Bifurcations and aggregation in large scale systems

Copyright © IFAC L arge Sc ale 51'Ste rn s: Th eo ry and App li ca ti ons 1986. Zurich. Switzerland. 1986 BIFURCATIONS AND AGGREGATION IN LARGE SCALE SYSTEMS M. Toro and J. Aracil Departa l/l flltu de A utu lll ti t ic a. Escuela Superiur de i llgfllinoJ ill du strial es. AI' d a. Reil/a "'·f era d es 5/ 11. Set'ilia, Spaill ABSTRACT This paper deals with the following problem: assume that a qualitative analysis (behaviour modes, bifurcation points, type of atractors .•• ) of a nonlinear dynamical system has been carried out and that afterwards this dynamical system is transformed into a large scale system through a disaggregating process of some (or all) of its variables. The problem at stake is to analyze whether the disagaregation gives rise to new behaviour modes, as a consequence of the appearance of new bifurcations in the disaggregated dynamical system. That leads us to study whether the original system and the disaggregated one are "equivalents" or whether the second one is richer in behaviours than the first one . The paper develops general results for standard disaggregation forms. Furthermore, practical applications of the proposed methodology to urban dynamics models is included . INTRODUCTION Consider a dynamical system given by the equations: where that a to (1), 1,r) is with • z '(z,q) (1) z E Rr and q E Rt. It is assumed disaggregation process is applied in such a way that every zi (i = deco~posed in parts Xj such that: z = !. x (2) i ... j i-I 0( = ~ n k=l k 0( +n i being ni the number of parts in which z~ has been decomposed. The set {xll.<j<~) will be called module i , associated to zi. After disagaregation, the dynamical system (1) will lead to a new one, of the form : . x = f(x,p) (3) where x ERn, p E RS being n = r n It should be noted that p is differen! from q, due to the greater richness in the description of (3) relative to (1). The model (3) will be considered a model refinement of the model (1). The problem is to study if the system (3) will show behaviour modes not shown by (1) . From a qualitative point of view that means to study if (3) will exhibit bifurcations not appear i ng in (1). The answer to those questions will be found through the qualitative analysis of (1) and (3). However, system (3) is a large scale system and, therefore, the qualitative analysis can be a very difficult task. In this paper we consider only dynamical 135 systems with point atracto r s; that is, we are restricted to dynamical systems with static bifurcations. For these systsms we propose a method wich allows to analyze if the behaviour modes of (3) are the same as those of (1); that is, if , as a consequence of the disaggregation process, there appear bifurcations in (3) not shown by (1) . This kind of results has practical interest because the process starts normally with dimension model (Randers 1980) disaggregated later on . a lot of modellina a small which is The paper proposed models. ends with applications of ths method to some urban dynamics BIFURCATION ANALYSIS Considering only static bifurcation analysis of reduced to the study of the equation: 'f'(z,q) = 0 b i furcations, the the model (1) is the solutions to (4) when the parameters q are varied, and to the stability study of each one of those solut i ons. The graphical representation of these solutions versus every parameter q gives rise to the bifurcation diagram of (4). These d i agrams can be obtained numerically with the help of continuation methods (~ubi~ek 1976). Let (zo,qO) be a solution of Eq. (4). If varying q around qo the number of solutions of (4), or just the stability of any of them, are chanaed, then it is said that (zo , qO) is a bifurcation point. These points are fundamental in the qualitative analysis of system (1) , since they supply all the informat i on needed to determine the qualitative shape of the bifurcation 136 M. Toro and J. Aracil diagram. For static bifurcations the bifurcation points are given by Eq. (4) and (5) since in the bifurcation points an eigenvalue of Jacobian matrix Dz'(z,q) is zero. REDUCIBLE DTNAnICAL STSTEftS Consider that we are interested on the bifurcation analysis of a large scale dynamical system. This problem could be greatly simplified if we can find a subsystem of the large scale one that would "concentrate" all the bifurcations. Theorems 1 and 2 below help to cope with that problem. Suppose we are given a dynamical system: · u = h(u,a) large scale that can be partitioned into the form: • u h (u ,u , a) 1 1 1 2 • u h (u ,u ,a) 2 2 1 2 where u = (u 1 ,u 2 ). Equilibria solutions to [he equations: h(u,u,a) 0 112 h(u,u,a) 0 212 Then, the stated. Tb.ore. 1 following theorems (6.1 ) (6.2) of (6) are (7.1) (7.2) can be If Eq. (7) can be transformed into the form h (u ,u ,a) 0 (8.1) 112 u = F(a) 2 (8.2) then system (6) has the same bifurcation diagram than the associated reduced system • u = h (u ,u ,a) (9) 1 2 where u2 is now a (constant) parameter, but related to parameters a by Eq. (8.2). Tbeore. 