ERGODIC THEORY APPROACH TO CHAOS: REMARKS AND COMPUTATIONAL ASPECTS

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1 In. J. Appl. Mah. Compu. Sci.,, Vol., o., 9 7 DOI:.7/v--9- ERGODIC THEORY APPROACH TO CHAOS: REMARKS AD COMPUTATIOAL ASPECTS PAWEŁ J. MITKOWSKI, WOJCIECH MITKOWSKI Faculy of Elecrical Engineering, Auomaics, Compuer Science and Elecronics AGH Universiy of Science and Technology, al. Mickiewicza 3/B-, 3-9 Cracow, Poland pawel.mikowski@gmail.com Deparmen of Auomaics AGH Universiy of Science and Technology, al. Mickiewicza 3, 3-9 Cracow, Poland wojciech.mikowski@agh.edu.pl We discuss basic noions of he ergodic heory approach o chaos. Based on simple examples we show some characerisic feaures of ergodic and mixing behaviour. Then we invesigae an infinie dimensional model (delay differenial equaion) of eryhropoiesis (red blood cell producion process) formulaed by Lasoa. We show is compuaional analysis on he previously presened heory and examples. Our calculaions sugges ha he infinie dimensional model considered possesses an aracor of a nonsimple srucure, supporing an invarian mixing measure. This observaion verifies Lasoa s conjecure concerning nonrivial ergodic properies of he model. Keywords: ergodic heory, chaos, invarian measures, aracors, delay differenial equaions.. Inroducion In he lieraure concerning dynamical sysems we can find many definiions of chaos in various approaches (Rudnicki, ; Devaney, 97; Bronszejn e al., ). Our cenral issue here will be he ergodic heory approach. Ergodic heory in general has is origin in physical sysems of a large number of paricles, where microscopic chaos leads o macroscopic (saisical) regulariy. As he beginning of ergodic heory, he momen when Bolzmann formulaed his famous ergodic hypohesis, in (see, e.g., adzieja, 99; Górnicki, ) or in 7 (Lebowiz and Penrose, 973), can probably be considered. For more informaion abou he ergodic hypohesis, consul also he works of Birkhoff and Koopman (93) as well as Dorfman ().. Ergodic heory and chaos: Basic facs One of he mos fundamenal noions in ergodic heory is ha of invarian measure (see Lasoa and Mackey, 99; Fomin e al., 97; Bronszejn e al., ; Rudnicki, ; Dawidowicz, 7), which is a consequence of Liouville s heorem (see, e.g., Szlenk, 9; Landau and Lifszyc, 7; Arnold, 99; adzieja, 99; Dorfman, ). Transformaions (or flows) wih an invarian measure display hree main levels of irregular behaviour, i.e., (ranging from he lowes o he highes) ergodiciy, mixing and exacness. Beween ergodiciy and mixing we can also disinguish ligh mixing, mild mixing and weak mixing (Lasoa and Mackey, 99; Silva, ) and, on he level similar o exacness, he ype of K-flows (or K- propery, K-auomorphism) (cf. Rudnicki 9a; 9b; ; Lasoa and Mackey,99). In his aricle we will consider only ergodiciy and mixing. Firs we formalize hese noions and show some simple examples of ergodic and mixing ransformaions. Then in Secion 3. we analyze an infinie dimensional sysem which addiionally has ineresing medical (hemaological) inerpreaions. By {S } we denoe a semidynamical sysem or a semiflow on he meric space X, i.e., (i) S (x) =x for all x X; (ii) S (S (x)) = S + (x) for all x X,and, R + ; (iii) S : X R + X is a coninuous funcion of (, x). By a measure on X we mean any probabiliy measure defined on he σ-algebra B(X) of Borel subses of X. A

2 P.J. Mikowski and W. Mikowski measure μ is called invarian under a semiflow {S }, if μ(a) =μ(s (A)) for each and each A B... Ergodiciy. ABorelseA is called invarian wih respec o he semiflow {S } if S (A) =A for all. We now denoe by (S, μ) asemiflow{s } wih an invarian measure μ. Thesemiflow(S, μ) is ergodic (we say also ha he measure is ergodic) if he measure μ(a) of any invarian se A equals or. Le us now consider wo simple examples. Example. Le S :[, π) [, π) be a ransformaion generaing roaion hrough an angle φ on a circle wih uni radius (see Lasoa and Mackey, 99; Bronszejn e al., ; Devaney, 97; Dorfman, ): S(x) =x + φ (mod π). () If φ/π is raional, we can find invarian ses which have measure differen from or, and hus S is no ergodic. However, if φ/π is irraional, hen S is ergodic (for a proof, see he work of Lasoa and Mackey (99, p. 7) or Devaney (97, p. )). If we ake, e.g., φ = and pick an arbirary poin on he circle, we can observe ha successive ieraions of his poin under he acion of S will densely fill he whole available space (circle) (see Fig. ). Example. To undersand beer he ypical feaures of ergodic behaviour, le us consider he following ransformaion (see Lasoa and Mackey, 99, p. ): S(x, y) =( +x, 3+y) (mod ). () This is an exension of he roaional ransformaion () on he space [, ] [, ] [, ] [, ]. InFig. we can observe he resul of he acion of S on he ensemble of 3 poins disribued randomly in he area [,.] [,.]. The ransformaion () shifs he iniial area and does no spread he poins over he space. When we measure he Euclidean disance during ieraions beween wo arbirarily chosen close poins, we noice ha i is consan (Fig. ). Thus he popular crierion of chaos, i.e., sensiiviy o iniial condiions, is no a propery of ergodic ransformaions. Their propery is he dense rajecory (we formalize his fac in he las paragraph of his secion). One of he mos imporan heorems in ergodic heory is he Birkhoff individual ergodic heorem (Birkhoff, 93a; 93b; Birkhoff and Koopman, 93; Lasoa and Mackey, 99, Fomin e al., 97; Szlenk 9; Dawidowicz, 7; adzieja, 99; Gornicki, ; Dorfman, ). Here we cie a popular exension of his heorem (see Lasoa and Mackey, 99, p. ; Fomin e al., 97, p. ). Recall ha by (S, μ) we denoe a semiflow {S } wih an invarian measure μ. Theorem. (Exension of he Birkhoff heorem) Le (S, μ) be ergodic. Then, for each μ-inegrable funcion f : X R, he mean of f along he rajecory of S is equal almos everywhere o he mean of f over he space X, ha is, T lim f(s (x)) d = f(x) μ(dx), (3) T T μ(x) X μ-almos everywhere. If we subsiue f = A in Eqn. (3) ( A is he characerisic funcion of A) (see Lasoa and Mackey, 99; Rudnicki, ; Dawidowicz, 7), hen he lef-hand x ieraion Fig.. ormalized (o he probabiliy densiy funcion) round hisogram (bars inside he circle) showing ha a single poin under he acion of he ergodic ransformaion () wih φ = fills densely he whole circle. Fig.. Ieraions of he ergodic ransformaion () acing on an ensemble of 3 poins randomly disribued in [,.] [,.]: s ieraion, nd ieraion, 3rd ieraion, Euclidean meric beween wo arbirarily chosen close poins from he ensemble.

3 Ergodic heory approach o chaos: Remarks and compuaional aspecs side of (3) is he mean ime of visiing he se A and he righ-hand side is μ(a), and his corresponds o ergodiciy in he sense of Bolzmann, which roughly speaking is he mean ime ha a paricle of a physical sysem spends in some region and i is proporional o is naural probabilisic measure (Dawidowicz, 7; Dorfman, ; adzieja, 99; Górnicki, ; Birkhoff and Koopman, 93; Lebowiz and Penrose, 973) We can see ha ergodic behaviour in he pure form does no need o be very irregular and unpredicable. In fac, an invarian and ergodic measure should have some addiional properies o be ineresing from he poin of view of dynamics. Briefly speaking, i should be nonrivial for example, we inuiively undersand ha o have ineresing dynamics he measure should no be concenraed on a single poin. According o our knowledge, wo approaches o his problem appear in he lieraure. In he main ideas, boh seem o be similar, bu in he lieraure exis separaely. One is he heory of Prodi (9) (and Foias (973)), which says ha saionary urbulence occurs when he flow admis nonrivial invarian ergodic measure. This heory was srongly developed by Lasoa (979; 9) (see also Lasoa and Yorke, 977; Lasoa and Myjak, ; Lasoa and Szarek, ) and furher by Rudnicki (9a; 9; 9) (see also Myjak and Rudnicki, ) as well as Dawidowicz (99a; 99b) (see also Dawidowicz e al., 7). Anoher one uses he noion of SRB (Sinai, Ruelle, Bowen) measures (see, e.g., Bronszejn e al., ; Dorfman, ; Taylor, ; Tucker, 999). Roughly speaking, boh he approaches say ha o have ineresing dynamics he suppor of he measure should be possibly a large se. Le us now assume ha X is a separable meric space and μ is a probabiliy Borel measure on X such ha supp μ = X. We can sae ha (see Rudnicki,, p. 77, Proposiion ), if a semiflow (S, μ) is ergodic, hen for μ-almos all x he rajecory S (x), is dense. (see Lasoa and Mackey, 99, p. 7, pp. ) S(x, y) =(x + y, x +y) (mod ). () This is an example of he Anosov diffeomorphism (Anosov, 93) (see also Bronszejn e al.,, p. 93). In Fig. 3 we can see he firs he fifh and he enh ieraion of he mixing ranformaion () acing on he ensemble of 3 poins disribued randomly in he area [,.] [,.]