Dynamical P obabilis ic P Sys ems: De ini ions
and Applica ions
Da io Pescini1, Daniela Besozzi2, Claudio Zand on1, Gianca lo Mau i1
1Uni e si `a degli S udi di Milano-Bicocca, DISCo
Dipa imen o di In o ma ica, Sis emis ica e Comunicazione
Via Bicocca degli A cimboldi 8, 20126 Milano, I aly
E-mail: pescini/zand on/[email p o ec ed]
2Uni e si `a degli S udi di Milano, DICo
Dipa imen o di In o ma ica e Comunicazione
Via Comelico 39, 20135 Milano, I aly
E-mail: [email p o ec ed]
Summa y. We in oduce dynamical p obabilis ic P sys ems, a a ian whe e p obabili-
ies associa ed o he ules change du ing he e olu ion o he sys em, as a new app oach
o he analysis and simula ion o he beha io o complex sys ems. We de ine he no ions
o he analysis o he dynamics and we show some applica ions o he in es iga ion o he
p ope ies o he B ussela o (a simple scheme o he Belouso -Zabo hinskii eac ion),
he Lo ka-Vol e a sys em and he decay p ocess.
1 In oduc ion
P sys ems [4] a e a class o dis ibu ed pa allel compu ing de ices, inspi ed by
he s uc u e and he unc ioning o cells. The basic model consis s o a cell-like
memb ane s uc u e, composed by se e al compa men s whe e mul ise s o ob-
jec s e ol e acco ding o gi en ules, in a nonde e minis ic and maximally pa allel
manne . A compu a ion de ice is ob ained s a ing om an ini ial con igu a ion
and le ing he sys em e ol e. In he ollowing, we assume ha he eade is
amilia wi h he basic no ions and he e minology unde lying P sys ems. We
e e , o de ails, o [5]. Upda ed in o ma ion abou P sys ems can be ound a
h p://psys ems.disco.unimib.i /.
Many esea ch s udies a ound P sys ems concen a es on compu a ional powe
aspec s. In his pape , we p opose a new app oach o he in es iga ion and he
applica ion o P sys ems, which consis s in in e p e ing hem as ools o he de-
sc ip ion and he analysis o he dynamical beha io o complex sys ems. A simila
app oach is conside ed also in [1, 6]. As said, memb ane sys ems a e inspi ed om
he unc ioning o he cell, hence i is na u al o conside hem o modelling di e -
en cellula p ocesses and na u al li ing sys ems, wi h he inal goal o p oducing
276 D. Pescini, D. Besozzi, C. Zand on, G. Mau i
new ools and acqui ing use ul in o ma ion o he scien is s (mainly, biologis s)
wo king on he modelled sys em. Some i s s eps in his di ec ion ha e al eady
been made, see [3] o a ious applica ions.
Since we a e in e es ed in desc ibing he e olu ion o a complex sys em, and
since changes o many di e en condi ions can ha e di ec in luence on he eac ion
pa ame e s and beha io , he basic model o P sys ems is no sui able o desc ibe
hese kind o p ocesses. Fo his eason we in oduce, in Sec ion 2, dynamical p ob-
abilis ic P sys ems, whe e a p obabili y is associa ed o each ule and i changes
du ing he whole p ocess (we will alk abou e olu ion ins ead o compu a ion). In
Sec ion 3 we in oduce some no ions which will hen be used o analyze he beha -
io o such sys ems. In Sec ion 4 we show some applica ions o he B ussela o , a
simpli ied heo e ical scheme which desc ibes he Belouso -Zhabo inskii eac ion
(BZ, in sho ), he Lo ka-Vol e a and decay p ocesses. In pa icula , we show we
can simula e he beha io o chemical oscilla o eac ions. Indeed, he in e ac ion
o wo o mo e oscilla ing sys ems is o in e es o many biological p ocesses and
sys ems, as i cons i u es an impo an ac o o keep ali e an o ganism o a com-
plex sys em cons i u ed by se e al sub-componen s o di e en ypes. Finally, in
Sec ion 5 we p esen he conclusion and gi e some pe spec i e o u u e wo k.