2 If the hypotheses of theorem are fullfilled and, furthermore, the same conditions that guarantee the stability in every branch of the bifurcation diagram of (9), can guarantee the stability of the corresponding branches in the bifurcation diagram of (6), then system (6) is reducible to (9). These theorems will be proved generalized in a forthcoming paper. and APPLICATION TO THE DISAGGREGATION PROCESS Take Eqs. (3) and reorder them in such a way that the following partition could be made: x f (x ,x ,w ) 1 1 1 2 1 . (10) x = f (x ,x ,w ) 2 2 1 2 2 where xl E Rr and x2 E Rn-r. The vector x is formed by a "representative" componeJt of every module i. The components of vector x2 are sorted in blocks coming from the different modules obtained by disaggregation of the variables zi It is convenient to transform vector (xl' x2) into vector (y,k), where Yi (that is, Yi = zi) will be the addition of all the x. variables belonging to module i and k tHe rate of variables x 21 relative to y ~ This transformation is carried out by: i where, being B an (r x if x2. belongs otherloli se, and matrix with c JJ module i. (11 ) (12) n-r) matrix, with b iJ = 1 to module i and b ij = 0, where C is a dragonal = l/Yi if x 2j belongs to Transformation (11) has an inverse, which is meaningful for studying the equilibria of (10) whether Yi ~ 0, or whether Yi = 0, and the form of the equations causes y disappear from the denominator. Thl~ happens when Eq. (3) has the form: • x i f (x,w)x (13 ) i i After applying transformations (11) to Eq. (10) we get: y f (x ,x ,w) ) + Bf (x ,x ,w) 1 1 2 2 1 2 (14) k Cf (x ,x ,w) 2 1 2 If Eq. (3 ) takes the form (13 ) then Eq. (14.2) will take the form: k f (x ,x ,w)k 2 1 2 i Bifur cations and Aggregation in La rg e Scale Systems 137 In Eqs. (14) xl and x2 are functions of y and I<.i' and are gIven by Eq. (11). Reordering parameters w, Eqs. (14) can be written: y g (y,l<.,p ) 1 1 • (15) k g (y,k,p ) 2 2 The transformation of parameters w into (Pl,P2) should be made in order to 1001<. for a correspondence between the variables and parameters of Eq. (16) below and the ones of Eq. (1). If the hypoteses of theorems 1 and 2 are fullfilled, then the dynamical system (15) is reduced to: y g (y,l<.,p ) (16) 1 1 where k is now a constant parameter. SOnE SPECIAL CASES Previous results can be kinds of disaggregation, used. applied to two which are widely a) Linear dl ... are.atlon. Suppose that functions f21 appearing in Eq. (10) are linear functions of variables x beloging to the same module as x 21 • Tfien the disaggegation is called linear. In such a case, functions g2 of (15) do not depend on y, due to [he form of transformation (11). Indeed, transformation (11) can be considered as an application of two successive transformations. The first one transforms (x 1, x 2) into (y,xi) through matrix T. Tfie second one, transforms (y,x ) into (y,k) by means of matrix T1• Function f2i is transformed into ffl throueh T2• This last function f2i s linear in y and in variables x 2j ' where the latter 6elong to module i. Through Tl functions e 2j tal<.e the form = f2 /Yi' Since f 2j is linear in Yi x 2j ,jex p ressions f 2j /Yi only depend on If apart from it beine a lin.ar disaegregation, matrix D g2 in (15) is stable then the above t~eorems can be applied and the disaegregation does not add new bifurcations (new behaviour modes) . In next section an example of this case will be presented. b) Dl .... r •• atlon wltb kernel This disaggregation occurs when the non linearities that appear in fl and f2 have as the only argument the variables Yi (that is, the additions of all the variables x belonging to module i) and, furthermore~ when, after transformation (11) , theorems 1 and 2 can be applied. APPLICATIONS TO URBAN DYNAftICS In urban dynamics (Alfeld and Graham, 1976) the evolution of the housing, or of the business structures, is described by a model of the form z = z(qlT(hz)-q2) (17) If the case of the housing evolution is considered, then z stands for housing, ql for the rate of housing demolition and qlT(hz) for the rate of housing construction. Function T(hz) represents the housing-land multiplier and its shape is show in Fig. 1. 