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On the global asymptotic stability of a system of N generalized chemical rate equations
Volume 113A, number 5
PHYSICS LETTERS
23 December 1985
ON THE GLOBAL ASYMPTOTIC STABILITY OF A S Y S T E M OF N G E N E R A L I Z E D C H E M I C A L RATE E Q U A T I O N S George T S A R O U H A S Department of Theoretical Mechanics, University of Thessalonikh Thessaloniki, Greece
Received 10 October 1985; accepted for publication 23 October 1985
It is proved that a system of N generalizedchemical rate equations, which for N = 2 describes the kinetics of irradiation-produced point defects, has a unique equilibrium point in the part of phase space where all dependent variables are positive. By finding an appropriate Lyapunov function, it is also shown that this equilibrium point is globally asymptotically stable, for all positive initial conditions of the system.
1. Introduction. The kinetics of irradiation-pro-
duced interstitials and vacancies are described by the non-linear coupled differential equations b i = k i - P t ~ i - rc 1 c2,
i = 1, 2
(la)
[here (.) = d/dt denotes the time derivative], where c i ('~ 0) are the atomic concentrations of interstitials and vacancies, k i (> 0) are their production rates per lattice, Oi ( > 0) are the reaction rate constants describing the total point-defect losses to the sinks, and r ( > 0) is the bulk recombination coefficient [ 1 - 3 ] . As is known eqs. (la) have a unique fixed point which is globally asymptotically stable [4]. Here we consider the system of non-linear differential equations N Ci = ki - PiCi - ci ~ rO'ci ]~i (i = 1 , 2 ..... N , N >
2),
(lb)
which is the N-dimensional generalization of the twodimensional system (la) [5]. In the case N = 3, the third component of the system, c 3 say, can be considered as the concentration of voids, which can interact directly with vacancies and interstitials [ 1 - 3 ] . For N >/2, system ( l b ) can describe the interaction of different chemical rate equa. tions [51. In (Ib') the rq = rji ( > 0) represent the bulk recombination coefficients corresponding to the species i and ] (N is finite). 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In this letter we prove that the system (lb), for positive ki, Pi, r# has a unique equilibrium point with positive coordinates ~i and show that this point is globally asymptotically stable by finding an appropriate Lyapunov function for it [6]. This result clearly demonstrates that system ( l b ) exhibits regular motion for all positive initial conditions and has no chaotic behavior whatsoever [7], at least in the physically interesting case c i >>-O. 2. Existence and uniqueness o f a n equilibrium point.
From (lb) we find that the coordinates ~i of the equilibrium point of system ( l b ) satisfy the relation
(
N
~i = ki Oi + ~ r f ~ i i4,i
)1
(2)
•
Proposition 1. For positive ki, Pi and to. , eqs. (2) have a unique non-negative solution ~i/> 0, i = 1 ..... N. Proof. (i) Let X, Y E RN, vectors in RN, with components x i , Y i (i = 1, 2 ..... N), respectively. In the following, vectors will be represented by capital letters and their components by lower case letters. We define X I> Y by the relations x i >t Yi (i = 1 , 2 , .... N) and X I> 0 by the relations x i >1 0 (i = 1 , 2 ..... N). Next we define a function F: RN ~ RN by the following relation, satisfied by it s components:
N
)-1
239
Volume 113A, number 5
PHYSICS LETTERS
23 December 1985
Since F is a continuous function, X+ = lim X2n+l = lim F(X2n ) = F ( l i m n__,~. X2n ) = F ( X _ ) ,
and similarly X_ ---F(X+). Eqs. (3), for x i = xi+ and x i = x i - become: ~ °
"
J
Xi,2n
xi+ = ki
i +.
. r¢'xj-
,
x i - -- ki oi + r xj÷ \ j4=i Eliminating k i from the above two equations we find
Fig. 1. The two subsequences Xi,2n and xi,2n+l.
N
Then one can easily show that for each X , Y E R N, 0 ~< X ~< Y implies F(X) ~> F(Y) >~ 0. Let { X n } be an sequence defined recursively b y . X 0 = O,
Xn+ 1 = F ( X n ) .
lirn S2n = X+
exist. Similarlyxi,2n+l is a decreasing and bounded sequence in R + and lim xi,2n+l =xi+,
lira X2n+l =X_
also exist (see fig. 1). From the above we conclude that xi+ ~ x i_ and X+ ~> X_. We shall now prove that X+=X_.
