________
‘~‘.
/ +
0
~ x4
6
Fig. 3. Numerical analysis ofeqs. (4a)—(4d) for c= 10: (a) stable limit cycle for m=5.65 not involving the saddle point~(b) aperiodicbehavior (seemingly strange attractor) for m= 5.8; (c) aperiodic behavior (seemingly strange attractor) for m=6. Crosses depict the steady state values given by eqs. (5)—(7) in the text.
371
Volume 143, number 8
PHYSICS LETTERSA
seemingly strange attractor is observed (see figs. 3b and 3c. The transition from birhythmicity to turbulence in the model of eqs. (4a)—(4d) is a very peculiar one as it occurs without a cascade of period-doubling bifurcations, unlike what has been observed in other systems of biochemical interest [3—5].It should also be pointed out that this route to turbulence has a bearing on the mechanism postulated by Arneodo et al. [10]. This mechanism holds for a general class of equations that are invariant under the transformation x —x, y —y and z z in P3, as shown by means of an ad hoc model [101, and implies a route to turbulence through bifurcations of stable homoclinic orbits formed by fusion of two symmetric limit cycles. When the system of equations (4a)—(4d) is rewritten in terms of independent variables in ~ as ~
~
~
dx/dt=x[c—x— (1 + m)y—z+y2] dy/dt=y(mx—xy—yz)
,
,
(9a) (9b)
dz/dt=z[y— (m—c+x+y+z)(c—x—y—z)], (9c) the two steady state solutions given by eqs. (6a)— (6c) and (7a)—(7c) are no longer equidistant from the saddlepoint (eqs. (5a)and (Sb)), and hence no
372
29 January 1990
invariance properties other than those of eqs. (4a)— (4d), namely x z and y c—x—y—z, exist. Therefore the model discussed here seems to imply existence of a route to turbulence through limit cycle fusion independent of symmetry conditions such as those postulated by Arneodo et a!. [10]. ~
~-
This work was supported by MPI and CNR.
References [1] R.M. May, Nature 269 (1977) 471. [2] May, and Nature 261 (1976) [3) R.M. L.F. Olsen H. Degn, Q. Rev.459. Biophys. 18 (1985) 165. [4] 0. Decroly and A. Goldbeter, C.R. Acad. Sci. 11298 (1984) 779. [5] 0. Decroly and A. Goldbeter, Proc. Nat!. Acad. Sci. USA 79(1982)6917. [6] E. Di Cera, P.E. Phi!!ipson and J. Wyman, Proc. Nat!. Acad. Sci. USA 85 (1988) 5923. [7]T.R. Cech and B.L. Bass, Annu. Rev. Biochem. 55 (1986) [8] M. Eigen and P. Schuster, The hypercycle (Springer, Berlin, 1979). [9] E. Di Cera, P.E. Phillipson and J. Wyman, Proc. Nat!. Acad. Sci. USA 86 (1989) 142.
[lO]A. Arneodo, (1981) 197.
P. Coullet and C. Tresser, Phys. Lett. A 8!
PHYSICS LETTERS A
29 January 1990
BIRHYTHMICITY AND A ROUTE TO TURBULENCE THROUGH LIMIT CYCLE FUSION IN A SIMPLE AUTOCATALYTIC SYSTEM Simone MORI Dipartimento di Fisica, Università di Parma, 43100 Parma, Italy
and Enrico DI CERA’ Istituto di Fisica, Università Cattolica, Largo F. Vito 1, 00168 Rome, Italy Received 27 July 1989; revised manuscript received 13 October 1989; accepted for publication 20 November 1989 Communicated by A.P. Fordy
A simple system of autocatalytic reactions shows bistability, birhythmicity, and a route to turbulence through “fusion” of two independent limit cycles.