15 ~-- ----~~---- --------, 10 05 (J) 0' 06 o. 10 Fig. 1 The qualitative analysis of this model can be found elsewhere (Aracil, 1981), and some related material in (Aracil, 1984). Model (17) describes the housing evolution. However, a disaggregation of the housing sector, tal<.ing into account the connection between housine units and the socioeconomic status of their occupants, can lead to a model refinement. This is done in (Alfeld and Graham, chap. 9) where the following model is proposed as a disaggreeation of (17). ~ n (x +n x >t(x +x +x )-n x 12122 123 5 1 . (18) x = n x -x 2 5 1 2 x = n x -x 362 3 where the total number of houses z has been disaggregated into variables x1 ,x 2 ,x 3 correspondine to upper income, middle income and lower income houses. The disaggregation from (17) to (18) is of the same type as the one from (1) to (3). A transformation of type (11) can be applied to this model, giving: • y n (1 -I<. -I<. +n I<. )~(y)y-n I<. y 2 2 3 2 2 7 3 I<. n (1 -I<. -k )-n I<. 2 5 2 3 6 2 I<. = n I<. -n k 36273 It should be noticed disaggregation is of linear (19) is reducible provided D g is stabl •. In this case, I<. 2 (19) that the type. System that matrix we have: 138 M. To ro a nd J. Aracil D g = I<. 2 [ -n 5 -n 6 n 6 whose stability is guaranted provided that ni>O. Consequently, system (19) is reducible to: y n (l -I<. -I<. +n I<. )T(y)y-n I<. y (20) 2 2 3 2 2 7 3 It should be noticed that (20) is equivalent to (l 7) so that a correspondence between the parameters of (l8) the and those of (17 ) can be form: q ~ n (1-1<. -I<. +n I<. ) 1 223 2 2 q~nl<. 273 stated in ( 21 ) where I<. and 1<.3 can be expressed as functiona of parameters n from equilibrium equations of system (i9). The disaggregation process has not eupplied new bifurcations, but it has aiven a more detailed way of computing the parametere of the aggregated model. If now expresion (17) models the evolution of business structures, then in (Alfeld and Graham 1976 , chap.8) a different form of disagaregation is considered. Tal<.ing into account the aging and obsolesc~nce of the business structures a disaggregated model of the following form is proposed • x = x (n ,(hex +x +x )- n ) (22) ,I 1 1 1 2 3 4 x = n x +x (n ,(~hex +x +x )-n -n ) 2 4 1 2 6 1 2 3 7 8 · x = n x + x (n "(h(x +x +x )-n ) 3 7 2 3 9 2 3 10 where now z in Eq . (17) stands for the number of business, and x2 ,x 1 and x3 in Eq. (22) for new busrness, mafure business, and deteriorating business. Reordering (22) and applying obtain: into the form (X 1 ,X 2 ,X 3) transformation (1), we If we are given ni then Eq. (24) are ~~:~:rissy:t~:~ct~~~ ~~' =a~~~) t~:~:~~r:~ required by theorem 1. Eq . (23.1) can be written: (25) The condition for an equilibrium y ~ 0 of equation (25) to be stable is T'(hy)(O. This condition guarantees the stability of the corresponding branch in the bifurcation diagram of (23), with y ~ 0 and I<.f ~O, since the jacobian matrix of this ast system worl<.s out to be: 1 • c "C(hy) 2 • c 4 o [ C '('( hy) c 't (hy) 3 - c 6 where all parameters ci are positive if nf are within the range of values meaningfu to the model. The condition for (26) to be stable is T'(hy)<O as it is easily shown. Comparing Eq. (23.1) easy to deduce the from Eq. (22) which parameters from Eq. with Eq, (17) it is grouping of parameters are equivalent to the (17) . ACQlOVLEDGtlENT This worl<. was supported by CAICYT under pr oject 1102/84 REFERENCES Alfeld, L. and A. Graham (1976) . Introduction to urban dyn .. ic •• Wright-Allen Press. Aracil , J. (1981). Structural stability of low-order system dynamics models. Int. J. Sy.te. Science. 12,423-441. Aracll, J, (1984) . Qualitative analysis and bifurcations in system dynamics models. IEEE-S"C-14, 4, 688-696, Kubicel<., H. (1976). Algorithm 502. Dependence of solutions of Nonlinear Systems on a parameter. AC" Tran •• "-th. Software 2, 98-107 Randers, J. (1980), Ele .. nt. of tbe .y.t •• dyn .. ic ... thod. HIT Press, y (I<. n +(1-1<. -I<. )n +1<. n )"C(hy)y-(n (1-1<. -I<. )+n I<.)y (23.1) . I<. .. • 1 I<. = 3 1 1 1 3 6 3 9 8 1 3 10 3 (n "C(hy)-n )1<. 1 4 1 n (1I<. -I<. )+1<. (n "C(hy)-n ) 7 1 3 3 9 10 Taking into account the equilibria of (23), the valuss of ki at the equilibrium point can be obtained from paramsters ni throuah the equations: (I<. n +(1-1<. -k )n +1<. n )n In -en (1-1<. -I<. )+n 1<.)=0 1 1 1 3 6 3 9 4 1 8 1 3 10 n (1-1<. -I<. )+1<. (n n In -n )=0 7 1 3 3 9 4 1 10 (23.2) (23.3) (24 . 1) (24.2)