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Mathematical problems arising in qualitative simulation of a differential equation
Artificial Intelligence 55 (1992) 61-86 Elsevier
61
Mathematical problems arising in qualitative simulation of a differential equation Olivier Dordan* Universit~ de Bordeaux 11, ISHA, 146 rue L~o Saignat, 33016 Bordeaux Cedex, France
Received June 1990 Revised December 1991
Abstract
Dordan, O., Mathematical problems arising in qualitative simulation of a differential equation, Artificial Intelligence 55 (1992) 61-86. Kuiper's QSIM algorithm is an expert system which studies the qualitative evolution of a differential inclusion. This paper addresses several issues raised by QSIM-type algorithms attached to landmark values, by using the concept of QQqualitative cells/Q/Q. Only the systems which share the same behavior on the boundary between two cells can be analyzed qualitatively. The principal purpose of this paper is to associate with a differential equation its QQqualitative projection/Q/Q which assigns to each qualitative cell its QQsuccessors/ Q/Q, to show how this algorithm can be implemented, and to illustrate the results with QQreplicator systems/Q/Q: a class of differential equation used in biochemistry and biology.
1. Introduction
One of the first qualitative simulation approaches was done by Kuipers in 1986 (see [13]). The purpose of QSIM was to derive a qualitative model from a quantitative one and then to simulate a qualitative behavior of some classes of real functions. Those functions represent the evolution of physical parameters. Each of these functions was supposed to have a finite ordered set of landmark values, Lr,
lo
0004-3702/92/$05.00 © 1 9 9 2 - Elsevier Science Publishers B.V. All rights reserved
O. Dordan
62
a n d a finite ordered set of distinguished time-points, T1i,
a= t,
= b,
where t E [a, b] is a distinguished time-point of f if t is an element of the set:
{t E [a, b] [ f(t) = x, where x is a landmark value of f} (see [13, p. 297]). Since we generally consider a physical system described by several functions, we define the global set of distinguished time-points, and the global set of landmarks as: T = U ~ 1 Tti, L = UP I L,;. The qualitative states are defined as pairs:
QS(f, t) := ( qval, qdir) , where
{l~.
if f ( t ) = l~ E L ,
qval= (1,,l i+,), [inc, qdir = ~std, ~dec,
(1)
i f f ( t ) C(l,,l~+~),
if f ' ( t ) > 0 , if f'(t) = 0 , iff'(t)<0.
For example: QS(water-level, tl) = ((10m, 20m), dec). The algorithm uses several rules which determine those qualitative states which are potential successors of the current qualitative state. I A network of constraints is added to include the functional dependencies between the different phy.sical parameters. The purpose of this paper consists in pointing out and addressing some mathematical problems raised by this algorithm. We define a qualitative behavior of an ordinary differential equation ,~ = f(x) where the state x ranges over a closed subset K of a finite-dimensional vector space X := ~", taking the same framework of landmark values introduced by Kuipers.2 First of all, for a given qualitative property (1), we define the concept of qualitative cell which is the set of states which satisfy the qualitative property. We want to study the behavior of the system from one qualitative cell to another. This amounts to finding the transition rules from one cell to another. The system needs to satisfy "collective" properties since the behavior of the vector field must be the same on the "boundary" separating two qualitative ceils. This is a first requirement of "qualitative filterability". Indeed, if this ~The non-intersection constrain about trajectories of autonomous systems was included in QSIM's package (see [14]). 2 In this paper we shall only treat the case of the evolution of the velocity's sign. Then we shall only have a qualitative behavior QS(x, t) := (qdir) where the value of qdir ranges over the set { - , 0, + }". The complete algorithm is treated in [7].
Qualitative simulation of a differential equation
63
property does not hold true, it is impossible to proceed with a meaningful qualitative analysis of the evolution of the system. (One can try to change the initial qualitative description to obtain a qualitative property like adding a new landmark in Kuiper's QSIM, and thus, in our framework, by adding new cells in such a way that the behavior of the system is the same on the boundary of two cells.) In other words, we shall construct the qualitative dynamical map q~3 associating with each qualitative cell its successor(s). It can be regarded as a
projection of the continuous dynamical system. We shall single out the concept of qualitative equilibrium, that is, a qualitative cell K, satisfying the following property: for each initial point x 0 in K, at least one solution x(-) to the differential equation starting from x 0 remains in K for all t. 4 Fortunately, Nagumo's Theorem provides necessary and sufficient conditions to characterize this "viability property". It is also very powerful to deal with constrained systems. The paper is organized as follows: • • • • •
We shall describe the concept of qualitative cells in Section 2. In Section 3, we shall give our version of the QSIM algorithm. We shall give a complete example in Section 4. In Section 5, we shall study the class of linear systems. In Section 6, we describe an application of this mathematical version of the QSIM algorithm for the special class of "replicator-systems" arising in chemistry and biomathematics. We shall present software involving numerical and symbolic computation which provides images of the qualitative cells; a "symbolic" description for the transition matrix of the system associating with each qualitative cell its successors, both on the screen (for three dimensions); and a prepared LATEX report. • The mathematical tools are given in the last section. The proof of the main theorems required by the algorithm involve the definition of tangent cones and uses Nagumo's Theorem which will be defined among other basic definitions. We shall also give in this section a characterization of nonemptiness for a qualitative cell. 3 In fact, it will be set-valued. 4 This m e a n s that the qualitative cell enjoys the "viability property". This property can be e x t e n d e d to differential inclusions in the following way: we consider initial value problems associated to the differential inclusion, x(t) ~ F(x(t)), where F is a set-valued m a p from X to X, a finite-dimensional vector space. If K is a closed subset of the domain of F, we say that solution x(-) to a differential inclusion is viable if x(t) remains in K for all t. T h e scope of viability theory is to characterize those sets and to use this characterization for m a n y purposes. Differential inclusions are also a powerful way to model uncertainty in the description of a physical system, and a natural, although sophisticated, mathematical tool to use in qualitative physics (see [1] and [2] for a presentation of these tools).
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2. Landmarks and qualitative cells We present here the concept of qualitative cells. Let us first consider the differential equation:
2(0 = f(x(t)),
(2)
where the state x ranges over a closed subset K of a finite-dimensional vector space X := R". For each c o m p o n e n t i we define a set S i of " l a n d m a r k s " for the states:
S i : : { S l i , . . . ,Spii} ,
Sli
i•
In a similar way we define a set L i of " l a n d m a r k s " for the velocities: L i
: : {11,. • . , In~} ,
Ill < 12i < ' "
< lnl i •
We introduce the qualitative sets ~ " and ow" defined by:
S
: : {((~,,, ~12) . . . . .
,jQan := {((/~I1, Jill2) .....
(~°,, ~.2)) I ~,,,~,2 e s,, ~,, ~< ~',2}, (J[~,l' ~p2)) I [~il,[~i2 e Si, ~il ~/~i2} -
We also suppose that if there exists a in L~ such that: ail ~< a ~< a ~ then a = a i l or a = a~2. We assume the same hypothesis for the sets S i. We define two set-valued maps from 5¢" to R", the first one connects any m e m b e r a ~ 5~" to
P,,(a) := {v E ~" [ail
• o i < ai2 ,
if all ~ ai2 ;
and vi = a i l , otherwise}, and the second is defined by:
/3 ( a ) : = {v E [~"1 all <~ O i <~ ai2 } . We also define two set-valued maps from L#" to ~n, the first one connects any m e m b e r b E 5¢ n to the following convex set: Q . ( b ) := {v E ~" [ bi~ < v~ < b~2, if b~ ~ b~2 ; and vg = bi~, otherwise} , and the second is defined by:
O_,,(b) := {v ~ R" I b,, <~ v~ <- bi~} . W h e n a landmark is equal to ---~ we shall define P.(a) or Q . ( b ) with "~<" or We split the viability domain K of the differential equation into "qualitative cells" and "large qualitative cells" defined by
K(~,b) := {x E K [ x E P.(a), f(x) E Q . ( b ) } ,
Qualitative simulation of a differential equation
I((a.b) : = {X ~ K I x E fin(a), f(x) E
65
an(b)}.
For clarity of exposition we shall present here a simplified version: the confluence case. For that we take: L i := { - ~ , 0, +co}
and
Si : = { - ~ , + ~ } .
In this case the qualitative state of a solution to the differential equation (2) at a given time t is the knowledge of the monotonicity property of each component xi(t ) of a solution x(.) to this differential equation, i.e., the knowledge of the sign of the derivatives :~i(t). Hence the qualitative behavior is the evolution of qualitative states of the solution, i.e., the evolution of the vector of signs of the components of Yc(t) =f(x(t)), which must be determined without solving the differential equation. In the following definition we shall introduce some notations and recall the definition of the qualitative addition. 2.1. We define ~ n := { - , 0 , +}n as the n-dimensional confluence space (see [6]). • For any a in ~ n we set Definition
R: := {v ~ R n I sign(v/)
=
ai}
and aR+ := {v E I~n [ sign(v/)
=
a i or O} .
• For any x in R n we shall denote by sn(x ) the sign vector: (sign(x 1),... . , sign(xn)). • The following table defines "standard qualitative addition".
0 +
-
0
-
-
+
(-,0,+)
-
0
+
(-,o,+}
+
+
2.2. We set " { - , 0, + } " instead of " ? " because a general treatment of the qualitative operator yields qualitative set-valued operators (see [2, Chapter 5; 7]).
Remark
For studying the qualitative behavior of the differential equation (2), i.e., the evolution of the functions t~-->s,(Yc(t)) associated to solutions x(.) of the differential equation, we split the viability domain K of the differential equation into 3 n "qualitative cells" K a and "large qualitative cells" k a defined
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by K a := {x E K I fix) ~ ~,i},
(3)
K. := { x ~ KI f l x ) E a ~
}.
Let us give an example of qualitative cells. We have chosen the following Van der Pol equation (see Fig. 1):
21 =Xl _X2_XE(X2+y2), )C2 ~- Xl q- X2 -- X2( X2 q- y 2 )
.
Indeed, the quantitative states x(.) evolving in a given qualitative cell K,~ share the same monotonicity properties because, as long as x(t) remains in K,, Vi=I .....
n,
sign(~)=a,.