240
r¢.xj_ l , v
N = P i x i - + X i - "j~i rq'xJ+
(i = 1,..., N ) .
(4)
Since the composite function F o F is monotone, and X 2 ~>X 1 ~>0, cf. (3), (4), it follows that the subsequence (X2n }, n E N, is monotonically increasing. On the other hand, b y virtue o f the form o f F in (3), the subsequence (X2n+l}, n E N, must necessarily be monotonically decreasing. Thus, since X 1 ~> 0 we have X2n ~ m, k > ~ where l, m E N are arbitrary, we have (see also fig. 1) X2m <~ X2k ~ X2k+ 1 ~< X2/+ 1 • Def'ming now by xi, k the ith components o f X k we make the following statements: (ii) Each xi,2n is an increasing and bounded sequence in R+ (R + is the set o f the non-negative real numbers). Hence, lim xi,2n = x i _ ,
fiixi + + xi+
Summing over i and using the fact that r O. = rji we obtain N i=1
Oi(xi+ - x i _ ) = 0 .
(5)
Since (5) is a sum over positive terms and Pi > O, xi+ = x i_ and X+ = X _ is a fixed point of F. (iii) Let X ' = F ( X ' ) be another fixed point with X ' >~ 0. Then F(X'):X'<~F(O)=X1,
X'>~X 2 .
By induction we obtain X2n ~
where ~i is the ith coordinate o f the equilibrium point, eqs. (lb) take the form
Volume 113A, number 5
PHYSICS LETTERS N
N
regions:
~i = --PiZi - (zi + ~i) .]~i rO.z] - z i ]~i roll.
(6)
Since for c i = O, ci = ki > O, we have
gi + ~i = Ci >~ O .
(7)
Consequently, with Z = (z 1, z 2 ..... Zn), N a~ i div 2 = ~ = - - / . . d Pi + ~ r#(~] + z]) < O. i=1 ~t" i=1 i~]
(8)
Therefore the system (6) is dissipative with a nonconstant dissipation, which suggests that we should look for a Lyapunov function for it. Proposition 2. The function N
V=~lzil,
i = 1 , 2 ..... N w i t h z i 4 = 0 ,
i=1
is a Lyapunov function for eqs. (6). P r o o f Let s i = sign(z/), i.e. s i = + 1 ff z i > O, s i = - 1 if z i < 0. Then we can write
z i = Izil si,
(9)
and using (6) we have N N dti=l
N
23 December 1985
N
iz,
(lO) /~1
(12)
II = (zi: V > const),
(13)
III = {zi: V = const}.
(14)
The existence of the Lyapunov function V ensures that the phase flow on the faces of the polyhedron is directed toward region I. However, this function cannot give us information concerning the direction of the phase flow on the edges o f the polyhedron since l?is not defined there. We can obtain this information directly from eqs. (6) by considering the inner product £ between the vector normal to the faces o f the above polyhedron and ,~ = (Zl, z2 ..... Zn)" After a short calculation one can show that V is negative for all cases, including the situation where more than one o f the zi's are zero. F is equal to zero only when the system is on his equilibrium point. Therefore the phase flow on the edges is also directed toward region I. I would like to thank Professor Tassos Bountis, Department o f Theoretical Mechanics, University of Thessaloniki, Thessaloniki, Greece, for his useful remarks on the final version of the text.
"
References
where
t , =zi~i(si+sj)+zj~i(si+s])+ziz](si+s]).
I = {zi: V < const),
(11)
Since Pi > 0 we have Pizisi = pi[zil > O, while for all choices ofsi, s/it can be easily verified that t# >1 0. Therefore, with r¢ > 0, we conclude from (10) that I? < 0 and hence that V is indeed a legitimate Lyapunov function [6]. The equation V = const represents a closed hypersurface which has the form o f a polyhedron separating the phase space (z 1 , z 2 ..... Zn) in the following
[ 1 ] N.Q. Lam, S.J. Rothman and R. Sizmann, Radiat. Eft. 23 (1974) 53. [2] J.R. Mathews, Contemp. Phys. 18 (1977) 571. [3] L.K. Mansur, Nucl. Technol. 40 (1978) 5. [4] G. Tsarouhas, Phys. Lett. 103A (1984) 433. [5] G. Tsarouhas and T. Christidis, Phys. Lett. 109A (1985) 424. [6] N. Rouche, P. Habets and M. Laloy, Appl. Math. Sci. 22 (1977). [7] P. Cvitanovi~, ed., University in chaos (Hilger, Bristol, 1984).