Coexistence of multiple steady states is a dynamical feature often encountered in nonlinear systems [1]. Dynamical phenomena such as bistability, birhythmicity and chaos are commonly obtained as a result of multiple bifurcations ofthe variables in discrete evolution when dealing with difference equations [2]. On the other hand, these phenomena are less frequently observed in the case of differential equations ofbiological and biochemical interest [3]. For example, the presence of multiple periodic regimes in a set of differential equations modelling an enzymatic reaction has been reported only recently [4]. The connection between this dynamical behaviour and the occurrence of chaos through a cascade of period-doubling bifurcations has also been discussed [5]. We have recently drawn attention to a class of autocatalytic reaction networks subject to mass constraints [6] that may be of relevance in the description of “imperfect” catalytic activity, as seen in the case of self-splicing RNAs [7]. The properties of these networks can be described with relatively simple differential equations, such as those derived for the the hypercycle Underimperfect suitable conditions reactiontheory system[81. involving cat-
alysts undergoes a cascade of period-doubling bifurcations leading to chaos [91.We have now found that a simple extension of the reaction scheme leads to the occurrence of bistability, birhythmicity, as well as a very peculiar route to turbulence via limit cycle fusion, which is the purpose of this paper to report. The reaction scheme of interest here includes four “imperfect catalysts”, X’s, and two “cofactors”, Y’s, as follows: (Ia) X2+X3—2X3+Y,
(Ib)
,
Y23+X4~2X4,
(lc)
X4 +X~-+2X~+Y2.
(ld)
.
.
The underlying differential equations are derived by application of mass action kinetics, i.e., .
.
dx, /dt = x1 (k4x4
.
—
k1 Yi x2)
,
dx2idt=x2(i~1v~x1—i~2x3)
(2a) (2b)
dx3/dt=x3(k2x2 —k3v2x4) (2c) 1d k k ~ ~ t=x~( 3~v2x3 4x1) (2d) where the k’s are kinetic constants, and x’s and v’s refer to the concentration of catalysts and the cofac,
—
To whom correspondence should be addressed,
369
Volume 143, number 8
PHYSICS LETTERS A
tor respectively. The reaction scheme (1) is driven far from equilibrium by a constant energy source included in the kinetic constants [9]. Since the total concentration of each cofactor is taken to be constant, we must have the following conservation conditions,
29 January 1990
x? = (c—rn 1 )/x2, ..v~=[rn+l—~J(rn—3)(rn+l)]/2,
(6a) (6b)
x~=(c—rn—l)/x~,
(6c)
x~=[rn+ 1 +~J(rn_3)(m+1 )]/2,
(6d)
—
and
y1+x2=rn1=const,
(3a)
y2+x4=rn2=const,
(3b)
x?=(c—rn—l)/x2, where rn1 and rn2 are the total amounts of cofactors, and x2 and x4 correspond to the concentrations ofYi andY2 bound to X1 and X3 respectively. One sees from eqs. (2a)—(2d) that only three variables are truly independent due to conservation of the catalysts, i.e., ~ dx,/dt=O, and hence x~+x2+x3+x4= c=const. In a previous study [9] we have shown that in the presence of only one “unbuffered” cofactor, i.e., for rni~cand rn2>>c, the system of equations period-doubling bifurcations. Here we are interested (2a)—(2d) gives rise to chaos through a cascade of in the case of two unbuffered cofactors, i.e., for rni, rn2 and c of comparable magnitude. The salient dynamical properties ofthe system can be shown for a simplified case in which all the k’s are assumed to be equal and thus included in the time variable. A further assumption is made that rn1=rn2=rn, so that the system of equations (2a)— (2d) becomes a function of only two parameters, rn and c, and the reaction scheme becomes invariant under the transformations x1 x~and x2 x4, i.e., ~
(4a)
,
(4b)
dx3/dl=x3[x2—(rn—x4)x4]
,
(4c)
dx4/dt=x4[(rn—x4)x3—x1]
.
(4d)
2/dtx2[(mx2)xi~3]
x~=[rn+l+..,/(m—3)(rn+l)]/2,
(7b)
(c—rn 1) /x9, x~=[rn+l—~/(m—3)(rn+l)]/2.
(7c) (7d)
X~=
—
s
a
~
+
0
6 x4
s
b
x3
There are three sets of non-trivial solutions of eqs. (4a)—(4d) in the physically meaningful range O~x~c. The first one, x?=x~=(c—2m+2)/2, x9=x~=rn—l,
(5a) (5b)
is a saddle point. The other4,two are equidisandpoints are given by tant from the first one in P 370
I
. .