. The poins are being spread over he space and aferwards ha ransformaion is lierally mixing hese poins in he whole space. The Euclidean disance beween close poins firs grows quickly and hen flucuaes irregularly (Fig. 3 ). The difference beween he ergodic ransformaion () (cf. Fig. ) is noiceable. Typical for mixing is he sensiiviy o iniial condiions (we will formalize his fac furher on). We can say more abou he chaoiciy of mixing sysems. Firs le us recall he following definiion (Auslander and Yorke, 9) (see also Rudnicki, ). Definiion. The flow is chaoic in he sense of Auslander and Yorke if (i) here exiss a dense rajecory, and (ii) each rajecory is unsable. Insabiliy here means ha here exiss a consan η> such ha for each poin x X and for each ɛ > here exiss a poin y B(x, ɛ) and > such ha ρ(s (x),s (y)) > η,whereρis he meric in X and B(x, r) is he open ball in X wih cener x and radius r>. Insabiliy can be also described here as he sensiiviy o iniial condiions, which is a popular crierion of chaos. ow, wih he assumpion ha X is a separable meric space and μ is a probabiliy Borel measure on X such.. Mixing. ow we will consider he noion of mixing, which exhibis a higher level of irregular behaviour han ergodiciy. The lieraure says ha he concep of a mixing sysem was inroduced by J.W. Gibbs (see, e.g., Dorfman,, p., ). A semiflow (S, μ) is mixing (see, e.g., Lasoa and Mackey, 99; Rudnicki, ; Bronszejn e al., ) if. lim μ(a S (B)) = μ(a)μ(b) for all A, B B. () This means ha he fracion of poins which a =are in A and for large are in B is given by he produc of he measures of A and B in X. Mixing sysems are also ergodic. Example 3. Le us consider he mixing ransformaion ieraion Fig. 3. Ieraions of he mixing ransformaion () acing on an ensemble of 3 poins randomly disribued in [,.] [,.]: s ieraion, h ieraion, h ieraion, Euclidean meric beween wo arbirarly chosen close poins from he ensemble.

4 P.J. Mikowski and W. Mikowski ha supp μ = X, we can sae ha (see Rudnicki,, p. 77, Proposiion ), if a semiflow (S, μ) is mixing, hen he semiflow {S } is chaoic in he sense of Auslander and Yorke. Example. Once again le us consider he mixing ransformaion () from Example 3. Le us consider a correlaion coefficien in he form (see de Larmina and Thomas, 93) γ xy (τ) = c xy(τ), τ =,,,..., () σ x σy where c xy (τ) = lim (x i x )(y i+τ y (τ)), (7) and x = lim σ x = σ y = i= i= lim lim x i,y (τ) = lim i= y i+τ () (x i x ), (9) i= (y i+τ y (τ)). () i= Once again he ranformaion () is acing on he ensemble of poins (his ime for higher accuracy). Afer a few ieraions i reaches he saisical equilibrium on he ensemble and wih furher ieraions i is mixing he ensemble in he space. We ake a sequence x i of he euclidean norms for he ensemble in he equilibrium, so we have a sequence of values. y i+τ for τ =is he same as x i and for τ =,,...i forms a sequence for furher ieraions. So using he formula () we obain a correlaion funcion where for τ =we have correlaion x i wih x i (Fig. ) and for τ =,,... we have correlaion beween x i and y i+τ which is moving away in ime. The resul is visible in Fig.. We can see ha he correlaion funcion () for he ensemble decreases o a value near very quickly (already in he nd ieraion). When we draw he spread of he ensembles on he space for τ >, e.g., τ =, we can see ha poins are correlaed neiher linearly nor in any oher way (Fig. ). Since he mixing ransformaion is also ergodic, we can change averages over he ensemble o averages along a single rajecory. So insead of calculaing a correlaion funcion for he whole ensembles, we can calculae i for a single rajecory and is ime shifs. The resul is presened in Fig. ; we can see ha he correlaion funcions in boh cases (ensemble and single rajecory) are almos he same. Such a rapid decrease in correlaion is ypical for mixing sysems (see Bronszejn e al., ; Rudnicki, ; 9). correlaion.. 