241
PHYSICS LETTERS
23 December 1985
ON THE GLOBAL ASYMPTOTIC STABILITY OF A S Y S T E M OF N G E N E R A L I Z E D C H E M I C A L RATE E Q U A T I O N S George T S A R O U H A S Department of Theoretical Mechanics, University of Thessalonikh Thessaloniki, Greece
Received 10 October 1985; accepted for publication 23 October 1985
It is proved that a system of N generalizedchemical rate equations, which for N = 2 describes the kinetics of irradiation-produced point defects, has a unique equilibrium point in the part of phase space where all dependent variables are positive. By finding an appropriate Lyapunov function, it is also shown that this equilibrium point is globally asymptotically stable, for all positive initial conditions of the system.
1. Introduction. The kinetics of irradiation-pro-
duced interstitials and vacancies are described by the non-linear coupled differential equations b i = k i - P t ~ i - rc 1 c2,
i = 1, 2
(la)
[here (.) = d/dt denotes the time derivative], where c i ('~ 0) are the atomic concentrations of interstitials and vacancies, k i (> 0) are their production rates per lattice, Oi ( > 0) are the reaction rate constants describing the total point-defect losses to the sinks, and r ( > 0) is the bulk recombination coefficient [ 1 - 3 ] . As is known eqs. (la) have a unique fixed point which is globally asymptotically stable [4]. Here we consider the system of non-linear differential equations N Ci = ki - PiCi - ci ~ rO'ci ]~i (i = 1 , 2 ..... N , N >
2),
(lb)
which is the N-dimensional generalization of the twodimensional system (la) [5]. In the case N = 3, the third component of the system, c 3 say, can be considered as the concentration of voids, which can interact directly with vacancies and interstitials [ 1 - 3 ] . For N >/2, system ( l b ) can describe the interaction of different chemical rate equa. tions [51. In (Ib') the rq = rji ( > 0) represent the bulk recombination coefficients corresponding to the species i and ] (N is finite). 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In this letter we prove that the system (lb), for positive ki, Pi, r# has a unique equilibrium point with positive coordinates ~i and show that this point is globally asymptotically stable by finding an appropriate Lyapunov function for it [6]. This result clearly demonstrates that system ( l b ) exhibits regular motion for all positive initial conditions and has no chaotic behavior whatsoever [7], at least in the physically interesting case c i >>-O. 2. Existence and uniqueness o f a n equilibrium point.
From (lb) we find that the coordinates ~i of the equilibrium point of system ( l b ) satisfy the relation
(
N
~i = ki Oi + ~ r f ~ i i4,i
)1
(2)
•
Proposition 1. For positive ki, Pi and to. , eqs. (2) have a unique non-negative solution ~i/> 0, i = 1 ..... N. Proof. (i) Let X, Y E RN, vectors in RN, with components x i , Y i (i = 1, 2 ..... N), respectively. In the following, vectors will be represented by capital letters and their components by lower case letters. We define X I> Y by the relations x i >t Yi (i = 1 , 2 , .... N) and X I> 0 by the relations x i >1 0 (i = 1 , 2 ..... N). Next we define a function F: RN ~ RN by the following relation, satisfied by it s components:
N
)-1
239
Volume 113A, number 5
PHYSICS LETTERS
23 December 1985
Since F is a continuous function, X+ = lim X2n+l = lim F(X2n ) = F ( l i m n__,~. X2n ) = F ( X _ ) ,
and similarly X_ ---F(X+). Eqs. (3), for x i = xi+ and x i = x i - become: ~ °
"
J
Xi,2n
xi+ = ki
i +.
. r¢'xj-
,
x i - -- ki oi + r xj÷ \ j4=i Eliminating k i from the above two equations we find
Fig. 1. The two subsequences Xi,2n and xi,2n+l.
N
Then one can easily show that for each X , Y E R N, 0 ~< X ~< Y implies F(X) ~> F(Y) >~ 0. Let { X n } be an sequence defined recursively b y . X 0 = O,
Xn+ 1 = F ( X n ) .
lirn S2n = X+
exist. Similarlyxi,2n+l is a decreasing and bounded sequence in R + and lim xi,2n+l =xi+,
lira X2n+l =X_
also exist (see fig. 1). From the above we conclude that xi+ ~ x i_ and X+ ~> X_. We shall now prove that X+=X_.
