2 BASIC APPROXIMATION & MATRIX REPRESENTATION

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1 Sborík vědeckých rací Vysoké školy báňské - Techcké uverzty Ostrava číslo 2 rok 26 ročík II řada stroí čláek č.565 Xu XU * Stehe P. BNKS ** NONINER SYSTEMS: GOB THEORY OF TTRCTORS ND THEIR TOPOOGY NEINEÁRNÍ SYSTÉMY: VŠEOBECNÁ TEORIE TRKTORŮ JEJICH TOPOOGIE bstract I ths aer we shall study the global theory of varat sets of olear dyamcal systems by usg a tragulato of the mafold o whch the system s defed ad relacg the system by a lear model defed by a bary matrx. bstrakt V řísěvku e osáa všeobecá teore varatích sad eleárích dyamckých systémů s využtím tragulace agregátu a kterém e systém defová a výměou systému omocí leárího modelu defovaého bárí matcí. INTRODUCTION I ths aer we cosder a dyamcal system o a closed (comact) tragularsable mafold M ad show how to reduce the study of the toology of ts varat sets to a certa lear dyamcal system defed by a matrx. From ths matrx t wll be ossble to obta aroxmatos to the varat sets of the orgal dyamcs ad the to comute toologcal varats of these aroxmatos. We shall be terested studyg the lmts of the aroxmatos as the sze (mesh) of the tragulato teds to zero. There are varous other aroaches to ecewse-smooth aroxmatos-see for examle [Hsu 985] where cellular decomostos are used or [Baks ad Khathur 989] where ecewselear systems are studed va algebrac toology. The geeral structure theory of dyamcal systems o comact mafolds ca be foud [Shlkov et al 998] or [Baks ad Sog 26] where automorhc fucto theory s used to geerate systems o surfaces. The terretato gve here terms of a matrx ad the related lear dscrete system has several advatages. I artcular t s easy to detfy varat sets. 2 BSIC PPROXIMTION & MTRIX REPRESENTTION Suose that K s a smlcal comlex defg a tragulato of a -dmesoal comact mafold M. et K deote the -skeleto of K so that { T } K T * Xu XU Deartmet of utomatc Cotrol ad Systems Egeerg Uversty of Sheffeld Sheffeld UK co5xx@sheffeld.ac.uk ** Stehe P BNKS Deartmet of utomatc Cotrol ad Systems Egeerg Uversty of Sheffeld Sheffeld UK s.baks@sheffeld.ac.uk 243

2 where each T s a (closed) -smlex. We shall defe a dscrete dyamcal system o whch K aroxmates the orgal oe. To do ths let { b b } be the barycetres of the smlces of K ad let be the soluto of the dyamcal system through the we defe the ma ϕ b (Of course f ϕ b ever leaves T the FT T.) Thus we obta a ma b. Suose that whe ϕ leaves T t eters T b FT T () F : K K (2) defed by (). It s a fte dscrete dyamcal system. s such we ca make all the usual deftos (of lmt cycles varat sets etc.). Thus for examle Defto (forward) varat set of the dyamcal ma F s a subset S K such that F( S ) S. We are terested the questo of whe varat sets of the orgal system are aroxmated by those of the dscrete system (2). I order to aswer ths we eed to cosder the oto of verse system. Defto verse system s a collecto where ( Λ ) s a drected set s a ma for µ subect to the codtos such that ( X ) X Λ X Λ are some sets ad µ : X µ X d Λ µ ν. µ µν ν The lmt X of the verse system X s the subset of Note that there are atural roectos: µ ( x ) x for µ. µ : X. X Gve a attractor I a dyamcal system let Π X cosstg of all x ( x ) ΠX be a eghbourhood of I whch all traectores are attracted to I. Cosder the set of all oe covergs C of I whch are cotaed U I. et 244 U I

3 be a drected system where takes ay set X µ ( X ) X Λ ad X C Λ X µ s a refemet of X f µ : µ X µ X to oe cotag t. (We shall sst that the dameters of the sets o a cover decrease wth creasg.) We ca thk of the verse lmt as the tersecto of all X.e. X ( X ) I. It s ths sese that we say that a coverg of a set by some tragulate eghbourhood coverges to the set. Note that the verse lmt of such a system of oe covers of a metrc comactum coverges to the set tself. I fact a more detaled dscusso of ths theory could be gve terms of strog shae theory as gve the latter referece. Hece we have Theorem s the mesh δ of the tragulato K teds to zero a forward varat set of F teds to the verse lmt X of the above eghbourhood of the dyamcal attractor I. U I We wll troduce a matrx reresetato of the dscrete ma (). I fact we defe a matrx R by a f FT T otherwse th We wll also detfy wth the ut bass vector of T T ( ) R.e. T. Note that the matrx has a sgle each colum ad the other elemets of the colum k are zero. It s easy to see that all owers of k also have the same roerty. Clearly terms of we have M M M M f FT so that the system s effectvely learsed by. T Theorem 2 If { T } a matrx of the form T t s a varat set of F the s equvalet (by smlarty ) to ~ ~ ~ 2 ~