3 au. au=.. correlaion.. 3 au. au=.. Fig.. Rapid decrease in he correlaion for he mixing ransformaion()foranensembleof poins, correlaion for a single rajecory and is ime shif, spread of poins of he ensemble for τ =, i.e., correlaion of he iniial ensemble wih iself, spread of poins of he ensemble for τ =. 3. Infinie dimensional case Supremum norm L norm Fig.. Two rajecories of Eqn. () for consan iniial funcions differen by. of he absolue value of he disance beween he values, disance in he supremum norm, disance in he L norm. Le us now consider he delay blood cell producion model formulaed by Lasoa (977):

5 Ergodic heory approach o chaos: Remarks and compuaional aspecs 3 d = σ +(ρ ( h)) s e γ ( h). () d Biological inerpreaions of his equaion have heir origin in he famous research of Ważewska-Czyżewska and Lasoa (97) ino mahemaical modelling of he dynamics of eryhropoiesis, which is a process of red blood cells (eryhrocyes) formaion in he bone marrow. For furher insigh ino his research, consul he works of Ważewska-Czyżewska (93) and Lasoa e al. (9). R is a global number of eryhrocyes in blood circulaion, σ denoes he desrucion rae of cells, ρ is oxygen demand, γ is he coefficien describing sysem exciaion and h is he delay ime represening he ime of mauraion of eryhrocyes. The conribuion of parameer s o a biomedical inerpreaion can be found in he work of Mikowski (). According o he auhors knowledge, he biomedical meaning of his parameer has no been explained in he lieraure ye. The producion funcion of blood cells in Eqn. () (which can be inerpreed as a feedback) has he form of he so-called unimodal funcion. Briefly speaking, i is a funcion wih one smooh maximum. Because of such a form of he feedback, Eqn. () may display very complicaed dynamics including chaos (see Ważewska-Czyżewska, 93; Mackey, 7; Liz and Ros, 9; Mikowski, ). Biological delay models wih unimodal nonlineariies were considered also by Mackey and Glass (977) as well as Gurney e al. (9), who described experimenal daa of icholson (9). However, he nonlineariy in Eqn. () is more flexible and gives sronger possibiliies for applicaions (for a deailed discussion of his problem, see Mikowski (). 3.. Conjecure of Lasoa. Lasoa (977, p. ) formulaed a conjecure concerning ergodic properies of Eqn. (), i.e., le C h be he space of coninuous funcions v :[ h, ] R wih he supremum norm opology. For some posiive values of parameers ρ, h, s and σ, here exiss a coninuous measure on C h which is ergodic and invarian wih respec o Eqn. (). By a coninuous measure we undersand here a measure which vanishes a poins (see Lasoa, 977; Lasoa and Yorke, 977) and in his sense he measure is nonrivial. Thus, according o our previous discussion, he conjecure concerns he chaoic behaviour of Eqn. (). I migh be very difficul o solve his problem using only mahemaical ools. In general, according o he auhors knowledge, here are very few resuls where chaos for delay differenial equaions was proved using only mahemaical ools. One of such resuls was given by Walher (9). Our aim is o invesigae Eqn. () numerically in order o check if i exhibis nonrivial ergodic properies. There is also an ineresing hisorical conex of Lasoa s hypohesis. Ulam (9, p. 7) (see also Myjak, ) posed he problem of he exisence of nonrivial invarian measures for ransformaions of he uni inerval ino iself defined by a sufficienly simple funcion (e.g., a piecewise linear funcion or a polynomial) whose graph does no cross he line y = x wih a slope in an absolue value less han. Laer Lasoa and Yorke (973) solved he problem. The conjecure of Lasoa for Eqn. () looks like a generalizaion of Ulam s conjecure o firs order differenial delay equaions. This associaion comes up during numerical invesigaions of Lasoa s delay equaion, where we search for a proper shape of unimodal feedback o find nonrivial ergodic properies (see Fig. 7). 