240
r¢.xj_ l , v
N = P i x i - + X i - "j~i rq'xJ+
(i = 1,..., N ) .
(4)
Since the composite function F o F is monotone, and X 2 ~>X 1 ~>0, cf. (3), (4), it follows that the subsequence (X2n }, n E N, is monotonically increasing. On the other hand, b y virtue o f the form o f F in (3), the subsequence (X2n+l}, n E N, must necessarily be monotonically decreasing. Thus, since X 1 ~> 0 we have X2n ~
fiixi + + xi+
Summing over i and using the fact that r O. = rji we obtain N i=1
Oi(xi+ - x i _ ) = 0 .
(5)
Since (5) is a sum over positive terms and Pi > O, xi+ = x i_ and X+ = X _ is a fixed point of F. (iii) Let X ' = F ( X ' ) be another fixed point with X ' >~ 0. Then F(X'):X'<~F(O)=X1,
X'>~X 2 .
By induction we obtain X2n ~
where ~i is the ith coordinate o f the equilibrium point, eqs. (lb) take the form
Volume 113A, number 5
PHYSICS LETTERS N
N
regions:
~i = --PiZi - (zi + ~i) .]~i rO.z] - z i ]~i roll.
(6)
Since for c i = O, ci = ki > O, we have
gi + ~i = Ci >~ O .
(7)
Consequently, with Z = (z 1, z 2 ..... Zn), N a~ i div 2 = ~ = - - / . . d Pi + ~ r#(~] + z]) < O. i=1 ~t" i=1 i~]
(8)
Therefore the system (6) is dissipative with a nonconstant dissipation, which suggests that we should look for a Lyapunov function for it. Proposition 2. The function N
V=~lzil,
i = 1 , 2 ..... N w i t h z i 4 = 0 ,
i=1
is a Lyapunov function for eqs. (6). P r o o f Let s i = sign(z/), i.e. s i = + 1 ff z i > O, s i = - 1 if z i < 0. Then we can write
z i = Izil si,
(9)
and using (6) we have N N dti=l
N
23 December 1985
N
iz,
(lO) /~1
(12)
II = (zi: V > const),
(13)
III = {zi: V = const}.
(14)
The existence of the Lyapunov function V ensures that the phase flow on the faces of the polyhedron is directed toward region I. However, this function cannot give us information concerning the direction of the phase flow on the edges o f the polyhedron since l?is not defined there. We can obtain this information directly from eqs. (6) by considering the inner product £ between the vector normal to the faces o f the above polyhedron and ,~ = (Zl, z2 ..... Zn)" After a short calculation one can show that V is negative for all cases, including the situation where more than one o f the zi's are zero. F is equal to zero only when the system is on his equilibrium point. Therefore the phase flow on the edges is also directed toward region I. I would like to thank Professor Tassos Bountis, Department o f Theoretical Mechanics, University of Thessaloniki, Thessaloniki, Greece, for his useful remarks on the final version of the text.
"
References
where
t , =zi~i(si+sj)+zj~i(si+s])+ziz](si+s]).
I = {zi: V < const),
(11)
Since Pi > 0 we have Pizisi = pi[zil > O, while for all choices ofsi, s/it can be easily verified that t# >1 0. Therefore, with r¢ > 0, we conclude from (10) that I? < 0 and hence that V is indeed a legitimate Lyapunov function [6]. The equation V = const represents a closed hypersurface which has the form o f a polyhedron separating the phase space (z 1 , z 2 ..... Zn) in the following
[ 1 ] N.Q. Lam, S.J. Rothman and R. Sizmann, Radiat. Eft. 23 (1974) 53. [2] J.R. Mathews, Contemp. Phys. 18 (1977) 571. [3] L.K. Mansur, Nucl. Technol. 40 (1978) 5. [4] G. Tsarouhas, Phys. Lett. 103A (1984) 433. [5] G. Tsarouhas and T. Christidis, Phys. Lett. 109A (1985) 424. [6] N. Rouche, P. Habets and M. Laloy, Appl. Math. Sci. 22 (1977). [7] P. Cvitanovi~, ed., University in chaos (Hilger, Bristol, 1984).
241