The qualitative cell K o is then the set of equilibria of the system, because K o = { x ~ K ] f(x) = 0}. Studying the qualitative evolution of the differential equation amounts to knowing the laws (if any) which govern the transition from one qualitative cell K, to other cells without solving the differential equation. Remark 2.3. Instead of studying the monotonicity properties of each component xi(') of the state of the system under investigation, which can be too I
1 t
t
+
+
4-
+
+
4-
+
+
+
+
4-
+
+
+
+
_
+'~-
* 7
t
__5/" •
+
stands for
(+,+)
-
stands for
(-,-)
*
stands for
(+,-)
•
stands for
(-,+)
•
Fig. 1. Qualitative cells of Van der Pol equations.
Qualitative simulation of a differential equation
67
numerous, one can only study the monotonicity properties of m functionals V~(x(.)) on the state (for instance, energy or entropy functionals in physics, observations in control theory, various economic indexes in economics) which do matter.
3. The qualitative algorithm The purpose of this section is to present a version of the QSIM algorithm introduced by Kuipers for studying the qualitative evolution of solutions to differential equation (2). We first posit the following technical assumptions that will be explained in Section 7: f is continuous differentiable with linear growth ;
(4a)
K is a closed viability domain for f (see Definition 7.2).
(4b)
We say that f has a linear growth if there exists a positive constant c such that: Vx ~ D o m a i n ( f )
Ilf(x)ll-< c(llxll + 1).
Let us denote by S:K~--~ ~1(0, ~; X) the flow associating with each initial state x o E K the solution Sxo(. ) to the differential equation (2) starting at x 0. Definition 3.1. Let us consider a map f from K to X, f : K ~ X. We denote by 9 ( f ) , the subset of qualitative states a E ~ n such that the associated qualitative cell K a is not empty. We shall say that a qualitative state c E ~ ( f ) is a successor of b E ~ ( f ) if for all initial states x 0 E k~ M/~,., there exits ~- E (0, +~) such that Sxo(s ) E K, for all s ~ (0, ~-). A qualitative state a E @ ( f ) is said to be a qualitative equilibrium if it is its own successor. Our main objective is to express the fact that c is a successor of b through a set-valued map q~, one can construct ci9 using only the knowledge off. For that purpose, we shall set h(x) := f'(x)f(x), that is the acceleration of the system. We introduce the notation g i a := {X ~ ga I f ( x ) i = 0).
(Naturally, g a = gia whenever a i = 0 . ) We shall denote by F the set-valued map from ~ n to itself defined by
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O. Dordan
Va E ~ " ,
(l'(a))~ is the set of signs of h i ( x )
(5)
when x E / ~ i , . We also set lo(x ) := {i = 1 . . . .
, m I flx)i
~--
O}
and
We introduce the operation A on ~ " defined by bi , (b ^ c)i := [ 0 ,
if b i = c i , if b i ¢ c i ,
and the set-valued operation v where b v c is the subset of qualitative states a such that a i :=
b i or c i
.
We set a#b
¢~ V i = l , . . . , m ,
a i # b i.
Definition 3.2. We shall associate with the system (2) the discrete dynamical system on the confluence set ~ " defined by the set-valued map 4~ : ~ ' , , , ~ Yt" associating with any qualitative state b the subset
• (b) : : {c E ~ ( f ) [ r(b ^ c) c c v 0}.
(6)
We begin with necessary conditions for a qualitative state c to be a successor of b:
Proposition 3.3. L e t us a s s u m e that f is c o n t i n u o u s l y differentiable a r o u n d the viability d o m a i n K. I f c E ~ ( f )
is a successor o f b, then c belongs to ~ ( b ) .
For proving this proposition we use L e m m a 7.3 (last section). We shall now give an intuitive idea of the construction of F and q~ with a graphic in ~2 (see Fig. 2). Suppose we are interested in the transition between the two sets ( + , + ) and ( + , - ) . (We are in the velocity space, so the vector field on this space is an acceleration vector field and for this reason we have to compute h in our algorithm.) Thus we consider the vector field on ( + , + ) A ( + , --) ----( + , 0). For instance, we see that a vector (v,, v2) points towards ( + , - ) if and only if v 2 is positive. T o have a qualitative transition between ( + , + ) and ( + , - ) the whole vector field should satisfy a similar property of sign. We have a transition between ( + , + ) and ( + , - ) if the sign of v2 is always positive or always
Qualitative simulation of a differential equation
(-,÷)
69
(+,+) V2
l
mV3
v1 (--)
(÷,-)
Fig. 2. negative for all vectors on ( + , 0). This information is given by • through F which collects the signs of acceleration on the intersection of two cells.
3.1. Qualitative equilibrium T h e o r e m 3.4. Let us assume that f is continuously differentiable around the viability domain K. We posit the transversality assumption (11) discussed below.
b is a qualitative equilibrium if and only if b belongs to ~ ( b ) . 3.2. The Q S I M algorithm We shall now distinguish the 2" "full qualitative states" a # 0 from the other qualitative states, 5 the "transition states". When I is a non-empty subset of N := {1 . . . . . m}, we associate with a full state a # 0 the transition state a t defined by 1 fO, ai:=~ai,
if i E I , ifi~l.
If a#O is a full qualitative cell, there are 2 " - 2
nontrivial transition cells
aIEavO where IC{1,...
,m},
I~{1 .... ,m},
I¢=0.
L e m m a 3.5. Let a#O be a qualitative state which is not a qualitative equilib-
rium. Then there exist a solution starting at some x E K a and some t I > 0 such that x(t) E K, for t E [0, tl) and X(tl) E K,, for some non-empty subset I C N: ViEI,
x(t 1 ) ~ K i ~ .
5These sets are open in K when f is continuous.
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L e m m a 3.6. Let a#O be a full transition state. If F(a)#O (and thus, is reduced to a point), then, for any transition state a ~, there exists a unique successor b := qb(al)#O, i.e., for any initial states x in the transition cell K,,~, there exists t 2 > 0 such that, for all t E (0, t2), the solution x(t) remains in the full qualitative cell K~,.
Definition 3.7. We shall say that the system ( f ) is strictly filterable if and only if for any full state a E @ ( f ) # 0 , e i t h e r / ' ( a ) # 0 or a is a qualitative equilibrium or all transition states a / (I # ~) are qualitative equilibria. Theorem 3.8. Let us assume that f is continuously differentiable, and that the system ( f ) is "strictly filterable". Let a E ~ n be an initial full qualitative state. Then, for any initial state x in K,, the sign vector
is a solution to the Q S I M algorithm defined in the following way: There exist a sequence o f qualitative states a k satisfying
ao:=a, ak +1 ~ ~(ak
V
0)
(7)
and a sequence t o : = O < t l < . . . < t , , < . . , V t ~ ( t k , tk+j),
such that
QS(f,t):=(qdir):=a(t)=ak, O S ( f , t,+.) := ( qdir) := a(tk+l) = a k ix ak+ 1 .
In other words, we know that the vector signs of the variations of the solutions to differential equation (2) evolve according to the set-valued dynamical system (7) and stop when a k is either a qualitative equilibrium or all its transition states alk are qualitative equilibria. Remark 3.9. The solutions to the QSIM algorithm (7) do not necessarily represent the evolution of the vector signs of the variations of the observations of a solution to the differential equation. Further studies must bring answers allowing to delete impossible transitions from one full qualitative cell /(a to some of its transition cells Ka,. This is the case for a qualitative equilibrium, for instance, since a is the only successor of itself. Therefore, the QSIM algorithm requires the definition of the set-valued map F : ~ n , ~ 5~n by computing the signs of the m functions hi(. ) on the qualitative cells Ki~ for all i ~ N and a E @ ( f ) # 0 . This can be implemented and we give some examples in the following section.
Qualitative simulation of a differential equation
71
If, by doing so, we observe that the system is strictly filterable, then we know that the set-valued dynamical system (7) contains the evolutions of the vector signs of the variations of the solutions to the differential equation (2).
4. A basic example
To illustrate the qualitative simulation algorithm we take the basic simple example chosen in [13]: a ball thrown upward in a constant gravitation field. Here K = E2 and the differential equation :f = f(x) of this simple system is given by: =X2~
=-g,
with g > 0 .
Remark 4.1. In this case where many cells are empty, it is obvious to filter the empty cells. It is also trivial to observe that there is no equilibrium. We have only to consider three non-empty qualitative cells: two of them are "full" and the other one is a transition cell. We also have VX~
2
ft(x)~-(O0
10).
Computation o f the sets K- i, , i = 1, 2 If a = (0, - ) , If a = ( + , - ) ,
= g(0,->,
/ 2o =0.
If a = ( - , - ) ,
Computation of the sets Io(x) If x ~ K(+,_),
Io(x) = O.
If x E K(0_ ~ ,
Io(x ) = {1}.
If x ~ K ( _ _ ) ,
lo(x) = 0.
Computation of the sets If x E K ( + _ ) ,
~I(+)(x)~--~2
If x E K(0_ ) ,
[]~/+O(x) = {U ~ []~2 I O 1 ~ 0 )
If x E K(_
~(x)
),
,
= R2 .
.
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Computation o f the set-valued map F(a) F(O, - ) = ( { - } , (O}). F ( + , - ) = ( { - } , {0}).
r ( - , -) = ({-}, {~}). Since for any a in @ ( f ) we have F(a)#O, we deduce that the system is strictly filterable. We obtain: q,(+,-) q,(o,-)
=
{(-, -)}.
= ((-,-)}.
• (-, - ) = ((-, -)}. We remark that ( - , - ) is a qualitative equilibrium, 6 the qualitative solution is given by (see Fig. 3).
(+, -)--, (o, - ) ~ (-, -).
5. L i n e a r case
Let us consider a differential system given by .f = A ( x ) ,
where A E ~ ( R ' , ~ " ) .
We shall take K = ff~". • We observe that the qualitative cells are convex cones and that the sets Ko when a is a "full qualitative state" are open in En. • If we suppose that A is also invertible, then for any qualitative state a in { - , 0, + }" its associate qualitative cell K a is not empty. (In this case 0 is the only element of K0.) We shall give the proof of this result in Section 7.
(+, -i
Fig. 3. Phase portrait of the ball system. We can easily check the transversality condition.