2 Dynamical P obabilis ic P Sys ems
In his sec ion we gi e he de ini ion o a p obabilis ic P sys em whe e he p oba-
bili ies associa ed o he ules a y du ing he e olu ion o he sys em. The me hod
o e alua ing p obabili ies and he way he sys em wo ks a e explained in de ails.
Then, we ex end he de ini ion o conside amilies o P sys ems o his ype, whose
membe s di e among each o he o he choice o some pa ame e s, bu no o
he main s uc u e.
We assume he eade o be amilia wi h he basic no ions and no a ions o P
sys ems [5]. Some p e equisi es abou mul ise s a e he e ecalled.
Le Vbe an alphabe , we deno e by V∗ he se o all s ings o e V, by λ
he emp y s ing and by V+=V∗ {λ} he se o non-emp y s ings. A mul ise
o e Vis a map M:V→N, whe e M(a) is he mul iplici y o any symbol
a∈V,Nis he se o na u al numbe s. A mul ise Mo e V={a1, . . . , al}can
be explici ly ep esen ed by he s ing x=aM(a1)
1aM(a2)
2. . . aM(al)
l, o all aisuch
ha M(ai)6= 0, and by all i s possible pe mu a ions. By in e p e ing a mul ise
in he co esponding o m o a s ing x, we can deno e by |x|i s leng h and by
|x|a he numbe o occu ences o a symbol ain x. The se o symbols om V
occu ing in xis deno ed by alph(x). Mo eo e , o e e y s ing x∈V∗we can
associa e he Pa ikh ec o ΨV(x)=(|x|a1,|x|a2,...,|x|al) ( he o de o symbols
occu ing in xma e s).
De ini ion 1. Adynamical p obabilis ic P sys em (DPP, in sho ) o deg ee nis
a cons uc Π= (V, O, µ, M0, . . . , Mn−1, R0, . . . , Rn−1, I;E),whe e:
Dynamical P obabilis ic P Sys ems: De ini ions and Applica ions 277
•Vis he alphabe o he sys em, O⊆Vis he se o analyzed symbols;
•µis a memb ane s uc u e consis ing o nmemb anes labelled wi h he numbe s
0, . . . , n −1. The skin memb ane is labelled wi h 0;
•Mi,i= 0, . . . , n−1, is he mul ise o e Vini ially p esen inside memb ane i;
•Ri,i= 0, . . . , n −1, is a ini e se o e olu ion ules associa ed wi h memb ane
i. An e olu ion ule is o he o m :uk
−→ , whe e uis a mul ise o e V,
is a s ing o e V×({he e, ou } ∪ {inj|1≤j≤n−1}) and k∈Ris a
cons an associa ed o he ule;
•I⊆ {0, . . . , n −1} ∪ Eis he se o labels o he analyzed egions;
•E={VE, ME, RE}is called he en i onmen , i consis s o an alphabe VE⊆
V, a eeding mul ise MEo e VEand a ini e se o eeding ules REo he
ype :u→( , in0), o u, mul ise s o e VE.
The alphabe Oand he se Ispeci y which symbols and egions a e o peculia
impo ance in Π, namely hose elemen s whose e olu ion will be analyzed and
simula ed.
De ini ion 2. Le Πbe a DPP. We call he pa ame e s o Π he se Pconsis ing
o : (1) he mul ise s M0, . . . , Mn−1, MEini ially p esen in µand in E, (2) he
cons an s o all ules in R0, . . . , Rn−1.
No e ha he alphabe s V, O, VE, he memb ane s uc u e µ, he o m o he
ules in R0, . . . , Rn−1, REand he se Io analyzed egions do no belong o he
se o pa ame e s o Π. We call hese componen s he main s uc u e o Π. We can
now ex end De ini ion 1 and conside a amily o DPPs, whe e he main s uc u e is
equal o all membe s o he amily, while he pa ame e s can change om membe
o membe . Fo ins ance, one can choose o analyze he same DPP wi h some
di e en se ings o ini ial condi ions, such as di e en ini ial mul ise s and/o
di e en ule cons an s ( his can be use ul when no all o hem a e p e iously
known) and/o di e en eeding mul ise s.
