Synthesizing Chaotic Maps with Prescribed Invariant Densities

Syn hesizing Chao ic Maps wi h P esc ibed In a ian Densi ies Alan Roge s a,∗Robe Sho en bDaniel M. He e nan c,1 aDepa men o Elec onic Enginee ing, NUI Maynoo h, Co. Kilda e, I eland bHamil on Ins i u e, NUI Maynoo h, Co. Kilda e, I eland cDepa men o Ma hema ical Physics, NUI Maynoo h, Co. Kilda e, I eland Abs ac The In e se F obenius-Pe on p oblem (IFPP) conce ns he c ea ion o disc e e chao ic mappings wi h a bi a y in a ian densi ies. In his no e, we p esen a new and elegan solu ion o he IFPP, based on posi i e ma ix heo y. Ou me hod allows chao ic maps wi h a bi a y piecewise-cons an in a ian densi ies, and wi h a bi a y mixing p ope ies, o be syn hesized. Key wo ds: Chaos, Chao ic Maps, Chaos Con ol, In e se F obenius-Pe on P oblem PACS: 05.45.-a, 02.50.- 1 In oduc ion The syn hesis, o cus om-design o chao ic maps, is a na u al ex ension o he esea ch ca ied ou on nonlinea dynamical sys ems o e he pas hi y yea s. Ulam and Von Neumann had e en s udied he logis ic map in he 1940s [1]. No wi hs anding Von Neumann’s amous poin abou using de e minis ic p ocesses o gene a e andomness, syn he ic chao ic maps do ha e many po- en ial applica ions, especially in ha dwa e-based andom-numbe gene a o s, and digi al noise gene a o s [2–4]. They may also be used o a i ically ec ea e physical da a om eal-wo ld sys ems [5]. In his pape , we p esen an elegan ∗Co esponding Au ho . Email add ess: [email p o ec ed] (Alan Roge s). 1Also a : School o Theo e ical Physics, Dublin Ins i u e o Ad anced S udies, Dublin 4, I eland P ep in submi ed o Physics Le e s A 6 Augus 2004 way o c ea ing designe chao ic maps wi h a bi a y in a ian densi ies. The syn hesis me hod p esen ed is based on he heo y o posi i e ma ices, and o igina es in wo k on synch onised communica ion ne wo ks. When a chao ic map is i e a ed, di e en ini ial condi ions exponen ially di e ge leading o comple ely di e en long- un beha iou . Howe e , when looked a s a is ically, many chao ic maps possess a single physically ele an in a ian densi y, which emains s able when andom noise is added o he p o- cess [6]. Gi en an a bi a y ini ial condi ion, he in a ian densi y desc ibes whe e i e a es end up on a e age. Fo simple maps such as he en map, o he Be noulli (2xmod 1) map, i e a es ha e an equal p obabili y o land- ing anywhe e in he s a e-space: he na u al in a ian densi y ρis cons an , and equals one [7]. Many maps, howe e , do no possess a simple in a ian measu e. The F obenius-Pe on ope a o Pτis an ope a o on he space o p obabili y densi y unc ions [8]. The in a ian densi y is a ixed poin o he F obenius- Pe on equa ion: Pτ (x) = d dxZ dλ τ−1[a,x] (1) While i is ela i ely s aigh o wa d o calcula e he in a ian densi y ρo piecewise a ine maps, mos con inuous maps do no possess a closed- o m so- lu ion o he F obenius-Pe on equa ion (especially whe e he in a ian densi y is a ac al). Clea ly, i i is e y di icul o analyse a map o ind i s in a ian densi y, hen he syn hesis p oblem - gene a ing a map which possesses a desi ed in a ian densi y - mus seem qui e a daun ing p ospec . Howe e , Ulam conjec u ed ha he F obenius-Pe on ope a o could be app oxima ed by he ac ion o a Ma ko map ac ing on a pa i ion o he in e al conce ned [9]. The Ulam conjec u e was p o ed in 1976 by Li [10]. The Ulam ma ix is a column s ochas ic ma ix which gi es he p obabili y o mo ing om any pa icula in e al in he pa i ion o any o he in e al. The p inciple eigen ec o o he Ulam ma ix is he in a ian densi y o he map. Thus i a Ma ko ma ix can be syn hesized o a