~
,
dx1/dt=x1[x4—(rn—x2)x2]
(7a)
___________
+
0
6
x4
Fig. I. Numerical analysis ofeqs. (4a)—(4d) forc = 10: (a) two stable foci for the m=5; (b)state two values stable given limit by cycles m=5.5. Crosses depict steady eqs. for (5)—(7) in thetext.
Volume 143, number 8
PHYSICS LETTERS A
5
x3
x3
:
29 January 1990
5
~
~
6
x3
6
x3
:
o
6
:
6
Fig. 2. Numerical analysis of eqs. (4a)—(4d) for c= 10 and m=5.556085 showing a homoclinic orbit in the x 3—x4 plane (a) and in the x2—x4 plane (b) with a direct involvement of the sad die point Crosses depict the steady state values given by eqs (5)— (7)inthetext x3
+
They are responsible for bistability (see fig la) and give rise to birhythmicity (see fig ib) for values of m above the Hopf bifurcation m= [c+2+~J(c_2)2+32]/4
(8)
which can be shown by normal mode analysis to be exactly the same for both fixed points. By further increasing the value of~nthe two independent periodic orbits coalesce into a homoclinic orbit (see figs. 2a and 2b) involving the saddle point. This orbit then evolves toward a unique, stable orbit (see fig. 3a) which, beyond a critical value of m, is destabilized toward an aperiodic regime whereby formation of a
_________
‘~‘.
/ +
0
~ x4
6
Fig. 3. Numerical analysis ofeqs. (4a)—(4d) for c= 10: (a) stable limit cycle for m=5.65 not involving the saddle point~(b) aperiodicbehavior (seemingly strange attractor) for m= 5.8; (c) aperiodic behavior (seemingly strange attractor) for m=6. Crosses depict the steady state values given by eqs. (5)—(7) in the text.
371
Volume 143, number 8
PHYSICS LETTERSA
seemingly strange attractor is observed (see figs. 3b and 3c. The transition from birhythmicity to turbulence in the model of eqs. (4a)—(4d) is a very peculiar one as it occurs without a cascade of period-doubling bifurcations, unlike what has been observed in other systems of biochemical interest [3—5].It should also be pointed out that this route to turbulence has a bearing on the mechanism postulated by Arneodo et al. [10]. This mechanism holds for a general class of equations that are invariant under the transformation x —x, y —y and z z in P3, as shown by means of an ad hoc model [101, and implies a route to turbulence through bifurcations of stable homoclinic orbits formed by fusion of two symmetric limit cycles. When the system of equations (4a)—(4d) is rewritten in terms of independent variables in ~ as ~
~
~
dx/dt=x[c—x— (1 + m)y—z+y2] dy/dt=y(mx—xy—yz)
,
,
(9a) (9b)
dz/dt=z[y— (m—c+x+y+z)(c—x—y—z)], (9c) the two steady state solutions given by eqs. (6a)— (6c) and (7a)—(7c) are no longer equidistant from the saddlepoint (eqs. (5a)and (Sb)), and hence no
372
29 January 1990
invariance properties other than those of eqs. (4a)— (4d), namely x z and y c—x—y—z, exist. Therefore the model discussed here seems to imply existence of a route to turbulence through limit cycle fusion independent of symmetry conditions such as those postulated by Arneodo et a!. [10]. ~
~-
This work was supported by MPI and CNR.
References [1] R.M. May, Nature 269 (1977) 471. [2] May, and Nature 261 (1976) [3) R.M. L.F. Olsen H. Degn, Q. Rev.459. Biophys. 18 (1985) 165. [4] 0. Decroly and A. Goldbeter, C.R. Acad. Sci. 11298 (1984) 779. [5] 0. Decroly and A. Goldbeter, Proc. Nat!. Acad. Sci. USA 79(1982)6917. [6] E. Di Cera, P.E. Phi!!ipson and J. Wyman, Proc. Nat!. Acad. Sci. USA 85 (1988) 5923. [7]T.R. Cech and B.L. Bass, Annu. Rev. Biochem. 55 (1986) [8] M. Eigen and P. Schuster, The hypercycle (Springer, Berlin, 1979). [9] E. Di Cera, P.E. Phillipson and J. Wyman, Proc. Nat!. Acad. Sci. USA 86 (1989) 142.
[lO]A. Arneodo, (1981) 197.
P. Coullet and C. Tresser, Phys. Lett. A 8!