Qualitative simulation of a differential equation
73
Lemma 5.1. Suppose now we have a differential system given by: Yc=f(x)= A(x)-v
,
wherevE~n.
For any qualitative state b E @(A) and for any i in { 1 , . . . , n}, F~(b) is given by the following formula (using the definition o f the standard qualitative addition): Fi(b ) =
(~
k=l,k~i
sign(aik(bk V 0)).
Proof. Since f~(x) = ~ aijxj - o i , j=l
we have hi(x ) =
aij
aikXk -- Ok ,
]=1
-k=l
I k~__laikX k -- Ui -~ 0 sign
atkx k -- Vt = b l v 0 .
Then sign(hi(x))=
k=l,k#i
sign(aik(b k v 0)),
-i
where x E K b .
[]
We deduce easily the following proposition: Proposition 5.2. We always consider the differential system described in the previous lemma. We assume that the matrix A is bijective. (a) I f all the entries o f A satisfy aij>~O ,
i# j ,
then a = (+ . . . . . +) and b = ( - , . . . . - ) are qualitative equilibria. (b) I f the entries o f A satisfy
aq=O,
icy,
all the qualitative cells are qualitative equilibria.
Proof. If the transversality assumption holds true, we know that a is a qualitative equilibrium if and only if a E ~(a). Here the transversality condition is assumed (it can be proved easily). If a E q~(a), then by definition, we have F(a) C a v O. In case (a) we obtain, using the preceding lemma, ( r ( a ) ) i = (o,
+).
Thus, a ~ @(a). We have a similar proof for b.
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74
For case (b) we obtain for each qualitative state c: (r(c))i
Thus c E O(c).
=
{0}.
~3
Remark 5.3. (1) If one had chosen a simple linear system ~/= A x and a qualitative description based on the qualitative cells,
/£~ := { x E K l x E a ~ + }
,
we would have found the first part of Proposition 5.1. We shall give the demonstration of this result in Section 7. (2) In the nonlinear case, we have the same kind of properties when the Jacobian matrix f ' ( x ) has equivalent good sign properties on the boundary of a set /7u. (3) Suppose we have a differential equation x = f ( x ) defined on K C R n. If the Jacobian matrix f ' ( x ) satisfies for all x E K conditions (a) of Proposition 5.2, the equation is called cooperative (the growth of every component is enhanced by an increase in any other component). (4) We can build many other examples. The qualitative state ( - , +, - ) is a qualitative equilibrium whenever the matrix A has the following form:
+)
(-,o,+}
-
-
{-,o,+}
.
Remark 5.4. Suppose we have a linear system ~/= A x . We can try to find a linear change of coordinates y = P x for the matrix of the system into a diagonal form. Then 3~-- P A P - l y enjoy property (b) of Proposition 5.2. If we can find such a real transformation, we find again the fact that the subspaces spanned by eigenvectors are invariant. The case K = Ez is very particular since we have the following result:
Proposition 5.5. For any two distinct full qualitative states a and b in { - , 0, +}z such that a A b ~ 0 we have a E 4~(b) or b E ga(a). Proof. We set c = a ^ b. Using Lemma 5.1 we remark that {0, E(c) =
= ciaic~,2 ~ or 0,
for one i E {1,2}, for the other o n e .
Ci{l,e~ denotes here the complementary of i in {1,2}. Then we get either
Qualitative simulation
of a
differential equation
75
a E th(b) or b ~ 4,(a) depending on the sign of a (if a is equal to O, then c is a qualitative equilibrium). [] We can now consider the four following types of matrices:
1:
({-,0,+} +
+ ) {-,0, +} '
2:
({-,0,+) _
(,0+, +
{_, o , +} ) ,
4:
(,0+, _
+)
{-,0, +)
'
Their corresponding qualitative behaviors are: 7
1:
{(+, ((+, {(-, ((-,
+)} c a,(+, +), +), (-, -)} c q,(+, - ) , -), (+, +)} C ~ ( - , +), -)} c q,(-, -) ;
3:
((-, ((+, {(-, ((+,
+)} c a,(+, +), +)} c ~(+, - ) , -)} C ¢b(-, +), -)} c a,(-, -) ;
2:
{(+, {(+, {(-, ((-,
4:
((+, ((-, ((+, {(-,
-)) c -)} c +)} c +)} c
a,(+, a,(+, q,(-, q,(-,
+), -), +), -) ;
-), (-, +)) C a,(+, +), -)} C a,(+, - ) , +)} C a,(-, +), +), (+, -)) c a,(-, - ) .
Examples of linear phase portraits are given in Fig. 4.
6. A p p l i c a t i o n to r e p l i c a t o r s y s t e m s
We have applied this "confluence qualitative dynamical analysis" to a special class of differential systems called "replicator systems". Suppose we have a system for which we know the growth rates gi(') of the evolution without constraints (also called "specific growth rates"): Vi
= l ....
, n ,
xi(t)
= xi(t)gi(x(t))
.
We correct this system by subtracting from each initial growth rate the common "feedback control ~7(.)" (also called "global flux" in many applications) defined as the weighted mean of the specific growth rates: Vx e S ~ ,
~7(x):= ~
xjg~(x).
j=l
7 We only consider here the transitions between two "full qualitative" states a and b different from 0 such that a A b # 0 .
O. Dordan
76
(+-)
(+ +> it _ (+,-)
t
,
(-,+)
~
-
~-
-) ~.__
-1
÷
it
(-,-)
c- I+)---
1
-1
1
-1
-I
!
A=
A ~
1
1
\
\
J
J
3
4,
tt
(-
+)
I I ~
i+,-)-
t
(-,+)
.
(+, +)
. -
(-,-) ,,
t
(+ +)
l
(
,t
t
'~ -1
/
-1
A =
\ -1
-1
1
-1
A=
1
-1
Fig. 4. E x a m p l e s o f l i n e a r p h a s e p o r t r a i t s .
Qualitative simulation of a differential equation
77
The new system is called "replicator system" (or system "under constant organization'r): Vi = 1 ....
, n,
Ai(t) = xi(t)(gi(x(t))
= xi(t)(gi(x(t))-
- ff(x(t)))
~
xj(t)gj(x(t))).
(8)
j=l
We can show that the probability simplex S n defined by
i=1
is a viability domain of the new dynamical system. This is an important example, because, in many problems, it is too difficult to describe mathematically the state of the system. Then assuming there is a finite number n of states, one rather study the evolution of their frequencies, probabilities, concentrations, mixed strategies (in games), etc. instead of the evolution of the state itself. Those systems are used in chemistry--we cite for example the model of catalytic growth of selfreproducing molecules: the hypercycle equation (see Eigen [9]). They are also used in many ecological models (see Hofbauer and Sigmund [11]). We can filter out many empty cells and one can show that the maximum number of qualitative cells which are not empty is: 3 n - 2 n+l q- 2. Software has been built to simulate this new "QSIM" algorithm applied to the "replicator system": • In this software we first use a symbolic calculus to correct the system introduced by the user and for computing the value f ' ( x ) f ( x ) necessary for the computation of the set-valued map @. • The different symbolic results are rewritten in FORTRAN syntax to be used in a FORTRAN program which provides the transition matrix and qualitative equilibria. It also draws the qualitative cells in the twodimensional simplex S 3. • A LATEX report is created such as in the examples. 8 We only have computed the image of full qualitative states by @ and in the image of those states we only have mentioned full qualitative states. In fact all the properties we can find depend on the choice of the system and on the qualitative frame. s
In the following examples the functions gi are linear functions:
gi(x) = ~_, aqxj . j=l
78
O. Dordan
Figures 5-7 aim to illustrate the set-valued map q~. The triangle is the set S 3 (which can be represented on the plane) and the "colored" subsets inside this triangle are the different sets S 3 where a is in @(f). The tree on the right symbolizes the map q~; the different elements of ~ 3 like for example (+, - , +) are represented by "colors". The root of each tree represents the element of which we want to define the image by 4~. An arrow between two "colors" means that there is a transition from the corresponding subsets in S 3. Example 6.1 (Fig. 5). Let the matrix A involved in the replicator system be: //1.00 A=[O.O0 \2.00
2.00 0.00 0.00
-1.00~ -200). 1100
• There are two non-empty "full" qualitative cells. Computation of the qualitative system q~:
•
~ ( - , - , +) = ( - , - , + ) , • (+, -, +) = (-, -, +).
• The following qualitative set is a qualitative equilibrium:
( - , - , +). •
Computation of the set-valued map
F.
C ( - , - , +) = ({0, - } , {0}, {0}), r ( o , - , +) : ({o, - } , {o}, {o}), r(o, o, 0) = ({0), {o}, {o}), r ( + , - , +) = ({o, - } , {o}, {o}).
1 2-1 0 0 -2
~..,
. ~ _ ~ .......
i i i i I i i I i i ~ . . _ _ J .......
..~
i
t tf
, *~
Fig. 5. Example 6.1.
~.
Qualitative simulation of a differential equation
79
Example 6.2 (Fig. 6). Let the matrix A involved in the replicator system be: /1.00 2.00 -1.00'~ A = ~ 3 . 0 0 0.00 - 2 . 0 0 ~ . \2.00 0.00 1.00/ • There are six non-empty "full" qualitative cells. • Computation of the qualitative system q~: ¢,(-, -, +) = (-, -, +), ¢,(+, -, +) = (-, -, +),
• (-, +, -) = (+, +, -),
• ( - , +, + ) =
, +,-),
~,(+, - , - ) = (+, +, - ) , ,p(+, +, - ) = (+, +, - ) . • The following qualitative sets are qualitative equilibria:
( ,
,+),
(+, + , - ) . • Computation of the set-valued map F :
r ( - , - , +) = ({-, o), {-, o}, (o)), r ( - , o, +) = ((o), {-, o), {o)), r ( - , +, - ) = ({o, +), (o), {-, o)), r ( - , +, o) = ((o), {o), ( - , o)), r ( - , +, +) = ((o), {-, o), ( - , o}), 1 2-1 3 0 -2
i
"~N
g:~i "?,,
P
/
t
•
Fig. 6. Example 6.2.