De ini ion 3. A amily o DPPs is de ined as F={(Π, P)|Πis a DPP and
Pis he se o pa ame e s o Π}. Gi en wo elemen s (Π1,P1),(Π2,P2)∈ F, i
holds Π1=Π2 o he main s uc u e and P16=P2 o he choice o (all o some)
elemen s in P1and P2.
In he ollowing, we will alk abou he e olu ion, no compu a ion, o a DPP,
since we a e no in e es ed in gene a ing languages bu in simula ing biological
o chemical sys ems. The amily Fdesc ibes a gene al model o he biological o
chemical sys em o in e es and, o any choice o he pa ame e s, we can in es i-
ga e he e olu ion o he co esponding ixed DPP.
A ixed ini ial con igu a ion o Πdepends on he choice o P, hence i consis s
o he mul ise s ini ially p esen inside he memb ane s uc u e, he chosen ule
cons an s and he eeding mul ise , which is gi en as an inpu o he skin mem-
b ane om he en i onmen a each s ep o he e olu ion by applying he eeding
ules. Di e en s a egies in he eeding p ocess can be used: o ins ance, one can
278 D. Pescini, D. Besozzi, C. Zand on, G. Mau i
use i o keep a a cons an alue he concen a ions o chemicals in ol ed in a
ce ain eac ion (see Sec ion 4.1 o an applica ion o he BZ), o o inc ease he
concen a ions o subs ances mimicking he biological anspo om he ex a-
cellula space. We assume ha , as long as he sys em e ol es, he en i onmen
con ains as many symbols as hey a e needed o con inuously eed he sys em.
A each s ep o he e olu ion, all applicable ules a e simul aneously applied
and all occu ences o he le -hand sides o he ules a e consumed, hence he
pa allelism is maximal a bo h le els o objec s and o ules. Fo simplici y, in
his pape we assume ha he sys em e ol es acco ding o a uni e sal clock, ha
is all memb anes and he applica ion o all ules a e synch onized. In he u u e,
his condi ion will be ex ended o conside ing also non-synch onized e olu ions.
The applied ules a e chosen acco ding o he p obabili y alues dynamically as-
signed o hem; he ules wi h he highes no malized p obabili y alue will be
mo e equen ly ossed. In simula ions, he ossing p ocess is ob ained by means
o a andom numbe gene a o , as desc ibed below. I some ules compe e o ob-
jec s and ha e he same p obabili y alues, hen objec s a e nonde e minis ically
assigned o hem.
The p obabili y associa ed o each ule in any se Ri,i= 0, . . . , n −1, is a
unc ion o i s cons an and o he cu en mul ise occu ing in memb ane i, and
i is e alua ed as ollows. Le V={a1, . . . , al},Mibe he mul ise inside memb ane
i, :uk
−→ a ule in Ri; le alph(u) = {a1, . . . , as}and u=aα1
1. . . aαs
s. To ob ain
he ac ual no malized p obabili y pio applying wi h espec o all o he ules
ha a e applicable in memb ane ia he same s ep, we need o e alua e he non-
no malized p obabili y epi( ) o , which depends on he cons an associa ed o
and on he le -hand side o , namely:
epi( ) = k·
s
Y
h=1
Mi(ah)!
αh!(Mi(ah)−αh)! =
=k·
s
Y
h=1
Mi(ah)(Mi(ah)−1) . . . (Mi(ah)−αh+ 1)
αh!(1)
ha is epi( ) is dynamically de ined, acco ding o he cu en mul ise occu ing
inside memb ane i, since we choose αhcopies o symbols ahamong all i s Mi(ah)
copies cu en ly a ailable in he memb ane i sel . I Ri={ 1, . . . , m}, he no -
malized p obabili y o any ule jis
pi( j) = epi( j)
Pm
j=1 epi( j).(2)
In he simula ions, he pa allel applica ion o he ules is done by spli ing one
pa allel s ep in o se e al sequen ial sub-s eps. I is possible o sepa a e each single
pa allel s ep in o wo s ages, exploi ing he ac ha he p obabili y dis ibu ion
and he applicabili y o he ules a e unc ions only o he le -hand side o he ules
and hei cons an s. In he i s s age objec s a e assigned o ules by means o a
Dynamical P obabilis ic P Sys ems: De ini ions and Applica ions 279
andom numbe gene a o , while in he second one he mul ise is upda ed using
a s o ed ace o he ules p e iously ossed. I should be poin ed ou ha , du ing
he i s s age, he p obabili y dis ibu ion o he ules has o be kep cons an ,
o he wise he applica ion o he ules would become sequen ial.