3.. Calculaions. umerical invesigaions show ha Eqn. () exhibis nonrivial ergodic properies for ρ korelacja ensemble of (9)..... RL977,korelacjaod9co.,corr. 3 au RL977,rozrzuod9 ensemble of (9) correlaion ensemble of (9+)..... RL977,korelacjaraj,corr. au RL977,rozrzuod9 ensemble of (9) Fig.. Rapid decrease in he correlaion for Eqn. () for an ensemble of rajecories, correlaion for a single rajecory and is ime shif, spread of poins of he ensemble for τ =, i.e., correlaion of he iniial ensemble wih iself, spread of poins of he ensemble for τ =. F() F()= Fig. 7. Range of parameers (shaded area) which generae he nonrivial ergodic behaviour of he righ-hand side of (), unimodal feedback funcion in reference o he linear desrucion rae of he red blood cells. In boh he cases he lower bound corresponds o ρ =. and he upper o ρ =.. f() 3 3 sigma*

6 P.J. Mikowski and W. Mikowski [.,.], σ =., s =, γ =andadelayof h>9. In Fig. 7, we can see he range of he righ-hand side F () of (), wih he line F () =. Thelower bound of he shaded area corresponds o ρ =. and he upper bound o ρ =.. In Fig. 7 here is he same range of parameers bu presened in he form of he unimodal feedback funcion in reference o he linear desrucion rae of red blood cells. Ergodic properies susain for large values of h (like h =); however, he more h increases, he more rajecory is araced o and ergodic properies decay. We will show now some numerical experimens indicaing ergodic properies of Eqn. (). We choose ρ =., σ =., s =, γ =, i.e., he lower bounds from Fig. 7 and. Equaion () is solved using he MATLAB solver de3 (see Shampine e al., ). Many imporan aspecs concerning numerical invesigaions of probabilisic properies of delay differenial equaions were presened by Taylor (). Useful direcions for compuaional analysis of ergodic properies were presened by Lasoa and Mackey (99), Kudrewicz (99; 993, 7) as well as O (993). I is obvious ha numerically we canno show ergodic properies on he whole infinie dimensional space. We wan o show ha on some subspaces, Eqn. () has a smooh invarian densiy, which for a large ensemble (see Fig. 9) of rajecories is equal o he average along all single rajecories. Tha would indicae ha he sysem exhibis basic ergodic properies. Afer ha, using correlaion echniques and examining he unsabiliy of rajecories, we wan o invesigae mixing properies. As he sae of Eqn. () we will consider a funcion of an inerval of lengh h (delay) (see Fig. ). We will analyze is behaviour in subspaces of an infinie dimensional space of is values. A graphical example of such a subspace is presened in Fig.. I is a six-dimensional space consruced by aking six arbirary poins of he funcional sae of Eqn. (). Anoher soluion is o equip he space C h wih a proper norm; however, in his aricle apar, from one excepion (see Fig. ), we shall no consider his case. Resuls of compuaional analysis of Eqn. () in such spaces can be found in he work of Mikowski () Ergodiciy of he flow. Consider Fig. 9, showing a bunch of rajecories of Eqn. (). Firs hey evolve quie regularly bu afer some ime he flow becomes very irregular, we could even say urbulen. Addiionally, rajecories are bounded. Le us ake wo arbirary subspaces form he infinie dimensional space we have inroduced previously, e.g., he mos naural space of values R and he space ( h) (which is ofen used for delay differenial equaions). In Fig. we can observe chosen momens of evoluion of consan iniial funcions of Eqn. () disribued exponenially on some inerval. Figure,,(e),(g) shows he evoluion on he space of R and Fig.,, (f), (h) on he space ( h). Afer some ime he normalized (o he probabiliy densiy) hisograms (couning he number of poins of he ensemble in he subinervals of he space) end o invarian hisograms, i.e., some ime afer simulaions hey almos do no change heir shape. This may indicae ha we have reached some invarian densiy. In order o check if his densiy ends o be smooh, we could calculae a significanly larger ensemble of ra- 3 ( h) ( 9h/) ( h/) ( h/) ( h/) ( h) 9. ( 9h/) ( h/) ( h/) 9. ( h/) Fig.. Geomerical represenaion of sae evoluion given by Eqn. (): an arbirary sae, an example of is represenaion in he six-dimensional space ( h/) ( h) ( h/) ( h/) ( 9h/) [ h, ] Fig. 9. Bunch of rajecories of Eqn. (). Firs he flow is regular, hen i becomes urbulen.