'~
IIIIIIIIIl#~''"
O, Dordan
80
r(o, - , +) = ( { - , o}, {o), (o)), r(o, o, o) = ({o), {o}, ( o ) ) ,
r(o, +, - ) = ((o, +}, {o), {o}). r(+,
, ) = ((o), {o, +), {-, o, + ) ) ,
r ( + , - , o) = ((o}, (o}, {-, o, + )), + ) = ( ( - , o), (o), { - , o, + ) ) ,
r(+,-,
r ( + , o, - ) = ((o}, {o, +}, {o}), r ( + , +, - ) = ((o, +), (o, +), {o}). Example 6.3 (Fig. 7). Let the matrix A involved in the replicator system be A=
5.00 -4.00 7.00
-1.00 0.00 -8.00
0.00~ 3.00 / . 0.50/
(9)
/
• There are six non-empty "full" qualitative cells. • Computation of the qualitative system @:
,t,(-, - , +) = ( - , +, + ) , ~ , ( - , +, - ) = f), ~ , ( - , +, +) = ( - , +, - ) , • (+, - , - ) = (+, - , + ) , • (+, - , +) = ( ~,(+, + , - ) - - ( + ,
-, +), , ).
-y
5-10
-403
/~
t
• ~
Fig.
7.
•
.
.
Example
•
•
,
6.3.
iiiillllll.....___.~ ..........
Qualitative simulation of a differential equation •
Computation of the set-valued map
81
F:
r ( - , - , +) = ( ( - , o}, (o, +}, (o}), r ( - , o, +) = ((o}, {o, +}, (o}), r ( - , +, - ) = ({-, o, +}, {o}, {-, o}), r ( - , +, o) = ({o}, {o}, {-, o}), r ( - , +, +) = ( ( - , o}, (o, +}, {-, o}), r(o, - , +) = ({-, o}, {o}, {o}),
(ao)
r(o, o, o) = ({o}, (o}, (o}), r(o, +, - ) = ({-, o, +}, (o}, {o}), r(+,-,-)
= ({o}, {-, o}, {o, +}),
r ( + , - , o) = ({o}, {o}, (o, +}), r ( + , - , +) = ({-, o}, {o}, (o, +}), r ( + , o, -) = ((o}, ( - , o}, {o}), r ( + , + , - ) = ({-, o, +}, {-, o}, {o}). Remark 6.4. In some example, such as Example 6.3, we can have a connected component of a qualitative cell which is a qualitative equilibrium and another one which is nothing particular. Remark 6.5. Examples 6.1 and 6.2 suggest to investigate some extension of the "bifurcation theory" for qualitative cells. We can check that the two given matrices are quite similar: only the entry a2~ is different. We can see that the results are quite different since, for example, the number of qualitative cells is different. Figure 8 illustrates this phenomenon.
.~.,,.illh, ,,,,.,,,,------"-
I
IIIIUIIIIIIIIIIII111,t,, . ~
.'-~,t~__.--.~'"
Stllll1111~.--.-.
~'~*
Fig. 8.
,
.~.~4~0
IIIIIIIIII.~'"
-13~-- "~'t""
82
O. Dordan
7. The mathematical tool kit
We now have to explain where those results come from. Viability theory aims to characterize different qualitative properties of dynamical systems (like invariance and viability). We generally consider differential inclusions rather than differential equations: we associate to a point a set of velocities. Differential inclusions can be used to model systems which are not well known (this is one of the goals of qualitative physics). Suppose we want to model the fact that the velocity v(x) at point x is between two landmarks a(x) and b(x). We set v(x) E [a(x), b(x)] so we define a differential inclusion. Since there is no reason for qualitative cells to be smooth sets (and generally they are not), we introduce first the concept of tangent cones for nonsmooth sets, which replaces the concept of tangent space in differential geometry (Fig. 9). This notion will be crucial to express the fact that a trajectory is going into a given qualitative cell and to characterize invariant or viable sets which are the analogues of equilibria for points. There are many ways to describe technically the concept of tangency to a given subset K at a point x in K. We regard the subset K as the subset of elements of the whole space X obeying given constraints. Pick any direction v in X and start from x in the direction v, ranging over the line x + hv when h > 0. We would like to distinguish those directions which, for small h, do not lead too far away from K. Such directions encapsulate the idea of tangency. We shall use two different cones: the Bouligand contingent cone and the Clarke tangent cone (which is always convex and a subset of the Bouligand contingent cone). K x
x -t- TK(X)
+ CK(~)
TK(X)
CK(X)
\1) Fig. 9. Tangent cones.
Qualitative simulation o f a differential equation
83
Definition 7.1. Let K C X be a subset of a normed vector space X and x ~ / ~ belong to the closure of K. The contingent cone T~(x) is the closed set of elements v which satisfy: lim inf d(x + hv, K ) / h = O. h~O+
The Clarke tangent cone CK(x) is the closed convex cone of elements v which satisfy: lim
K
d( y + hv , K ) / h = O .
h---~O+,y~x
Observe that when x is an element of the interior of K, TK(X ) = X. The Bouligand contingent cone is the most intuitive; it means that v is in the tangent cone if we can find two sequences hn~-~0 + and v n ~ - v such that the sequence x + h , v , is in K for all n. This cone is not generally convex, and since we sometimes need convexity for technical assumptions we also introduce the Clarke tangent cone which is a convex set in any case. Let us now consider the differential equation (2):
Definition 7.2. A subset K is a viability domain of f if and only if VxEK,
f(x)~TK(X ).
Fortunately, Nagumo's T h e o r e m defined below provides necessary and sufficient conditions to characterize this "viability property". It is also very powerful to deal with constrained systems. We shall denote by Xxo(.) a solution of the differential equation (2) starting at the point x 0. The following lemma shows right away why we need such tangent cones. L e m m a 7.3. Suppose we have a closed set K and x o belonging to K. If there exists T such that: VtE[0, T],
Xxo(t) ~ K ,
then f(Xo) belongs to TK(Xo). Proof. For any h in [0, T] we have
dK(Xo + hf(xo)) <~ IIx0 + hf(xo) - Xxo(h)H . Then
1 d, (Xo + hf(xo)) <<-
Xo(0) -
Xxo(h) _ f(Xo )
We conclude by letting h converge to 0.
[]
o. Dordan
84
This necessary condition is very intuitive; it means that the vector field points towards K. Under certain assumptions the converse is true; this is due to Nagumo in 1941. (See a proof and further development in [2].)
Theorem 7.4 (Nagumo). Suppose f & continuous with linear growth, then K enjoys the following viability property: for any initial state x o C K, there exists a solution x(.) to differential equation (2), which is viable in the sense that x(t) remains in K for all t >10 if and only if K is a viability domain. We can see in Fig. 10 that this theorem is not obvious. We see that the trajectory starting from x 0 exits from K even if the vector v I is tangent to K. We also have the analogue of Nagumo's Theorem for differential inclusions. In Section 5, we observed that we can find another qualitative frame which satisfies the first part of Proposition 5.2. To prove it, we use Nagumo's T h e o r e m . We only have to check the viability condition:
vET~,,,(x) ¢:5 vii>0
if x j = 0
but:
xE~"+ ~
AxE~"+ ~
2ETR,~(x ).
The calculus of tangent cones sometimes requires transversality assumptions; hence for T h e o r e m 3.4 we have to impose the following condition: Vx ~ K , ,
f'(x)CK(x ) - a~/+°(x~ = ~ " .
(11)
Figure 11 (in ~2) illustrates the geometrical intuition of transverse sets. Suppose we want to compute the tangent cone to the set C 1 n C~ at the point x. Since C1 n C~ is equal to x, the tangent cone Tqnc~(x ) is equal to {(0, 0)}, on the other hand Tq(x) n T~. (x) is equal to the line which is the intersection of
VI
Fig. 10.
Qualitative simulation of a differential equation Half plane Pl
85
Half plane P2
Fig. 11. the two half-planes: P1 n P2. We d o n ' t have equality because the intersection C 1 n C 2 is not " b i g " e n o u g h in relation to C 1 and C 2, so we say that the two sets are not transverse. This is a similar condition to ensure the non-emptiness of a qualitative cell as we can see with the following t h e o r e m : T h e o r e m 7.5. Let us assume that f is continuously differentiable around the viability domain K. Let £ belong to the qualitative cell K o. We posit the transversality condition:
f'(£)C~(£)
- a~"+ = ~ " .
(12)
Then the qualitative cell K a is n o n - e m p t y and £ belongs to its closure. In particular, if f ' ( £ ) C A £ ) = ~" ,
then the 3 n qualitative cells K a are n o n - e m p t y . In Section 5, we asserted that all the qualitative cells were not e m p t y . In the t r e a t e d case we r e m a r k that, for all x in R", A ' ( x ) is equal to A. Since A is bijective we have I m ( A ) is equal to ~" and then we apply T h e o r e m 7.5.