pa icula eigen ec o , hen we ha e a solu ion o he In e se F obenius-Pe on p oblem (IFPP). The IFPP has been ackled and sol ed by a ious g oups (see [11] and e e ences he ein). The me hod mos ele an o his wo k is he solu ion o G´o a and Boya sky [12], [5]. They show how o gene a e a 3-band ans o ma ion om any gi en in a ian densi y. Howe e , ou me hod is mo e di ec in ha bo h he eigen ec o and he Ulam ansi ion ma ix a e pa ame e ized. No wo k is equi ed o gene a e he ma ix - he p ocedu e is comple ely mechanical. Ou me hod also gi es comple e con ol o e he mixing p ope ies o he map, independen o he in a ian densi y. 2 In he ollowing sec ion, we ou line he syn hesis me hod. We hen gi e a simple example o he applica ion o he me hod o syn hesize a map wi h a p esc ibed in a ian densi y. Some p ope ies o he ansi ion ma ix a e also gi en, ollowed by conclusions. The appendix gi es a li le ex a backg ound on he o igins o he ma ix used. 2 Syn hesis Me hod The ollowing ma ix a ises na u ally in he dynamical analysis o synch onised communica ion ne wo ks based on TCP (T ansmission Con ol P o ocol). Fu - he p ope ies o he ma ix a e gi en in he appendix, and he in e es ed eade should e e o [13], o [14], o he o igins o his ma ix. A=           β10· · · 0 0β20 0 . . . 0 ...0 0 0 · · · βn           +1 Pn i=1 αi           α1 α2 . . . αn           µ1−β11−β2· · · 1−βn¶(2) The ma ix Ais a column s ochas ic ma ix, and is s ic ly posi i e when αi≥0 and 0 < βi<1∀i∈ {1,· · · , n}, and so Acan ep esen a Ma ko p ocess. F om he heo y o posi i e ma ices, i is well-known ha he ma ix Ahas a leading eigen alue ρ(A) = 1, and a single eigen ec o in he posi i e o han , called he Pe on eigen ec o [15]. The Pe on eigen ec o , xpo he Ama ix has he ollowing o m: xT p=µα1 1−β1, . . . , αn 1−βn¶(3) Clea ly, we can con ol he dominan eigen ec o o he ma ix h ough choice o he αiand βi. Fo ou pu poses, he ma ix Ais he Ulam ma ix, and he Pe on eigen ec o ep esen s he in a ian densi y o he p ocess go e ned by A. The Ulam ma ix may be ep esen ed as a one-dimensional chao ic map i we le each en y o he ma ix deno e he ansi ion p obabili y om one in e al o ano he . Mo e o mally, pa i ion he uni in e al in o Nequal sub- in e als, {I1, . . . , IN}. Le en y aji o he Ama ix deno e he p obabili y o a ansi ion om in e al Ii o in e al Ij, deno ed pij. To cons uc he map, place a line segmen o slope ±1/pij in he squa e de ined by he in e als Ii, Ij, o each en y o he ma ix. I is con enien o s a a he poin (0,0) and place he line segmen s end o end, al hough he e a e many possibili ies, as illus a ed in Figu e 1. The map in Figu e 1 is one possible implemen a ion o he ollowing ansi ion ma ix: 3 01 1 x n x n+1 I 1 I 2 I 3 I 4 I 1 I 2 I 3 I 4 p 42 =1 p 11 =1/4 p 12 =1/4 p 13 =1/2 p 14 =0 Fig. 1. A possible one-dimensional map co esponding o a 4 ×4 ansi ion ma ix A=           1/4 1/3 1/4 0 1/4 1/6 1/4 1 1/2 1/4 1/4 0 0 1/4 1/4 0           (4) We can w i e he ma ix Ain he ollowing, mo e compac , o m, whe e we ha e no malized he alphas: Pαi= 1. A=           β1+α1(1 −β1)α1(1 −β2)· · · α1(1 −βn) α2(1 −β1)β2+α2(1 −β2) . . .... αn(1 −β1)βn+αn(1 −βn)           (5) I is possible o choose he αiand βiso ha he eigen ec o ep esen s any desi ed in a ian densi y. Fo simplici y, we can se all o he βi= 0.1 say, and hen de e mine he αi o he desi ed densi y. Once he αiand βia e de e mined, he ma ix Ais ully de e mined, and can hen be implemen ed as a map. 