7 Ergodic heory approach o chaos: Remarks and compuaional aspecs jecories, bu hen numerical calculaions ake a lo of ime and become useless. However, we can examine if he flow exhibis he ergodic propery, i.e., if hisograms for single rajecories are similar o ha of he ensemble. If ha were rue we could consruc a hisogram for a very long single rajecory and ha would reflec also he average over he ensemble (see Theorem ). Indeed, numerical simulaions indicae ha Eqn. () exhibis his ypical propery of ergodic flows; in Figs.,, we have more accurae hisograms for single rajecories. We can see ha he higher he accuracy he smooher he hisograms. Each rajecory is also irregular (see Figs., ), which is in accordance wih he heory discussed in previous secions. The behaviour of he ensemble on he space ( h) (see Figs.,, (f), (h)) as well as ha of he single rajecory on his space (see Figs.,) may sugges rl977,ewgeszbpoczrwna[.,.3],= RL977,ewzbpoczrwna[.,.3],si=.,r=.,g=,s=,h=,podphis3d,pk ha here exiss an aracor, which has a significan volume, supporing he invarian ergodic measure. 3.. Mixing properies of he flow. The flow generaed by Eqn. () exhibis also properies ypical for mixing sysems. umerical simulaions indicae ha each rajecory is unsable. In Fig. we can see ha he absolue value of he disance beween he values of wo rajecories saring from very close iniial funcions is flucuaing irregularly. We have marked before ha we will no consider any specific norm in he space, bu here we will make an excepion, because he unsabiliy for Eqn. () is much beer visible when we equip he space wih he supremum or L norm (see Fig.,). Addiionally, he correlaion for he ensemble and for he single rajecory and is ime shifs decreases rapidly (see Fig. ), which is characerisic for mixing sysems (see Secion.). The lack of correlaion suggess ha he aracor does no have a simple srucure. I may also indicae ha each rajecory is urbulen in he sense of Bass (Bass, 97; Rudnicki, ; 9). Compuaional resuls concerning he problem of urbulence for Eqn. () can be found in he work of Mikowski () = rl977,ewgeszbpoczrwna[.,.3] = rl977,ewgeszbpoczrwna[.,.3] ( ) () RL977,ewzbpoczrwna[.,.3],si=.,r=.,g=,s=,h=,pk,podphis3d 3 3 () () RL977,ewzbpoczrwna[.,.3],si=.,r=.,g=,s=,h=,pk,podphis3d. Concluding remarks We have presened numerical compuaions suggesing ha he delay differenial equaion () posseses an aracor of a nonsimple srucure, supporing an invarian mixing measure. This verifies he conjecure of Lasoa which, using he language of ergodic heory, poses he problem of he chaoic behaviour of Eqn. (). More compuaional analysis concerning ergodic properies of Eqn. () as well as new conribuions o is = (e) rl977,ewgeszbpoczrwna[.,.3],= = () () (f) RL977,ewzbpoczrwna[.,.3],si=.,r=.,g=,s=,h=,pk,podphis3d (99) () Average along he rajecory raj,pk,podphis,si=.,r=.,g=,s=,sar=,h= 9 7 RL977,raj,si=.,r=.,g=,s=,sar=,h= ( h) RL977,raj,si=.,r=.,g=,s=,sar=,h=,=[,]co.,podphis3d (g) Fig.. Evoluion of he iniial exponenial disribuion of iniial consan funcions on [ h, ] in he space of a ime =, ime =, ime = (e), ime = and in he space ( h) (g), ime =, ime =, ime = (f), ime = (h). (h) Fig.. Average along a single rajecory in he space of, in he space ( h). Time evoluion of a single rajecory, he projecion ono he space ( h).