Rema k 1. A di e en p obabili y dis ibu ion o e ules could be ob ained by using
he classical a e law o Chemis y, hough he app oach used in equa ion (1) is
mo e accu a e om he combina o ial poin o iew [2]. I is well known om
Chemis y ha he a e o a eac ion ρa any ime is go e ned by he concen a ion
o he chemicals in ol ed, namely
ρ=k·Y
j∈J
[Aj]σj,
whe e he index j a ies o e all chemicals in ol ed in he eac ion, [Aj] ep esen s
he concen a ion o each chemical Ajand kis called he a e cons an . The alue
σjis always expe imen ally de e mined bu in he elemen a y eac ions, whe e i
is assumed o be he s oichiome ic coe icien .
Indeed, a high concen a ions (mul iplici ies) he wo app oaches a e undis in-
guishable, bu a lowe ones ou choice is p e e able since i accoun s o he exac
numbe o all possible uples o e ol ing objec s.
3 Analysis o he Dynamics in DPP
In his sec ion we in oduce some no ions ha will be used o he analysis o he
beha io o a DPP. The inal goal is o in oduce an app op ia e de ini ion o he
phase space. Usually, he e olu ion o a physical sys em is comple ely de e mined
by means o he mo ion equa ions, a se o di e en ial equa ions in e ed by he
sys em p ope ies. In he case o P sys ems his ole should be accomplished by
he e olu ion ules, which c ea e a one- o-one mapping be ween he applica ion o
each ule and he ela i e displacemen o he sys em in he phase space.
Fi s o all, o keep ace o he sys em e olu ion we ex end he de ini ion o
he alphabe V={a1, . . . , al}o Πby in oducing he pa ame e ime, ha is we
de ine he space e
V:= V×N=V× { ime}.
De ini ion 4. Le M={aα1
1, . . . , aαl
l}be a mul ise o e V, whe e αi≥0 o
all h= 1, . . . , l. We call a -mul ise he s uc u e M={aα1
1, . . . , aαl
l, }o e he
space e
V.
By abuse o no a ion, we will deno e bo h he mul ise o e Vand he -mul ise
in e
Vwi h he same symbol M, being i clea when one conside s also he ime
componen o no . To ep esen a -mul ise in he space e
Vwe de ine i s posi ion
ela i ely o he -mul ise O={0, . . . , 0}o e
V( he i s lcomponen s o Oa e
he null mul iplici ies o he symbols om V). We need also o ex end he no ion

280 D. Pescini, D. Besozzi, C. Zand on, G. Mau i
o Pa ikh ec o o he space e
Vas Ψe
V(M) = (α1, . . . , αl, ). This is necessa y i
we wan o dis inguish among wo mul ise s ha ing he same o al numbe s o
symbols bu di e en mul iplici ies o (a leas ) one symbol om V.
De ini ion 5. The posi ion o a -mul ise Mo e e
Vis he ec o −→
M=Ψe
V(M).
The ec o −→
O=Ψe
V(O) is called he o igin o e
V.
F om De ini ion 5 i ollows ha he posi ions o -mul ise s −→
Oand −→
Ma e
ec o s in he space Nl×N. The nex s ep is o in oduce a scala p oduc in
Nl, o na u ally de ine he no ion o dis ance be ween -mul ise s, hus gi ing he
s uc u e o an euclidian space o Nl.
De ini ion 6. Le −→
Mi,−→
Mjbe wo posi ions in Nl×N. The dis ance be ween
−→
Mi,−→
Mjis a unc ion d:Nl+1 ×Nl+1 −→ R+de ined as d2(−→
Mi,−→
Mj) = Pm
k=1(αi,k −
αj,k)2.