References [1] J.-P. Aubin, Viability Theory (Birkhiiuser, Boston, MA, 1991). [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis (Birkhfiuser, Boston, MA, 1990). [3] D.G. Bobrow, ed., Qualitative Reasoning about Physical Systems (North-Holland, Amsterdam, 1984/MIT Press, Cambridge, MA, 1985); also: Artif. lntell. 24. [4] R. Davis, Diagnostic reasoning based on structure and behavior, Artif. Intell. 24 (1984) 347-410. [5] J. de Kleer and D.G. Bobrow, Higher-order qualitative derivatives, in: Proceedings AAAI-
86
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17]
O. Dordan 84, Austin, TX (1984) 86-91; also in: D.S. Weld and J. de Kleer, eds., Readings in Qaalitative Reasoning about Physical Svstems (Morgan Kaufmann, San Mateo, CA, 1990). J. de Kleer and J.S. Brown, A qualitative physics based on confluences, Artif. lntell. 24 (1984) 7-83. O. Dordan, Analyse qualitative, Thesis, Universit~ de Paris 9 Dauphine, Paris (1990). O. Dordan, Algorithme de simulation qualitative d'une ~quation differentielle sur le simplexe, C.R. Acad. Sci. Paris SOr. 1 310 (1990) 479-482. M. Eigen, Selforganization of matter and the evolution of biological macromolecules, Naturwissenschaften 58 (1971) 465-523. K.D. Forbus, Qualitative process theory, Artif. lntell. 24 (1984) 85-168. J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems (Cambridge University Press, Cambridge, 1988). Y. lwasaki and H.A. Simon, Theories of causal ordering: reply to de Kleer and Brown, Artif. Intell. 29 (1986) 63-72. B.J. Kuipers, Qualitative simulation, Artif. lntell. 29 (1986) 289-338. B.J. Kuipers and W. Lee, Non-intersection of trajectories in qualitative phase space: a global constraint for qualitative simulation, in: Proceedings AAA1-88, St. Paul, MN (1988). E.P. Sacks, Piecewise linear reasoning, in: Proceedings AAAI-87, Seattle, WA (1987) 655-659; also in: D.S. Weld and J. de Kleer, eds., Readings in Qualitative Reasoning about Physical Systems (Morgan Kaufmann, San Mateo, CA, 1990). P. Struss, Mathematical aspects of qualitative reasoning, in: Proceedings Workshop on Qualitative Physics, Urbana-Champaign, IL (1987). K.M.-K. Yip, Extracting qualitative dynamics from numerical experiments, in: Proceedings AAAI-87, Seattle, WA (1987),
61
Mathematical problems arising in qualitative simulation of a differential equation Olivier Dordan* Universit~ de Bordeaux 11, ISHA, 146 rue L~o Saignat, 33016 Bordeaux Cedex, France
Received June 1990 Revised December 1991
Abstract
Dordan, O., Mathematical problems arising in qualitative simulation of a differential equation, Artificial Intelligence 55 (1992) 61-86. Kuiper's QSIM algorithm is an expert system which studies the qualitative evolution of a differential inclusion. This paper addresses several issues raised by QSIM-type algorithms attached to landmark values, by using the concept of QQqualitative cells/Q/Q. Only the systems which share the same behavior on the boundary between two cells can be analyzed qualitatively. The principal purpose of this paper is to associate with a differential equation its QQqualitative projection/Q/Q which assigns to each qualitative cell its QQsuccessors/ Q/Q, to show how this algorithm can be implemented, and to illustrate the results with QQreplicator systems/Q/Q: a class of differential equation used in biochemistry and biology.
1. Introduction
One of the first qualitative simulation approaches was done by Kuipers in 1986 (see [13]). The purpose of QSIM was to derive a qualitative model from a quantitative one and then to simulate a qualitative behavior of some classes of real functions. Those functions represent the evolution of physical parameters. Each of these functions was supposed to have a finite ordered set of landmark values, Lr,
lo
0004-3702/92/$05.00 © 1 9 9 2 - Elsevier Science Publishers B.V. All rights reserved
O. Dordan
62
a n d a finite ordered set of distinguished time-points, T1i,
a= t,
= b,
where t E [a, b] is a distinguished time-point of f if t is an element of the set:
{t E [a, b] [ f(t) = x, where x is a landmark value of f} (see [13, p. 297]). Since we generally consider a physical system described by several functions, we define the global set of distinguished time-points, and the global set of landmarks as: T = U ~ 1 Tti, L = UP I L,;. The qualitative states are defined as pairs:
QS(f, t) := ( qval, qdir) , where
{l~.
if f ( t ) = l~ E L ,
qval= (1,,l i+,), [inc, qdir = ~std, ~dec,
(1)
i f f ( t ) C(l,,l~+~),
if f ' ( t ) > 0 , if f'(t) = 0 , iff'(t)<0.
For example: QS(water-level, tl) = ((10m, 20m), dec). The algorithm uses several rules which determine those qualitative states which are potential successors of the current qualitative state. I A network of constraints is added to include the functional dependencies between the different phy.sical parameters. The purpose of this paper consists in pointing out and addressing some mathematical problems raised by this algorithm. We define a qualitative behavior of an ordinary differential equation ,~ = f(x) where the state x ranges over a closed subset K of a finite-dimensional vector space X := ~", taking the same framework of landmark values introduced by Kuipers.2 First of all, for a given qualitative property (1), we define the concept of qualitative cell which is the set of states which satisfy the qualitative property. We want to study the behavior of the system from one qualitative cell to another. This amounts to finding the transition rules from one cell to another. The system needs to satisfy "collective" properties since the behavior of the vector field must be the same on the "boundary" separating two qualitative ceils. This is a first requirement of "qualitative filterability". Indeed, if this ~The non-intersection constrain about trajectories of autonomous systems was included in QSIM's package (see [14]). 2 In this paper we shall only treat the case of the evolution of the velocity's sign. Then we shall only have a qualitative behavior QS(x, t) := (qdir) where the value of qdir ranges over the set { - , 0, + }". The complete algorithm is treated in [7].
Qualitative simulation of a differential equation
63
property does not hold true, it is impossible to proceed with a meaningful qualitative analysis of the evolution of the system. (One can try to change the initial qualitative description to obtain a qualitative property like adding a new landmark in Kuiper's QSIM, and thus, in our framework, by adding new cells in such a way that the behavior of the system is the same on the boundary of two cells.) In other words, we shall construct the qualitative dynamical map q~3 associating with each qualitative cell its successor(s). It can be regarded as a
projection of the continuous dynamical system. We shall single out the concept of qualitative equilibrium, that is, a qualitative cell K, satisfying the following property: for each initial point x 0 in K, at least one solution x(-) to the differential equation starting from x 0 remains in K for all t. 4 Fortunately, Nagumo's Theorem provides necessary and sufficient conditions to characterize this "viability property". It is also very powerful to deal with constrained systems. The paper is organized as follows: • • • • •
We shall describe the concept of qualitative cells in Section 2. In Section 3, we shall give our version of the QSIM algorithm. We shall give a complete example in Section 4. In Section 5, we shall study the class of linear systems. In Section 6, we describe an application of this mathematical version of the QSIM algorithm for the special class of "replicator-systems" arising in chemistry and biomathematics. We shall present software involving numerical and symbolic computation which provides images of the qualitative cells; a "symbolic" description for the transition matrix of the system associating with each qualitative cell its successors, both on the screen (for three dimensions); and a prepared LATEX report. • The mathematical tools are given in the last section. The proof of the main theorems required by the algorithm involve the definition of tangent cones and uses Nagumo's Theorem which will be defined among other basic definitions. We shall also give in this section a characterization of nonemptiness for a qualitative cell. 3 In fact, it will be set-valued. 4 This m e a n s that the qualitative cell enjoys the "viability property". This property can be e x t e n d e d to differential inclusions in the following way: we consider initial value problems associated to the differential inclusion, x(t) ~ F(x(t)), where F is a set-valued m a p from X to X, a finite-dimensional vector space. If K is a closed subset of the domain of F, we say that solution x(-) to a differential inclusion is viable if x(t) remains in K for all t. T h e scope of viability theory is to characterize those sets and to use this characterization for m a n y purposes. Differential inclusions are also a powerful way to model uncertainty in the description of a physical system, and a natural, although sophisticated, mathematical tool to use in qualitative physics (see [1] and [2] for a presentation of these tools).
64
O. Dordan
2. Landmarks and qualitative cells We present here the concept of qualitative cells. Let us first consider the differential equation:
2(0 = f(x(t)),
(2)
where the state x ranges over a closed subset K of a finite-dimensional vector space X := R". For each c o m p o n e n t i we define a set S i of " l a n d m a r k s " for the states:
S i : : { S l i , . . . ,Spii} ,
Sli
i•
In a similar way we define a set L i of " l a n d m a r k s " for the velocities: L i
: : {11,. • . , In~} ,
Ill < 12i < ' "
< lnl i •
We introduce the qualitative sets ~ " and ow" defined by:
S
: : {((~,,, ~12) . . . . .
,jQan := {((/~I1, Jill2) .....
(~°,, ~.2)) I ~,,,~,2 e s,, ~,, ~< ~',2}, (J[~,l' ~p2)) I [~il,[~i2 e Si, ~il ~/~i2} -
We also suppose that if there exists a in L~ such that: ail ~< a ~< a ~ then a = a i l or a = a~2. We assume the same hypothesis for the sets S i. We define two set-valued maps from 5¢" to R", the first one connects any m e m b e r a ~ 5~" to
P,,(a) := {v E ~" [ail
• o i < ai2 ,
if all ~ ai2 ;
and vi = a i l , otherwise}, and the second is defined by:
/3 ( a ) : = {v E [~"1 all <~ O i <~ ai2 } . We also define two set-valued maps from L#" to ~n, the first one connects any m e m b e r b E 5¢ n to the following convex set: Q . ( b ) := {v E ~" [ bi~ < v~ < b~2, if b~ ~ b~2 ; and vg = bi~, otherwise} , and the second is defined by:
O_,,(b) := {v ~ R" I b,, <~ v~ <- bi~} . W h e n a landmark is equal to ---~ we shall define P.(a) or Q . ( b ) with "~<" or We split the viability domain K of the differential equation into "qualitative cells" and "large qualitative cells" defined by
K(~,b) := {x E K [ x E P.(a), f(x) E Q . ( b ) } ,
Qualitative simulation of a differential equation
I((a.b) : = {X ~ K I x E fin(a), f(x) E
65
an(b)}.
For clarity of exposition we shall present here a simplified version: the confluence case. For that we take: L i := { - ~ , 0, +co}
and
Si : = { - ~ , + ~ } .
In this case the qualitative state of a solution to the differential equation (2) at a given time t is the knowledge of the monotonicity property of each component xi(t ) of a solution x(.) to this differential equation, i.e., the knowledge of the sign of the derivatives :~i(t). Hence the qualitative behavior is the evolution of qualitative states of the solution, i.e., the evolution of the vector of signs of the components of Yc(t) =f(x(t)), which must be determined without solving the differential equation. In the following definition we shall introduce some notations and recall the definition of the qualitative addition. 2.1. We define ~ n := { - , 0 , +}n as the n-dimensional confluence space (see [6]). • For any a in ~ n we set Definition
R: := {v ~ R n I sign(v/)
=
ai}
and aR+ := {v E I~n [ sign(v/)
=
a i or O} .
• For any x in R n we shall denote by sn(x ) the sign vector: (sign(x 1),... . , sign(xn)). • The following table defines "standard qualitative addition".