8 P.J. Mikowski and W. Mikowski biological meaning can be found in he work of Mikowski (). Acknowledgmen This work was parially financed wih sae science funds as a research projec (conrac no. 3 for he years, since coninued under he conrac ). References Anosov, D.V. (93). Ergodic properies of geodesic flows on closed Riemanian manifolds of negaive curvaure, Sovie Mahemaics Doklady : 3. Arnold, V.I. (99). Mahemaical Mehods of Classical Mechanics, nd Edn., Springer-Verlag, ew York, Y, (ranslaion from Russian). Auslander, J. and Yorke, J.A. (9). Inerval maps, facors of maps and chaos. Tohoku Mahemaical Journal. II. Series 3: 77. Bass, J. (97). Saionary funcions and heir applicaions o he heory of urbulence, Journal of Mahemaical Analysis and Applicaions 7: Birkhoff, G.D. (93a). Proof of a recurrence heorem for srongly ransiive sysems, Proceedings of he aional Academy of Sciences of he Unied Saes of America 7:. Birkhoff, G.D. (93b). Proof of he ergodic heorem, Proceedings of he aional Academy of Sciences of he Unied Saes of America 7:. Birkhoff, G.D. and Koopman, B.O. (93). Recen conribuions o he ergodic heory, Mahemaics: Proceedings of he aional Academy of Sciences : 79. Bronszejn, I.., Siemiendiajew, K.A., Musiol, G. and Muhlig, H. (). Modern Compendium of Mahemaics, PW, Warsaw, (in Polish, ranslaion from German). Dawidowicz, A.L. (99). On invarian measures suppored on he compac ses II, Universiais Iagellonicae Aca Mahemaica 9:. Dawidowicz, A.L. (99). A mehod of consrucion of an invarian measure, Annales Polonici Mahemaici LVII(3):. Dawidowicz, A.L. (7). On he Avez mehod and is generalizaions, Maemayka Sosowana :, (in Polish). Dawidowicz, A.L., Haribash,. and Poskrobko, A. (7). On he invarian measure for he quasi-linear Lasoa equaion. Mahemaical Mehods in he Applied Sciences 3: Devaney, R.L. (97). An Inroducion o Chaoic Dynamical Sysems, Addison-Wesley Publishing Company, ew York, Y. Dorfman, J.R. (). Inroducion o Chaos in onequilibrium Saisical Mechanics, PW, Warsaw, (in Polish, ranslaion from English). Foias, C. (973). Saisical sudy of avier Sokes equaions II, Rendiconi del Seminario Maemaico della Universia di Padova 9: 9 3. Fomin, S.W., Kornfeld, I.P. and Sinaj, J.G. (97). Ergodic Theory, PW, Warsaw, (in Polish, ranslaion from Russian). Górnicki, J. (). Fundamenals of nonlinear ergodic heory, Wiadomosci Maemayczne 37:, (in Polish). Gurney, W.S.C., Blyhe, S.P. and isbe, R.M. (9). icholson s blowflies revisied, aure 7: 7. Kudrewicz, J. (99). Dynamics of Phase-Locked Loops, WT, Warsaw, (in Polish). Kudrewicz, J. (993, 7). Fracals and Chaos, WT,Warsaw, (in Polish). Landau, L.D., Lifszyc, J.M. (7). Mechanics, PW, Warsaw, (in Polish, ranslaion from Russian). de Larmina, P. and Thomas, Y. (93). Auomaic Conrol Linear Sysems. Vol. : Signals and Sysems, WT,Warsaw, (in Polish, ranslaion from French). Lasoa, A. (977). Ergodic problems in biology, Sociéé Mahemaique de France, Aserisque : 39. Lasoa, A. (979). Invarian measures and a linear model of urbulence, Rediconi del Seminario Maemaico della Universia di Padova : 39. Lasoa, A. (9). Sable and chaoic soluions of a firs order parial differenial equaion, onlinear Analysis Theory, Mehods & Applicaions (): 93. Lasoa, A. and Mackey, M.C. (99). Chaos, Fracals, and oise Sochasic Aspecs of Dynamics, Springer-Verlag, ew York, Y. Lasoa, A., Mackey, M.C. and Wazewska-Czyzewska, M. (9). Minimazing heraupeically induced anemia, Journal of Mahemaical Biology 3: 9. Lasoa, A. and Myjak, J. (). On a dimension of measures, Bullein of he Polish Academy of Sciences: Mahemaics (): 3. Lasoa, A. and Szarek, T. (). Dimension of measures invarian wih respec o he Wazewska parial differenial equaion, Journal of Differenial Equaions 9:. Lasoa, A. and Yorke, J.A. (973). On he exisence of invarian measures for piecewise monoonic ransformaions, Transacions of he American Mahemaical Sociey :. Lasoa, A., and Yorke, J.A. (977). On he exisence of invarian measures for ransformaions wih sricly urbulen rajecories, Bullein of he Polish Academy of Sciences: Mahemaics, Asronomy and Physics (3): Lebowiz, J.L. and Penrose, O. (973). Modern ergodic heory, Physics Today : 7. Liz, E.and Ros, G. (9). On he global aracor of delay differenial equaions wih unimodal feedback, Discree and Coninuous Dynamical Sysems ():. Mackey, M.C. (7). Advenures in Poland: Having fun and doing research wih Andrzej Lasoa, Maemayka Sosowana : 3.

