ANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK

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1 Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION NETWORK TAKESHI SUGIMOTO Faculty of Egeerg Kaagawa Uversty Kaagawa Ward Yoohama, , Japa e-mal: tae@s.aagawa-u.ac.p Abstract Syergetc Iter-Represetato Networ s descrbed by -dmesoal ordary dfferetal equatos wth cubc olearty. The ovel fd s the fact that the basc equato set s trasformed to a set of -dmesoal Lota-Volterra equatos. The exstece ad stablty of fxed pots s show mathematcally rgorously by a seres of equalty codtos. For the sae of self-cosstecy ad umercal effcecy, t s proposed to use the -dmesoal Lota-Volterra equatos havg stable fxed pots.. Itroducto Syergetc Iter-Represetato Networ (SIRN short) s the paradgm for huma cogtve process that s frst proposed as IRN by Portugal ad the augmeted by Syergetcs (see for example [, 2]). Ther represetatos cover artfacts, atural ettes, behavors ad aythg our md [2]. Its mathematcal model s derved from the oe for Syergetc patter recogto []. The state varable vector s 2000 Mathematcs Subect lassfcato: 37F99, 9E0, 34D45. Keywords ad phrases: syergetcs, ter-represetato etwor, fxed pots. Receved October 24, 2008

2 2 TAKESHI SUGIMOTO expadable by represetatos such a maer: x : the state varable vector; x ξ v, () ξ v x : the order parameter for the th represetato; v : the patter vector of the th represetato; v : ts adot. Its dyamcs s descrbed by a autoomous ordary dfferetal equato wth respect to x wth cubc olearty: d dt 2 2 x ξ ξ v x x, (2) : the growth rate called the atteto parameter for th represetato; : the stregth of competto amog represetatos; : the parameter costrag the growth for all the represetatos. All these parameters are postve costats. Tag the er product of (2) ad v wth the ad of (), we obta the goverg equatos of the order parameters: d dt ξ 2 D ξ ξ, (3) D,.

3 ANALYSIS ON THE NATURE OF THE ASI EQUATIONS 3 2. Trasformato to Lota-Volterra Equatos Multplyg (3) by ξ ad substtutg for ξ, we obta 2 d dt 2 D. (4) It s a set of -dmesoal Lota-Volterra equatos, havg quadratc olearty. We should ote that the ew varable s o-egatve. 3. Nature of the Fxed Pots The fxed pots are obtaed by solvg or D 0 (5) 0 (6) for, 2,...,. Thus, the choce s 2, whch s equal to the total umber of the fxed pots. Let us deote (, l) as the lth fxed pot amog those havg o-zero compoets; l s [, ]. The frst s the total extcto of all the represetatos: ( 0, ) T ( 0, 0,..., 0). The secod s a set of sgle-represetato pots: (, ) ( 0,..., 0,, 0,..., 0) T for, 2,...,. I the rest of the fxed pots, several represetatos coexst. For example (, ) ( (, ) (,,..., ) T, 0,..., 0).

4 TAKESHI SUGIMOTO 4 Each compoet s obtaed mathematcally rgorously. Summg up (5) for to, we obta { },. (7) The equatos (5) ad (7) lead us to the soluto set:., (8) If ths value s egatve, such a fxed pot s ot feasble. Now, we shall show (9) for all the uder the codto of -represetatos coexstece: (0) for all the. Let us start from the followg equalty for : <. () Ths ca be show by straghtforward algebra as follows:

5 ANALYSIS ON THE NATURE OF THE ASI EQUATIONS 5 ( ) ( ) ( ) ( ) ( ). (2) I case, the last le of (2) s postve because of (0); hece () holds true; the mathematcal ducto leads us to the cocluso that () holds true for ay ; the (9) holds true for ay because of (0) ad (). The upper boud of (9) s gve by (3) for all the ad. Now, we shall chec stablty of the fxed pots. We eed to calculate the Jacoba of (4): I case of d J, (4) dt d 2 l, l dt 2( ). ( 0, ), d dt 2 0,. Therefore, J s postve defte ad hece 0, s a ustable ode.

6 6 TAKESHI SUGIMOTO I case of (, ), d dt 2 0.,, Therefore, J s egatve defte because of (3) ad hece all the are stable odes. (, ) for I case of (, ), (,) 2 ad d 2 l ad l dt, 2( ) ad, 0 ad.,, Ths Jacoba assumes the followg form: U ( (, ) A U J ), 0 A A covers the compoets wth to ad to, whle A L covers the compoets wth to ad to. The the determat of the Jacoba becomes as follows: L ( (, J ) ) A A. The lower part A L s egatve defte because ths dagoal matrx has all egatve compoets. To chec defteess of A U, we shall test t by usg two partcular vectors: y has y y ad zero for the rest; z has z z ad zero for the rest. The, we have U L ad y T A z U T,, y 2( 2 )( ) < 0 A U,, z 2( ) 0.

7 ANALYSIS ON THE NATURE OF THE ASI EQUATIONS 7 Therefore, A U s apparetly defte. Ths tur mples that ( (, J ) ) s also defte ad that ths Jacoba has some postve ad some egatve egevalues; hece (,) s a saddle pot. All such 2 pots are foud to be saddle pots the very same maer. I the vcty of these saddle pots, zero compoets wll ever grow aga, whle some of o-zero compoets wll grow further. Thus, represetatos are selected ths dyamcal system ad the ed oly oe survves. 4. oclusos To mae most of the Lota-Volterra formalsm, we mathematcally rgorously explore the ature of the fxed pots. Here s our proposal. As far as the state varable x s expaded by represetatos v for to, the total umber of the stable fxed pots should be. I ths sese, the Lota-Volterra formalsm s more atural tha the orgal formalsm, whch has 2 stable fxed pots: these have postve ad egatve order parameters. The orgal formalsm has cubc olearty, ad hece umercal scheme must have a fe tme terval to accurately tegrate steep chages. O the other had, the Lota-Volterra formalsm has quadratc olearty: the total umber of multplyg operatos s apparetly reduced; a less fe tme terval s allowed because of less steep chages. Acowledgemet I tha Professor Narta for ecouragg dscusso o ths topc. Refereces [] H. Hae ad J. Portugal, Syergetc tes I ad II, Self-Orgazato ad the tes, Sprger, Hederberg, (999), pp [2] J. Portugal, The Seve asc Propostos of SIRN, Nolear Pheomea omplex Systems 5(4) (2002), g