No e ha he wo posi ions −→
Mi,−→
Mjin De ini ion 6 need no o be necessa ily
one he e olu ion o he o he ( ha is, he mul ise inside he same memb ane
aken in o di e en ime s eps). In ac , gi en a amily Fo DPP and wo posi ions
−→
Mi,−→
Mj, he ollowing cases may hold:
1. −→
Mi,−→
Mjoccu in dis inc ime s eps, in he same memb ane o he same DPP
wi h equal se ing P;
2. −→
Mi,−→
Mjoccu in dis inc o equal ime s eps, in di e en memb anes o he
same DPP wi h equal se ing P;
3. −→
Mi,−→
Mjoccu in dis inc o equal ime s eps, in he same memb ane o he
same DPP wi h di e en se ings P1,P2;
4. −→
Mi,−→
Mjoccu in dis inc o equal ime s eps, in di e en memb anes o he
same DPP wi h di e en se ings P1,P2.
Tha is, we migh be in e es ed in looking a he mul ise occu ing inside a mem-
b ane du ing i s e olu ion, o compa ing wo mul ise s o di e en memb anes o
he same DPP (in equal o di e en ime s eps), o else wo mul ise s inside he
same (o e en a di e en ) memb ane bu analyzed in wo di e en e olu ions o
he amily o he DPP. In each o he ou cases, he dis ance gi es in o ma ion
abou “how a ” he s a es in he wo ajec o ies a e ( ha is, he -mul ise s in
he wo e olu ions).
In pa icula , gi en any couple o posi ions −→
Mi,−→
Mjo he same DPP ( o he
same o di e en se o ixed pa ame e s P), we can say ha hey a e simul aneous
i hey exis a he same ime s ep. This concep can be use ul mainly when one
conside s a memb ane s uc u e wi h deg ee n≥1, whe e many mul ise s a e
co-e ol ing.
De ini ion 7. Le −→
Mi,−→
Mjbe wo posi ions in Nl+1. The displacemen be ween
−→
Mi,−→
Mjis a unc ion −→
u:Nl+1 ×Nl+1 −→ Zlde ined as −→
u(−→
Mi,−→
Mj)=(αi,1−
αj,1, . . . , αi,l −αj,l).
Dynamical P obabilis ic P Sys ems: De ini ions and Applica ions 281
No e ha he displacemen can be ei he a posi i e o nega i e alue, and
i ells how he sys em “mo es”; in de ails, i ells how he mul iplici ies in he
posi ions −→
Mjdi e om hose in −→
Mi. Hence, i gi es mo e in o ma ion han he
dis ance, since i also conside s he di ec ion o he a ia ion. Indeed, i is also
possible o cons uc he e so bu:Nl+1 ×Nl+1 −→ Rlo he displacemen which
only gi es he in o ma ion abou he di ec ion o −→
u:
bu(−→
Mi,−→
Mj) = Ãαi,1−αj,1
d(−→
Mi,−→
Mj), . . . , αi,l −αj,l
d(−→
Mi,−→
Mj)!.
No e ha −→
u=bu·d, o cons uc ion.
The las s ep be o e a i ing o he de ini ion o he phase space consis s in
de ining he eloci y, which ca ies on he in o ma ion abou he ime he displace-
men be ween wo -mul ise s (in he same DPP, wi h equal ini ial se ings) needs
o ake place. Tha is, i ells how as he e olu ion om one s a e o he DPP o
he o he is.
De ini ion 8. Le −→
Mi,−→
Mjbe posi ions occu ing inside he same memb ane o a
DPP ( o a ixed choice o he pa ame e s) in dis inc ime s eps. The eloci y is
a unc ion −→
:Nl+1 ×Nl+1 −→ Rlde ined as
−→
(−→
Mi,−→
Mj) = µαi,1−αj,1
i− j
, . . . , αi,l −αj,l
i− j¶.