9 Ergodic heory approach o chaos: Remarks and compuaional aspecs 7 Mackey, M.C. and Glass, L. (977). Oscillaions and chaos in physiological conrol sysems, Science, ew Series 97(3): 7 9. Mikowski, P.J. (). umerical analysis of exisence of invarian and ergodic measure in he model of dynamics of red blood cell s producion sysem, Proceedings of he h European Conference on Compuaional Mechanics, Paris, France, pp.. Mikowski, P.J. (). Chaos in he Ergodic Theory Approach in he Model of Disurbed Eryhropoiesis, Ph.D. hesis, AGH Universiy of Science and Technology, Cracow. Mikowski, W. (). Chaos in linear sysems, Pomiary, Auomayka, Konrola (): 3 3, (in Polish). Mikowski, P.J. and Ogorzałek, M.J. (). Ergodic properies of he model of dynamics of blood-forming sysem, 3rd Inernaional Conference on Dynamics, Vibraion and Conrol, Shanghai-Hangzhou, China, pp Myjak, J. (). Andrzej Lasoa s seleced resuls. Opuscula Mahemaica (): Myjak, J. and Rudnicki, R. (). Sabiliy versus chaos for a parial differenial equaion, Chaos Solions & Fracals : 7. adzieja, T. (99). Individual ergodic heorem from he opological poin of view, Wiadomości Maemayczne 3: 7 3, (in Polish). icholson, A.J. (9). An ouline of he dynamic of animal populaion, Ausralian Journal of Zoology : 9. O, E. (993). Chaos in Dynamic Sysems, WT, Warsaw,(in Polish, ranslaion from English). Prodi, G. (9). Teoremi Ergodici per le Equazioni della Idrodinamica, C.I.M.E., Rome. Rudnicki, R. (9a). Invarian measures for he flow of a firs order parial differenial equaion, Ergodic Theory & Dynamical Sysems : Rudnicki, R. (9b). Ergodic properies of hyperbolic sysems of parial differenial equaions, Bullein of he Polish Academy of Sciences: Mahemaics 33( ): Rudnicki, R. (9). Srong ergodic properies of a firs-order parial differenial equaion, Journal of Mahemaical Analysis and Applicaions 33:. Rudnicki, R. (). Chaos for some infinie-dimensional dynamical sysems, Mahemaical Mehods in he Applied Sciences 7: Rudnicki, R. (9). Chaoiciy of he blood cell producion sysem, Chaos: An Inerdisciplinary Journal of onlinear Science 9(3):. Shampine, L.F., Thompson, S. and Kierzenka, J. (). Solving delay differenial equaions wih dde3, available a Silva, C.E. (). Lecure on dynamical sysems, Spring School of Dynamical Sysems, Będlewo, Poland. Szlenk, W. (9). Inroducion o he Theory of Smooh Dynamical Sysems, PW, Warsaw, (in Polish). Taylor, S.R. (), Probabilisic Properies of Delay Differenial Equaions, Ph.D. hesis, Universiy of Waerloo, Onario, Canada. Tucker, W. (999). The Lorenz aracor exiss, Compes Rendus. Mahémaique. Académie des Sciences, Paris 3(I): 97. Ulam, S.M. (9), A Collecion of Mahemaical Problems,Inerscience Publishers, ew York, Y/London. Walher, H.O. (9). Homoclinic soluion and chaos in ẋ() = f(x( )), onlinear Analysis: Theory, Mehods & Applicaions (7): Ważewska-Czyżewska, M. (93). Eryhrokineics. Radioisoopic Mehods of Invesigaion and Mahemaical Approach, aional Cener for Scienific, Technical and Economic Informaion, Warsaw. Ważewska-Czyżewska, M. and Lasoa, A. (97). Mahemaical problems of blood cells dynamics sysem, Maemayka Sosowana : 3, (in Polish). Paweł J. Mikowski received his M.Sc. in elecrical engineering in from he Faculy of Elecrical Engineering, Auomaics, Compuer Science and Elecronics of he AGH Universiy in Cracow, Poland. In he obained a Ph.D. degree upon compleing docoral sudies a he same universiy. His research is now mainly focused on applicaions of mahemaics in biology and medicine, as well as chaoic dynamics and numerical analysis of dynamical sysems. Wojciech Mikowski received his M. Sc. degree in elecrical engineering in 97 a he Faculy of Elecrical Engineering of he AGH Universiy in Cracow, where he currenly works in he Deparmen of Auomaics. A he same faculy in 97 he obained a Ph.D. degree and in 9 a D.Sc. degree in he field of auomaic conrol. In 99, he received he ile of a professor of echnical sciences. In he years 99 9 he was he dean of he faculy. Since 9 he has been a member of he Commiee on Elecrical Engineering, Compuer Science and Conrol of he Polish Academy of Sciences, Cracow Branch, and since 99 a member of he Commiee on Auomaic Conrol and Roboics of he Polish Academy of Sciences. In he years he was he presiden of he Cracow Branch of he Polish Mahemaical Sociey. His main scienific ineress are auomaic conrol and roboics, conrol heory, opimal conrol, dynamic sysems, heory of elecrical circuis, numerical mehods and applicaions of mahemaics. He has published 9 books and 97 scienific papers. Received: February Revised: 9 Sepember