0 +
-
0
-
-
+
(-,0,+)
-
0
+
(-,o,+}
+
+
2.2. We set " { - , 0, + } " instead of " ? " because a general treatment of the qualitative operator yields qualitative set-valued operators (see [2, Chapter 5; 7]).
Remark
For studying the qualitative behavior of the differential equation (2), i.e., the evolution of the functions t~-->s,(Yc(t)) associated to solutions x(.) of the differential equation, we split the viability domain K of the differential equation into 3 n "qualitative cells" K a and "large qualitative cells" k a defined
66
O. Dordan
by K a := {x E K I fix) ~ ~,i},
(3)
K. := { x ~ KI f l x ) E a ~
}.
Let us give an example of qualitative cells. We have chosen the following Van der Pol equation (see Fig. 1):
21 =Xl _X2_XE(X2+y2), )C2 ~- Xl q- X2 -- X2( X2 q- y 2 )
.
Indeed, the quantitative states x(.) evolving in a given qualitative cell K,~ share the same monotonicity properties because, as long as x(t) remains in K,, Vi=I .....
n,
sign(~)=a,.
The qualitative cell K o is then the set of equilibria of the system, because K o = { x ~ K ] f(x) = 0}. Studying the qualitative evolution of the differential equation amounts to knowing the laws (if any) which govern the transition from one qualitative cell K, to other cells without solving the differential equation. Remark 2.3. Instead of studying the monotonicity properties of each component xi(') of the state of the system under investigation, which can be too I
1 t
t
+
+
4-
+
+
4-
+
+
+
+
4-
+
+
+
+
_
+'~-
* 7
t
__5/" •
+
stands for
(+,+)
-
stands for
(-,-)
*
stands for
(+,-)
•
stands for
(-,+)
•
Fig. 1. Qualitative cells of Van der Pol equations.
Qualitative simulation of a differential equation
67
numerous, one can only study the monotonicity properties of m functionals V~(x(.)) on the state (for instance, energy or entropy functionals in physics, observations in control theory, various economic indexes in economics) which do matter.
3. The qualitative algorithm The purpose of this section is to present a version of the QSIM algorithm introduced by Kuipers for studying the qualitative evolution of solutions to differential equation (2). We first posit the following technical assumptions that will be explained in Section 7: f is continuous differentiable with linear growth ;
(4a)
K is a closed viability domain for f (see Definition 7.2).
(4b)
We say that f has a linear growth if there exists a positive constant c such that: Vx ~ D o m a i n ( f )
Ilf(x)ll-< c(llxll + 1).
Let us denote by S:K~--~ ~1(0, ~; X) the flow associating with each initial state x o E K the solution Sxo(. ) to the differential equation (2) starting at x 0. Definition 3.1. Let us consider a map f from K to X, f : K ~ X. We denote by 9 ( f ) , the subset of qualitative states a E ~ n such that the associated qualitative cell K a is not empty. We shall say that a qualitative state c E ~ ( f ) is a successor of b E ~ ( f ) if for all initial states x 0 E k~ M/~,., there exits ~- E (0, +~) such that Sxo(s ) E K, for all s ~ (0, ~-). A qualitative state a E @ ( f ) is said to be a qualitative equilibrium if it is its own successor. Our main objective is to express the fact that c is a successor of b through a set-valued map q~, one can construct ci9 using only the knowledge off. For that purpose, we shall set h(x) := f'(x)f(x), that is the acceleration of the system. We introduce the notation g i a := {X ~ ga I f ( x ) i = 0).
(Naturally, g a = gia whenever a i = 0 . ) We shall denote by F the set-valued map from ~ n to itself defined by
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Va E ~ " ,
(l'(a))~ is the set of signs of h i ( x )
(5)
when x E / ~ i , . We also set lo(x ) := {i = 1 . . . .
, m I flx)i
~--
O}
and
We introduce the operation A on ~ " defined by bi , (b ^ c)i := [ 0 ,
if b i = c i , if b i ¢ c i ,
and the set-valued operation v where b v c is the subset of qualitative states a such that a i :=
b i or c i
.
We set a#b
¢~ V i = l , . . . , m ,
a i # b i.
Definition 3.2. We shall associate with the system (2) the discrete dynamical system on the confluence set ~ " defined by the set-valued map 4~ : ~ ' , , , ~ Yt" associating with any qualitative state b the subset
• (b) : : {c E ~ ( f ) [ r(b ^ c) c c v 0}.
(6)
We begin with necessary conditions for a qualitative state c to be a successor of b:
Proposition 3.3. L e t us a s s u m e that f is c o n t i n u o u s l y differentiable a r o u n d the viability d o m a i n K. I f c E ~ ( f )
is a successor o f b, then c belongs to ~ ( b ) .
For proving this proposition we use L e m m a 7.3 (last section). We shall now give an intuitive idea of the construction of F and q~ with a graphic in ~2 (see Fig. 2). Suppose we are interested in the transition between the two sets ( + , + ) and ( + , - ) . (We are in the velocity space, so the vector field on this space is an acceleration vector field and for this reason we have to compute h in our algorithm.) Thus we consider the vector field on ( + , + ) A ( + , --) ----( + , 0). For instance, we see that a vector (v,, v2) points towards ( + , - ) if and only if v 2 is positive. T o have a qualitative transition between ( + , + ) and ( + , - ) the whole vector field should satisfy a similar property of sign. We have a transition between ( + , + ) and ( + , - ) if the sign of v2 is always positive or always
Qualitative simulation of a differential equation
(-,÷)
69
(+,+) V2
l
mV3
v1 (--)
(÷,-)
Fig. 2. negative for all vectors on ( + , 0). This information is given by • through F which collects the signs of acceleration on the intersection of two cells.
3.1. Qualitative equilibrium T h e o r e m 3.4. Let us assume that f is continuously differentiable around the viability domain K. We posit the transversality assumption (11) discussed below.
b is a qualitative equilibrium if and only if b belongs to ~ ( b ) . 3.2. The Q S I M algorithm We shall now distinguish the 2" "full qualitative states" a # 0 from the other qualitative states, 5 the "transition states". When I is a non-empty subset of N := {1 . . . . . m}, we associate with a full state a # 0 the transition state a t defined by 1 fO, ai:=~ai,
if i E I , ifi~l.
If a#O is a full qualitative cell, there are 2 " - 2
nontrivial transition cells
aIEavO where IC{1,...
,m},
I~{1 .... ,m},
I¢=0.
L e m m a 3.5. Let a#O be a qualitative state which is not a qualitative equilib-
rium. Then there exist a solution starting at some x E K a and some t I > 0 such that x(t) E K, for t E [0, tl) and X(tl) E K,, for some non-empty subset I C N: ViEI,
x(t 1 ) ~ K i ~ .
5These sets are open in K when f is continuous.
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L e m m a 3.6. Let a#O be a full transition state. If F(a)#O (and thus, is reduced to a point), then, for any transition state a ~, there exists a unique successor b := qb(al)#O, i.e., for any initial states x in the transition cell K,,~, there exists t 2 > 0 such that, for all t E (0, t2), the solution x(t) remains in the full qualitative cell K~,.
Definition 3.7. We shall say that the system ( f ) is strictly filterable if and only if for any full state a E @ ( f ) # 0 , e i t h e r / ' ( a ) # 0 or a is a qualitative equilibrium or all transition states a / (I # ~) are qualitative equilibria. Theorem 3.8. Let us assume that f is continuously differentiable, and that the system ( f ) is "strictly filterable". Let a E ~ n be an initial full qualitative state. Then, for any initial state x in K,, the sign vector
is a solution to the Q S I M algorithm defined in the following way: There exist a sequence o f qualitative states a k satisfying
ao:=a, ak +1 ~ ~(ak
V
0)
(7)
and a sequence t o : = O < t l < . . . < t , , < . . , V t ~ ( t k , tk+j),
such that
QS(f,t):=(qdir):=a(t)=ak, O S ( f , t,+.) := ( qdir) := a(tk+l) = a k ix ak+ 1 .
In other words, we know that the vector signs of the variations of the solutions to differential equation (2) evolve according to the set-valued dynamical system (7) and stop when a k is either a qualitative equilibrium or all its transition states alk are qualitative equilibria. Remark 3.9. The solutions to the QSIM algorithm (7) do not necessarily represent the evolution of the vector signs of the variations of the observations of a solution to the differential equation. Further studies must bring answers allowing to delete impossible transitions from one full qualitative cell /(a to some of its transition cells Ka,. This is the case for a qualitative equilibrium, for instance, since a is the only successor of itself. Therefore, the QSIM algorithm requires the definition of the set-valued map F : ~ n , ~ 5~n by computing the signs of the m functions hi(. ) on the qualitative cells Ki~ for all i ~ N and a E @ ( f ) # 0 . This can be implemented and we give some examples in the following section.
Qualitative simulation of a differential equation
71
If, by doing so, we observe that the system is strictly filterable, then we know that the set-valued dynamical system (7) contains the evolutions of the vector signs of the variations of the solutions to the differential equation (2).
4. A basic example
To illustrate the qualitative simulation algorithm we take the basic simple example chosen in [13]: a ball thrown upward in a constant gravitation field. Here K = E2 and the differential equation :f = f(x) of this simple system is given by: =X2~
=-g,
with g > 0 .
Remark 4.1. In this case where many cells are empty, it is obvious to filter the empty cells. It is also trivial to observe that there is no equilibrium. We have only to consider three non-empty qualitative cells: two of them are "full" and the other one is a transition cell. We also have VX~
2
ft(x)~-(O0
10).
Computation o f the sets K- i, , i = 1, 2 If a = (0, - ) , If a = ( + , - ) ,
= g(0,->,
/ 2o =0.
If a = ( - , - ) ,
Computation of the sets Io(x) If x ~ K(+,_),
Io(x) = O.
If x E K(0_ ~ ,
Io(x ) = {1}.
If x ~ K ( _ _ ) ,
lo(x) = 0.
Computation of the sets If x E K ( + _ ) ,
~I(+)(x)~--~2
If x E K(0_ ) ,
[]~/+O(x) = {U ~ []~2 I O 1 ~ 0 )
If x E K(_
~(x)
),
,
= R2 .
.