I should be poin ed ou he e ha , ac ually, his is he de ini ion o he a e age
eloci y, which becomes he “ins an aneous” eloci y when i− j= 1, which is he
minimal ime inc emen allowed in a disc e e- ime sys em, as a DPP is. No e ha i
−→
Mjis he posi ion e ol ed om −→
Miin he same memb ane, hen he ins an aneous
eloci y gi es he a ia ion o ha mul ise in a single ime s ep.
We a e now eady o de ine he phase space o a DPP, which is cons uc ed
as he ca esian p oduc o he phase spaces o all memb anes in he DPP. Le
−→
Mi= (α1, . . . , αl, ) be he posi ion o he -mul ise inside memb ane ia ime
, and le −→
(−→
Mi,−→
Mi)=( 1, . . . , l) be i s ins an aneous eloci y ( ha is, he
a ia ion o −→
Mia ime wi h espec o ime −1, o any ≥1).
De ini ion 9. We call a phase poin o −→
Mi he ec o −→
ϕ i= (α1, . . . , αl, 1,
. . . , l)∈Nl×Rl, o any ixed ∈N.
The phase poin ep esen s he s a e o memb ane ia any gi en ime . The
e olu ion o he mul ise in memb ane ican be desc ibed by he phase cu e, which
is a unc ion −→
ϕi:N−→ Nl×Rlsuch ha −→
ϕi( ) = −→
ϕ i.
The space Nl×Rlis he se o all he poin s −→
ϕ ico esponding o an e olu ion
o he mul ise inside any memb ane.
De ini ion 10. Le Πbe a DPP o deg ee n, o some n≥1. The space Φi=Nl×
Rlis called he phase space o he memb ane i. The space ΦΠ=Φ0×· · ·×Φn−1×ΦE
is called he phase space o he DPP.
282 D. Pescini, D. Besozzi, C. Zand on, G. Mau i
Hence, he phase space o a DPP desc ibes he e olu ion o he whole sys em,
wi h espec o bo h he change o all mul ise s and he passing o ime. Ac ually,
in analyzing he beha io o a gi en DPP, we will be in e es ed in conside ing only
he phase space es ic ed o he egions speci ied in he se I(see De ini ion 1).
Simila ly, only he e olu ion o symbols om Owill be analyzed o he mul ise s
p esen in he egions in I.
4 Applica ions
In his sec ion we p esen some applica ions o he DPP model o known p oblems
and he ela i e esul s ob ained om he co esponding simula ions.
4.1 The Belouso -Zhabo inskii Reac ion
The BZ chemical eac ion is conside ed he p o o ype oscilla o and exhibi s an
ex ao dina y a ie y o empo al and spa ial phenomena. I s oscilla ing beha -
io is one o he mos widely s udied, bo h heo e ically and expe imen ally, hus
making his eac ion a sui able wo kbench o he capabili ies o DPP. I s basic
mechanism consis s in he oxida ion o malonic acid, in acid medium, by b oma e
ions and ca alyzed by ce ium, which has wo s a es. The sus ained pe iodic oscil-
la ions a e obse ed in he ce ium ions. The B ussela o is a simpli ied heo e ical
scheme in oduced in [7] o explain he nonlinea oscilla ing beha io , and a e
ha was ca e ully s udied in, e.g., [8]. Despi e he ac ha i is physically un e-
alis ic, as i in ol es a imolecula s a e, i is ecognized o be he skele on o
he explana ion o he oscilla ing beha io in chemical eac ions. Mo eo e , i has
a e y simple desc ip ion: Ak1
−→ X, B +Xk2
−→ Y+D, 2X+Yk3
−→ 3X, X k4
−→ E.