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X n X n+1 Fig. 2. Map co esponding o he A ma ix in he example (6) 3 Example As an example, we will syn hesize a chao ic map whose in a ian densi y has a iangula shape: Fo simplici y, we pa i ion he uni in e al in o i e equal segmen s. Le ρdesi ed = [1,3,5,3,1]. We hen de e mine he αiwi h βi= 0.1. This gi es us he ollowing column-s ochas ic ansi ion ma ix: A=               0.1692 0.0692 0.0692 0.0692 0.0692 0.2077 0.3077 0.2077 0.2077 0.2077 0.3462 0.4462 0.3462 0.3462 0.3462 0.2077 0.2077 0.2077 0.3077 0.2077 0.0692 0.0692 0.0692 0.0692 0.1692               (6) A possible 1-D chao ic map co esponding o his ansi ion ma ix is shown in Figu e 2, and he in a ian densi y o he map a e 30000 i e a ions, ρac ual is shown in Figu e 3. The densi y has been scaled such ha he i s en- y o ρac ual = 1 o allow eady compa ison wi h he desi ed in a ian den- si y. I can be seen ha ρac ual is e y close o he desi ed in a ian densi y, ρ= [1,3,5,3,1]. Figu e 4 shows a ypical ou pu om he map. (MATLAB code o implemen he me hod is eely a ailable a h p://www.eeng.may.ie /˜a oge s/i pp.) 5 12345 0 1 2 3 4 5 6 In e al Numbe Rela i e numbe o i e a ions pe in e al Fig. 3. In a ian densi y o map in Figu e 2 a e 30000 i e a ions 0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I e a ion Numbe X n Fig. 4. Chao ic ime-se ies p oduced by he map in Figu e 2 4 Fu he P ope ies o he T ansi ion Ma ix The ma ix Aa ises na u ally in he s udy o synch onised communica ion ne wo ks, and is used o model he dynamics o ne wo ks employing TCP conges ion con ol [13] [14]. I has a numbe o in e es ing p ope ies, which we men ion he e. (1) The ma ix Ais s ic ly posi i e as de ined in sec ion 2, o s ic ly non- nega i e i we allow αi= 0. As such, i possesses a dominan eigen alue called he Pe on eigen alue λpo geome ic and algeb aic mul iplici y 1. The co esponding Pe on eigen ec o xT p=µα1 1−β1, . . . , αn 1−βn¶ ep esen s a ixed poin o he linea sys em W(k+ 1) = AW(k). Fo any o he eigen alue λi6=λp, we ha e |λi|< λp. (2) The a e o con e gence o he ixed poin is bounded by he second 6 la ges eigen alue o A. Fo ou applica ion, he second-la ges eigen- alue de e mines how quickly we ge o he desi ed in a ian densi y. The second-la ges eigen alue hus con ols he mixing p ope ies o he map. As we can make he second-la ges eigne alue as la ge o as small as we like, we ha e comple e con ol o e he mixing p ope ies, indepen- den o he in a ian densi y. I can be shown ha , apa om he Pe on eigen alue, all he he eigen alues o Alie wi hin he in e al [β1, βn]. (3) I he βia e dis inc , and a e o de ed such ha β1< β2<· · · < βn, hen i can be shown ha he eigen alues o Aa e in e laced wi h he βi hus: β1< λ1< β2< λ2<···< βn< λn=λp= 1.(7) (4) The Lyapuno exponen o he map esul ing om he ma ix Acan be shown o be Λ≃α1α2. . . αnln 1 α1α2. . . αn +α2 1ln 1 α1 +· · · +α2 nln 1 αn (8) in he limi βi→0. We ind his o be a e y good app oxima ion o small βi. 5 Conclusions F om an enginee ing iewpoin , cus om-design o chao ic maps is absolu ely necessa y o possible u u e applica ions. Maps mus be ailo ed o i he applica ion. In his pape , we ha e p esen ed a di ec way o c ea ing designe chao ic maps wi h a bi a y in a ian densi ies. The syn hesis me hod p e- sen ed is based on he heo y o posi i e ma ices, and o igina ed in wo k on synch onised communica ion ne wo ks. I s mos use ul p ope y is i s s aigh - o wa d implemen a ion. Mo eo e , ou me hod allows comple e con ol o he mixing p ope ies o he map, independen o he in a ian densi y. Acknowledgemen s We hank he e e ees o hei use ul commen s. This wo k was suppo ed by he I ish science and echnology agency, En e p ise I eland, unde esea ch g an No. SC-00-86 7 A T ansi ion ma ix - supplemen a y ma e ial The special o m o he ma ix Aa ises in he con ex o communica ion ne - wo ks, and i s o igins a e desc ibed ho oughly in [13] and [14]. Essen ially, he ma ix comes abou om a model o synch onised in o ma ion sou ces op- e a ing an Addi i e-Inc ease Mul iplica i e-Dec ease (AIMD) conges ion con- ol algo i hm. Ne wo ks o such de ices in he p esence o a bo leneck bu e may be modelled as a posi i e linea sys em, whence we ge he Ama ix. Mo e ecen esul s will be ound in [16]. We p esen some u he p ope ies o he ma ix Abelow, which a e p o ed in he a o emen ioned pape s. Lemma 1 Fo a posi i e column s ochas ic ma ix Ao he ollowing o m: diag(β1, . . . , βn)+(1/Pn i=1 αi)[α1, . . . , αn]T[1−β1, . . . , 1−βn], and βi∈(0,1), αi>0, hen he dominan (Pe on) eigen ec o o A, co esponding o he sole uni y eigen alue is gi en by xp=θ[α1 1−β1 ,..., αn 1−βn ], θ ∈R(A.1) Theo em 1 Conside he ma ix Ain Lemma 1. The ollowing s a emen s a e ue: (1) Excep o he Pe on eigen alue, all o he eigen alues lie in he in e al [β1, βn]. (2) I all he βs a e dis inc , hen β1< λ1< β2< . . . < βn< λn= 1 Ou line P oo : The ma ix Ais diagonally simila o a ma ix o he o m G−1(D+αβT)Gwhe e D=diag(β1, . . . , βn), and Gis a diagonal ma ix (simple calcula ion). Using Lemma 1 oge he wi h his esul , and s anda d esul s on he symme ic eigen alue p oblem (see o ins ance Theo em 8.6.2 in [17], o [18]), he p oo o (1) and (2) ollow di ec ly. Re e ences [1] S. Ulam, J. on Neumann, On combina ions o s ochas ic and de e minis ic p ocesses, Bulle in o he Ame ican Ma hema ical Socie y 53 (1947) 1120. [2] R. L. Kau z, Using chaos o gene a e whi e noise, Jou nal o Applied Physics 86 (10) (1999) 5794–5800. [3] L. Koca e , Chaos-based c yp og aphy: A b ie o e iew, IEEE Ci cui s and Sys ems Magazine 1 (3) (2001) 6–21. [4] M. Delgado-Res i u o, A. Rod iguez-Vazquez, In eg a ed chaos gene a o s, P oceedings o he IEEE 90 (5) (2002) 747–767. 8 [5] A. Boya sky, P. G´o a, Chao ic maps de i ed om ajec o y da a, Chaos 12 (1) (2002) 42–48. [6] H. G. Schus e , De e minis ic Chaos, VCH, 1989. [7] E. O , Chaos in Dynamical Sys ems, 2nd Edi ion, Camb idge Uni e si y P ess, 2002. [8] A. Laso a, M. Mackey, Chaos, F ac als, and Noise, Vol. 97 o Applied Ma hema ical Sciences, Sp inge -Ve lag, 1994. [9] S. M. Ulam, A Collec ion o Ma hema ical P oblems, Vol. 8 o In e science T ac s in Pu e and Applied Ma h, In e science, 1960. [10] T. Li, Fini e app oxima ion o he obenius-pe on ope a o : A solu ion o ulam’s conjec u e, Jou nal o App oxima ion Theo y 17 (1976) 177–186. [11] E. M. Boll , Con olling chaos and he in e se obenius-pe on p oblem: Global s abiliza ion o a bi a y in a ian measu es, In e na ional Jou nal o Bi u ca ion and Chaos 10 (5) (2000) 1033–1050. [12] P. G´o a, A. Boya sky, A ma ix solu ion o he in e se obenius-pe on p oblem, P oceedings o he Ame ican Ma hema ical Socie y 118 (2) (1993) 409–414. [13] R. Sho en, D. Lei h, J. Foy, R. Kildu , Analysis and design o synch onised communica ion ne wo ks, in: P oceedings o 12 h Yale Wo kshop on Adap i e and Lea ning Sys ems, 2003. [14] A. Be man, R. Sho en, D. Lei h, Posi i e ma ices associa ed wi h synch onised communica ion ne wo ks, Linea Algeb a and I s Applica ions (Accep ed). [15] D. G. Luenbe ge , In oduc ion o Dynamic Sys ems, Wiley, 1979. [16] D. Lei h, e al., S ochas ic equilib ia o aimd communica ion ne wo ks, submi ed o SIAM Jou nal on Ma ix Analysis and Applica ions. [17] G. Golub, C. an Loan, Ma ix Compu a ions, Johns Hopkins Uni e si y P ess, 1996. [18] R. Ho n, C. Johnson, Ma ix Analysis, Camb idge Uni e si y P ess, 1985. 9