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Computation o f the set-valued map F(a) F(O, - ) = ( { - } , (O}). F ( + , - ) = ( { - } , {0}).
r ( - , -) = ({-}, {~}). Since for any a in @ ( f ) we have F(a)#O, we deduce that the system is strictly filterable. We obtain: q,(+,-) q,(o,-)
=
{(-, -)}.
= ((-,-)}.
• (-, - ) = ((-, -)}. We remark that ( - , - ) is a qualitative equilibrium, 6 the qualitative solution is given by (see Fig. 3).
(+, -)--, (o, - ) ~ (-, -).
5. L i n e a r case
Let us consider a differential system given by .f = A ( x ) ,
where A E ~ ( R ' , ~ " ) .
We shall take K = ff~". • We observe that the qualitative cells are convex cones and that the sets Ko when a is a "full qualitative state" are open in En. • If we suppose that A is also invertible, then for any qualitative state a in { - , 0, + }" its associate qualitative cell K a is not empty. (In this case 0 is the only element of K0.) We shall give the proof of this result in Section 7.
(+, -i
Fig. 3. Phase portrait of the ball system. We can easily check the transversality condition.
Qualitative simulation of a differential equation
73
Lemma 5.1. Suppose now we have a differential system given by: Yc=f(x)= A(x)-v
,
wherevE~n.
For any qualitative state b E @(A) and for any i in { 1 , . . . , n}, F~(b) is given by the following formula (using the definition o f the standard qualitative addition): Fi(b ) =
(~
k=l,k~i
sign(aik(bk V 0)).
Proof. Since f~(x) = ~ aijxj - o i , j=l
we have hi(x ) =
aij
aikXk -- Ok ,
]=1
-k=l
I k~__laikX k -- Ui -~ 0 sign
atkx k -- Vt = b l v 0 .
Then sign(hi(x))=
k=l,k#i
sign(aik(b k v 0)),
-i
where x E K b .
[]
We deduce easily the following proposition: Proposition 5.2. We always consider the differential system described in the previous lemma. We assume that the matrix A is bijective. (a) I f all the entries o f A satisfy aij>~O ,
i# j ,
then a = (+ . . . . . +) and b = ( - , . . . . - ) are qualitative equilibria. (b) I f the entries o f A satisfy
aq=O,
icy,
all the qualitative cells are qualitative equilibria.
Proof. If the transversality assumption holds true, we know that a is a qualitative equilibrium if and only if a E ~(a). Here the transversality condition is assumed (it can be proved easily). If a E q~(a), then by definition, we have F(a) C a v O. In case (a) we obtain, using the preceding lemma, ( r ( a ) ) i = (o,
+).
Thus, a ~ @(a). We have a similar proof for b.
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74
For case (b) we obtain for each qualitative state c: (r(c))i
Thus c E O(c).
=
{0}.
~3
Remark 5.3. (1) If one had chosen a simple linear system ~/= A x and a qualitative description based on the qualitative cells,
/£~ := { x E K l x E a ~ + }
,
we would have found the first part of Proposition 5.1. We shall give the demonstration of this result in Section 7. (2) In the nonlinear case, we have the same kind of properties when the Jacobian matrix f ' ( x ) has equivalent good sign properties on the boundary of a set /7u. (3) Suppose we have a differential equation x = f ( x ) defined on K C R n. If the Jacobian matrix f ' ( x ) satisfies for all x E K conditions (a) of Proposition 5.2, the equation is called cooperative (the growth of every component is enhanced by an increase in any other component). (4) We can build many other examples. The qualitative state ( - , +, - ) is a qualitative equilibrium whenever the matrix A has the following form:
+)
(-,o,+}
-
-
{-,o,+}
.
Remark 5.4. Suppose we have a linear system ~/= A x . We can try to find a linear change of coordinates y = P x for the matrix of the system into a diagonal form. Then 3~-- P A P - l y enjoy property (b) of Proposition 5.2. If we can find such a real transformation, we find again the fact that the subspaces spanned by eigenvectors are invariant. The case K = Ez is very particular since we have the following result:
Proposition 5.5. For any two distinct full qualitative states a and b in { - , 0, +}z such that a A b ~ 0 we have a E 4~(b) or b E ga(a). Proof. We set c = a ^ b. Using Lemma 5.1 we remark that {0, E(c) =
= ciaic~,2 ~ or 0,
for one i E {1,2}, for the other o n e .
Ci{l,e~ denotes here the complementary of i in {1,2}. Then we get either
Qualitative simulation
of a
differential equation
75
a E th(b) or b ~ 4,(a) depending on the sign of a (if a is equal to O, then c is a qualitative equilibrium). [] We can now consider the four following types of matrices:
1:
({-,0,+} +
+ ) {-,0, +} '
2:
({-,0,+) _
(,0+, +
{_, o , +} ) ,
4:
(,0+, _
+)
{-,0, +)
'
Their corresponding qualitative behaviors are: 7
1:
{(+, ((+, {(-, ((-,
+)} c a,(+, +), +), (-, -)} c q,(+, - ) , -), (+, +)} C ~ ( - , +), -)} c q,(-, -) ;
3:
((-, ((+, {(-, ((+,
+)} c a,(+, +), +)} c ~(+, - ) , -)} C ¢b(-, +), -)} c a,(-, -) ;
2:
{(+, {(+, {(-, ((-,
4:
((+, ((-, ((+, {(-,
-)) c -)} c +)} c +)} c
a,(+, a,(+, q,(-, q,(-,
+), -), +), -) ;
-), (-, +)) C a,(+, +), -)} C a,(+, - ) , +)} C a,(-, +), +), (+, -)) c a,(-, - ) .
Examples of linear phase portraits are given in Fig. 4.
6. A p p l i c a t i o n to r e p l i c a t o r s y s t e m s
We have applied this "confluence qualitative dynamical analysis" to a special class of differential systems called "replicator systems". Suppose we have a system for which we know the growth rates gi(') of the evolution without constraints (also called "specific growth rates"): Vi
= l ....
, n ,
xi(t)
= xi(t)gi(x(t))
.
We correct this system by subtracting from each initial growth rate the common "feedback control ~7(.)" (also called "global flux" in many applications) defined as the weighted mean of the specific growth rates: Vx e S ~ ,
~7(x):= ~
xjg~(x).
j=l
7 We only consider here the transitions between two "full qualitative" states a and b different from 0 such that a A b # 0 .
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76
(+-)
(+ +> it _ (+,-)
t
,
(-,+)
~
-
~-
-) ~.__
-1
÷
it
(-,-)
c- I+)---
1
-1
1
-1
-I
!
A=
A ~
1
1
\
\
J
J
3
4,
tt
(-
+)
I I ~
i+,-)-
t
(-,+)
.
(+, +)
. -
(-,-) ,,
t
(+ +)
l
(
,t
t
'~ -1
/
-1
A =
\ -1
-1
1
-1
A=
1
-1
Fig. 4. E x a m p l e s o f l i n e a r p h a s e p o r t r a i t s .
Qualitative simulation of a differential equation
77
The new system is called "replicator system" (or system "under constant organization'r): Vi = 1 ....
, n,
Ai(t) = xi(t)(gi(x(t))
= xi(t)(gi(x(t))-
- ff(x(t)))
~
xj(t)gj(x(t))).
(8)
j=l
We can show that the probability simplex S n defined by
i=1
is a viability domain of the new dynamical system. This is an important example, because, in many problems, it is too difficult to describe mathematically the state of the system. Then assuming there is a finite number n of states, one rather study the evolution of their frequencies, probabilities, concentrations, mixed strategies (in games), etc. instead of the evolution of the state itself. Those systems are used in chemistry--we cite for example the model of catalytic growth of selfreproducing molecules: the hypercycle equation (see Eigen [9]). They are also used in many ecological models (see Hofbauer and Sigmund [11]). We can filter out many empty cells and one can show that the maximum number of qualitative cells which are not empty is: 3 n - 2 n+l q- 2. Software has been built to simulate this new "QSIM" algorithm applied to the "replicator system": • In this software we first use a symbolic calculus to correct the system introduced by the user and for computing the value f ' ( x ) f ( x ) necessary for the computation of the set-valued map @. • The different symbolic results are rewritten in FORTRAN syntax to be used in a FORTRAN program which provides the transition matrix and qualitative equilibria. It also draws the qualitative cells in the twodimensional simplex S 3. • A LATEX report is created such as in the examples. 8 We only have computed the image of full qualitative states by @ and in the image of those states we only have mentioned full qualitative states. In fact all the properties we can find depend on the choice of the system and on the qualitative frame. s
In the following examples the functions gi are linear functions:
gi(x) = ~_, aqxj . j=l
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Figures 5-7 aim to illustrate the set-valued map q~. The triangle is the set S 3 (which can be represented on the plane) and the "colored" subsets inside this triangle are the different sets S 3 where a is in @(f). The tree on the right symbolizes the map q~; the different elements of ~ 3 like for example (+, - , +) are represented by "colors". The root of each tree represents the element of which we want to define the image by 4~. An arrow between two "colors" means that there is a transition from the corresponding subsets in S 3. Example 6.1 (Fig. 5). Let the matrix A involved in the replicator system be: //1.00 A=[O.O0 \2.00
2.00 0.00 0.00
-1.00~ -200). 1100
• There are two non-empty "full" qualitative cells. Computation of the qualitative system q~:
•
~ ( - , - , +) = ( - , - , + ) , • (+, -, +) = (-, -, +).
• The following qualitative set is a qualitative equilibrium:
( - , - , +). •
Computation of the set-valued map
F.
C ( - , - , +) = ({0, - } , {0}, {0}), r ( o , - , +) : ({o, - } , {o}, {o}), r(o, o, 0) = ({0), {o}, {o}), r ( + , - , +) = ({o, - } , {o}, {o}).
1 2-1 0 0 -2
~..,
. ~ _ ~ .......
i i i i I i i I i i ~ . . _ _ J .......
..~
i
t tf
, *~
Fig. 5. Example 6.1.
~.