4 Proof By reumberg the tragles f ecessary we ca arrage that { T T l } s a varat set. Such a reumberg s equvalet to a smlarty trasformato o (comosed of elemetary oeratos). The form of gve the statemet the follows from stadard reresetato theory. Corollary If the dyamcal system has a fte umber of stable attractors the for large k eough k has the form ~ k O B ~ s B2 where the off-dagoal blocks are zero. 3 STBIITY THEORY & CONTRO THEORY We ca also use the matrx reresetato of a system defed by a tragulato of the sace to study stablty. If a system s globally asymtotcally stable the all the solutos ted to the org. Thus suose we tragulate the sace so that the org s cotaed tragle. The we have emma If the system s globally asymtotcally stable the the matrx assocated wth t as above has ts suffcetly hgh owers of the form Moreover the coverse s also true. ( ) k. I order that the matrx of the system satsfed the codtos of the lemma we see that t must be ossble to reumber the tragles so that s strctly uer tragular ad so we have a smle way of determg f a system s globally stable. Of course we ca also aly the dea to the case where the stablty s ot global. I that case the bas of attracto s a varat set ad the structure theorem 2 says that the matrx of the system must be block uer tragular. From lemma we see that the to left had block must be strctly uer tragular f the system has bas of attracto corresodg to the tragles ths art of the matrx. Cosder ext the alcato of the method to cotrol theory. Thus suose we have a olear cotrol system of the form x & f where the (scalar) cotrol u satsfes the hard costrat ( x u) u. If the system s defed o a mafold M wth a dstgushed ot (the org ) to whch t s desred to cotrol the system the we cosder as above a tragulato of M wth smlex cotag the dstgushed ot. For ay gve smlex the mafold we ow have a umber of ossble smlces to whch the dyamcs ca go from Ths s show Fg. below. 246 S S corresodg to a choce of cotrol.

5 Fg. Possble coe of cotrol drectos The matrx of the system ow aears the followg form: α α 2 α α where each α s or deedg o the choce of cotrol. (Note that of course α f there s o choce of cotrol whch drves the smlex k to smlex.) For each choce of cotrol at each smlex oly oe of the α s each colum wll be ozero. Combg these deas wth lemma we have the followg result whch gves a smle crtero for the stablzato of olear systems defed o mafolds: Theorem 3 system defed o a comact dfferetable mafold whch has a local reresetato the form x & f ( x u) ad a hard costrat as above s stablsable f the matrx of the tragulato for some choce of cotrols s equvalet uder smlarty trasformato to a strctly uer tragular matrx. k 4 EXMPE I ths secto we shall gve a smle examle of a system (a Va der Pol- lke system ths case) together wth a tragulato ad a matrx reresetato. The system s show (o a shere) Fg. 2. Fg. 2 Va der Pol- lke system 247

6 Rather tha show the ba em matrx to a hgh ower to llustra where ad sc matrx we wrte dow the syst te the varat cycle. I fact t s easy to see that the system matrx to a suffcetly hgh ower has the followg form: k CONCUSION ave show how to rereset a system the form of a lear matrx rerese REFERENCES ad Khathur S.. Structure ad Cotrol of Pecewse-lear Systems It. J. [2] Elltc ad utomorhc Dyamcal Systems o Surfaces to [3] ger-verlag Methods of qualtatve theory [5] ology of ttractors Nolear Dyamcal Revewer: rof. toí Víteček CSc. Dr.h.c. VŠB-Techcal Uversty of Ostrava I ths aer we h tato usg a tragulato of the mafold o whch the system s defed. From ths reresetato t s easy to determe varat sets ad ther toology. We have also show that the aroxmatos ted to the verse lmt of the coverg defed by the tragulato. The ma drawback wth the method s the large sze of the matrx reresetato artcularly for hghdmesoal systems. I a future aer we shall exame more effcet reresetatos whch cota the same formato. [] Baks S.P. Cotrol Baks S. P. ad Sog Y. aear It. J. Bfur. Chaos 26. Hsu Y Cell to Cell Mag Sr [4] Shlokov Shlkov Turaev D ad Chua olear dyamcs World Scetfc 998 Xu XU & Stehe P. Baks O the To Systems Proceedgs of 7th Iteratoal Caratha Cotrol Coferece (ICCC) 26 Ostrava Czech Reublc May ISBN