Qualitative simulation of a differential equation
79
Example 6.2 (Fig. 6). Let the matrix A involved in the replicator system be: /1.00 2.00 -1.00'~ A = ~ 3 . 0 0 0.00 - 2 . 0 0 ~ . \2.00 0.00 1.00/ • There are six non-empty "full" qualitative cells. • Computation of the qualitative system q~: ¢,(-, -, +) = (-, -, +), ¢,(+, -, +) = (-, -, +),
• (-, +, -) = (+, +, -),
• ( - , +, + ) =
, +,-),
~,(+, - , - ) = (+, +, - ) , ,p(+, +, - ) = (+, +, - ) . • The following qualitative sets are qualitative equilibria:
( ,
,+),
(+, + , - ) . • Computation of the set-valued map F :
r ( - , - , +) = ({-, o), {-, o}, (o)), r ( - , o, +) = ((o), {-, o), {o)), r ( - , +, - ) = ({o, +), (o), {-, o)), r ( - , +, o) = ((o), {o), ( - , o)), r ( - , +, +) = ((o), {-, o), ( - , o}), 1 2-1 3 0 -2
i
"~N
g:~i "?,,
P
/
t
•
Fig. 6. Example 6.2.
'~
IIIIIIIIIl#~''"
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80
r(o, - , +) = ( { - , o}, {o), (o)), r(o, o, o) = ({o), {o}, ( o ) ) ,
r(o, +, - ) = ((o, +}, {o), {o}). r(+,
, ) = ((o), {o, +), {-, o, + ) ) ,
r ( + , - , o) = ((o}, (o}, {-, o, + )), + ) = ( ( - , o), (o), { - , o, + ) ) ,
r(+,-,
r ( + , o, - ) = ((o}, {o, +}, {o}), r ( + , +, - ) = ((o, +), (o, +), {o}). Example 6.3 (Fig. 7). Let the matrix A involved in the replicator system be A=
5.00 -4.00 7.00
-1.00 0.00 -8.00
0.00~ 3.00 / . 0.50/
(9)
/
• There are six non-empty "full" qualitative cells. • Computation of the qualitative system @:
,t,(-, - , +) = ( - , +, + ) , ~ , ( - , +, - ) = f), ~ , ( - , +, +) = ( - , +, - ) , • (+, - , - ) = (+, - , + ) , • (+, - , +) = ( ~,(+, + , - ) - - ( + ,
-, +), , ).
-y
5-10
-403
/~
t
• ~
Fig.
7.
•
.
.
Example
•
•
,
6.3.
iiiillllll.....___.~ ..........
Qualitative simulation of a differential equation •
Computation of the set-valued map
81
F:
r ( - , - , +) = ( ( - , o}, (o, +}, (o}), r ( - , o, +) = ((o}, {o, +}, (o}), r ( - , +, - ) = ({-, o, +}, {o}, {-, o}), r ( - , +, o) = ({o}, {o}, {-, o}), r ( - , +, +) = ( ( - , o}, (o, +}, {-, o}), r(o, - , +) = ({-, o}, {o}, {o}),
(ao)
r(o, o, o) = ({o}, (o}, (o}), r(o, +, - ) = ({-, o, +}, (o}, {o}), r(+,-,-)
= ({o}, {-, o}, {o, +}),
r ( + , - , o) = ({o}, {o}, (o, +}), r ( + , - , +) = ({-, o}, {o}, (o, +}), r ( + , o, -) = ((o}, ( - , o}, {o}), r ( + , + , - ) = ({-, o, +}, {-, o}, {o}). Remark 6.4. In some example, such as Example 6.3, we can have a connected component of a qualitative cell which is a qualitative equilibrium and another one which is nothing particular. Remark 6.5. Examples 6.1 and 6.2 suggest to investigate some extension of the "bifurcation theory" for qualitative cells. We can check that the two given matrices are quite similar: only the entry a2~ is different. We can see that the results are quite different since, for example, the number of qualitative cells is different. Figure 8 illustrates this phenomenon.
.~.,,.illh, ,,,,.,,,,------"-
I
IIIIUIIIIIIIIIIII111,t,, . ~
.'-~,t~__.--.~'"
Stllll1111~.--.-.
~'~*
Fig. 8.
,
.~.~4~0
IIIIIIIIII.~'"
-13~-- "~'t""
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7. The mathematical tool kit
We now have to explain where those results come from. Viability theory aims to characterize different qualitative properties of dynamical systems (like invariance and viability). We generally consider differential inclusions rather than differential equations: we associate to a point a set of velocities. Differential inclusions can be used to model systems which are not well known (this is one of the goals of qualitative physics). Suppose we want to model the fact that the velocity v(x) at point x is between two landmarks a(x) and b(x). We set v(x) E [a(x), b(x)] so we define a differential inclusion. Since there is no reason for qualitative cells to be smooth sets (and generally they are not), we introduce first the concept of tangent cones for nonsmooth sets, which replaces the concept of tangent space in differential geometry (Fig. 9). This notion will be crucial to express the fact that a trajectory is going into a given qualitative cell and to characterize invariant or viable sets which are the analogues of equilibria for points. There are many ways to describe technically the concept of tangency to a given subset K at a point x in K. We regard the subset K as the subset of elements of the whole space X obeying given constraints. Pick any direction v in X and start from x in the direction v, ranging over the line x + hv when h > 0. We would like to distinguish those directions which, for small h, do not lead too far away from K. Such directions encapsulate the idea of tangency. We shall use two different cones: the Bouligand contingent cone and the Clarke tangent cone (which is always convex and a subset of the Bouligand contingent cone). K x
x -t- TK(X)
+ CK(~)
TK(X)
CK(X)
\1) Fig. 9. Tangent cones.
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83
Definition 7.1. Let K C X be a subset of a normed vector space X and x ~ / ~ belong to the closure of K. The contingent cone T~(x) is the closed set of elements v which satisfy: lim inf d(x + hv, K ) / h = O. h~O+
The Clarke tangent cone CK(x) is the closed convex cone of elements v which satisfy: lim
K
d( y + hv , K ) / h = O .
h---~O+,y~x
Observe that when x is an element of the interior of K, TK(X ) = X. The Bouligand contingent cone is the most intuitive; it means that v is in the tangent cone if we can find two sequences hn~-~0 + and v n ~ - v such that the sequence x + h , v , is in K for all n. This cone is not generally convex, and since we sometimes need convexity for technical assumptions we also introduce the Clarke tangent cone which is a convex set in any case. Let us now consider the differential equation (2):
Definition 7.2. A subset K is a viability domain of f if and only if VxEK,
f(x)~TK(X ).
Fortunately, Nagumo's T h e o r e m defined below provides necessary and sufficient conditions to characterize this "viability property". It is also very powerful to deal with constrained systems. We shall denote by Xxo(.) a solution of the differential equation (2) starting at the point x 0. The following lemma shows right away why we need such tangent cones. L e m m a 7.3. Suppose we have a closed set K and x o belonging to K. If there exists T such that: VtE[0, T],
Xxo(t) ~ K ,
then f(Xo) belongs to TK(Xo). Proof. For any h in [0, T] we have
dK(Xo + hf(xo)) <~ IIx0 + hf(xo) - Xxo(h)H . Then
1 d, (Xo + hf(xo)) <<-
Xo(0) -
Xxo(h) _ f(Xo )
We conclude by letting h converge to 0.
[]
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84
This necessary condition is very intuitive; it means that the vector field points towards K. Under certain assumptions the converse is true; this is due to Nagumo in 1941. (See a proof and further development in [2].)
Theorem 7.4 (Nagumo). Suppose f & continuous with linear growth, then K enjoys the following viability property: for any initial state x o C K, there exists a solution x(.) to differential equation (2), which is viable in the sense that x(t) remains in K for all t >10 if and only if K is a viability domain. We can see in Fig. 10 that this theorem is not obvious. We see that the trajectory starting from x 0 exits from K even if the vector v I is tangent to K. We also have the analogue of Nagumo's Theorem for differential inclusions. In Section 5, we observed that we can find another qualitative frame which satisfies the first part of Proposition 5.2. To prove it, we use Nagumo's T h e o r e m . We only have to check the viability condition:
vET~,,,(x) ¢:5 vii>0
if x j = 0
but:
xE~"+ ~
AxE~"+ ~
2ETR,~(x ).
The calculus of tangent cones sometimes requires transversality assumptions; hence for T h e o r e m 3.4 we have to impose the following condition: Vx ~ K , ,
f'(x)CK(x ) - a~/+°(x~ = ~ " .
(11)
Figure 11 (in ~2) illustrates the geometrical intuition of transverse sets. Suppose we want to compute the tangent cone to the set C 1 n C~ at the point x. Since C1 n C~ is equal to x, the tangent cone Tqnc~(x ) is equal to {(0, 0)}, on the other hand Tq(x) n T~. (x) is equal to the line which is the intersection of
VI
Fig. 10.
Qualitative simulation of a differential equation Half plane Pl
85
Half plane P2
Fig. 11. the two half-planes: P1 n P2. We d o n ' t have equality because the intersection C 1 n C 2 is not " b i g " e n o u g h in relation to C 1 and C 2, so we say that the two sets are not transverse. This is a similar condition to ensure the non-emptiness of a qualitative cell as we can see with the following t h e o r e m : T h e o r e m 7.5. Let us assume that f is continuously differentiable around the viability domain K. Let £ belong to the qualitative cell K o. We posit the transversality condition:
f'(£)C~(£)
- a~"+ = ~ " .
(12)
Then the qualitative cell K a is n o n - e m p t y and £ belongs to its closure. In particular, if f ' ( £ ) C A £ ) = ~" ,
then the 3 n qualitative cells K a are n o n - e m p t y . In Section 5, we asserted that all the qualitative cells were not e m p t y . In the t r e a t e d case we r e m a r k that, for all x in R", A ' ( x ) is equal to A. Since A is bijective we have I m ( A ) is equal to ~" and then we apply T h e o r e m 7.5.
References [1] J.-P. Aubin, Viability Theory (Birkhiiuser, Boston, MA, 1991). [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis (Birkhfiuser, Boston, MA, 1990). [3] D.G. Bobrow, ed., Qualitative Reasoning about Physical Systems (North-Holland, Amsterdam, 1984/MIT Press, Cambridge, MA, 1985); also: Artif. lntell. 24. [4] R. Davis, Diagnostic reasoning based on structure and behavior, Artif. Intell. 24 (1984) 347-410. [5] J. de Kleer and D.G. Bobrow, Higher-order qualitative derivatives, in: Proceedings AAAI-
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[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17]
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