In his sec ion we desc ibe he B ussela o in e ms o DPP and we show
he analysis and some esul s ob ained om he simula ions. Indeed, in o de
o desc ibe a chemical o a biological sys em e ol ing o e ime, a kind o ule
able o eac o he a ia ion o occu ences o symbols ( ha is, concen a ions o
subs ances) is needed. Fo his pu pose, we belie e ha he dynamical p obabilis ic
ules a e eally sui able, so we conside he amily o DPP ha , acco ding o
De ini ion 1, a e gi en by
ΠBZ = ({A, B, X, Y },{X, Y },[0]0, M0, R0,0; EBZ),
whe e M0={Am1, Bm2, Xm3, Y m4}and R0={ 1:Ak1
−→ X, 2:BX k2
−→ Y,
3:XXY k3
−→ XXX, 4:Xk4
−→ λ}, o some k1, . . . , k4∈R. No e ha , wi h
espec o he o iginal equa ions in he B ussela o , we choose no o conside he
chemicals Dand Esince hey a e no ele an o he sys em e olu ion. The en i-
onmen EBZ is gi en by he alphabe {A,B}, he mul ise MEBZ ={An1, Bn2},
o some n1, n2∈Nand he eeding ules REBZ ={ 5:A−→ (A, in0), 6:
B−→ (B, in0)}. Acco ding o De ini ion 2, he se o pa ame e s o ΠBZ is
Dynamical P obabilis ic P Sys ems: De ini ions and Applica ions 283
PBZ ={m1, . . . , m4, k1, . . . , k4, n1, n2}. A amily FBZ can be gi en by conside ing
di e en alues o he elemen s in PBZ.
The simula ions based on he DPP app oach ha e shown all he dynamical be-
ha io s which cha ac e ize he con inuously s i ed BZ (see o example [1, 2, 8]),
bu he e due o space limi s we only p esen he quasi pe iodic oscilla ions (in
Figu e 1, o Pqp
BZ ={100,100,1000,2000,50,0.5,5·10−5,5,100,100}) and he a -
ac o (in Figu e 2, o Pa
BZ ={100,100,1000,2000,1,1,1,1,100,100}). A ading
ansi ion om one o he o he is possible by uning he pa ame e s in PBZ . Since
in he li e a u e on he B ussela o he phase plane has been widely iden i ied wi h
he X-Yplane, ou a en ion is ocused on he dynamic o hese symbols. A i s
cha ac e iza ion o he sys em dynamic can be ob ained by looking di ec ly o he
empo al e olu ion o he wo a iables: Fig.1.(a) and Fig.2.(a) allow o disc imi-
na e he quasi pe iodic oscilla ion o he i s case om he a ac ed dynamic o
he second one. Fig.1.(b) and Fig.2.(b) show he phase space o memb ane 0: in
he i s case we ob ain a limi cycle, in he second case only he ini ial mul ise
(poin a igh -up co ne ) and he a ac o (poin a le -bo om co ne ) can be
displayed. Fig.1.(c) and Fig.2.(c) show he e olu ion o mul iplici ies o Xand Y;
he p ojec ion on X−Yplane o hese pic u es ob iously co espond o Fig.1.(b),
Fig.2.(b), espec i ely. Finally, Fig.1.(d) and Fig.2.(d) show he spec a: in he
i s case, he spec um shows he highes peak, co esponding o he p incipal
oscilla ion equency, and some o he ha monics, plus he s ochas ic con ibu e
which is sp ead all o e he o he equencies; in he second case (whe e he Yaxis
is in loga i hmic scale), he spec um co esponds o a δo Di ac cen e ed in he
0 equency ( he heigh o δis equal o he mean alue o he mul iplici ies o X
and Y), since his is he Fou ie ans o m o a cons an (in ime) signal.
Rema k 2. To make clea he de ini ions o Sec ion 3, we gi e some ex-
amples by ex ac ing h ee -mul ise s om he simula ed e olu ion o
(ΠBZ ,Pqp
BZ ). Chosen he -mul ise s M39 ={100,100,1921,1029,39},M40 =
{100,100,2701,262,40},M53 ={100,100,109,1055,53}, hei posi ions a e
M39 = (100,100,1921,1029,39),M40 = (100,100,2701,262,40),M53 =
(100,100,109,1055,53). The dis ance be ween M53 and M39 is d(M53,M39) =
(0 + 0 + (−1812)2+ 262)1/2≈1812.19, while he displacemen is u(M53,M39) =
(0,0,−1812,26). The e so associa ed o his displacemen is ˆu(M53,M39) =
(0,0,−1812/1812.19,26/1812.19) ≈(0,0,−0.99,0.0014), which says ha he p e-
dominan di ec ion o he mo ion is along he Xaxes ( ha is, he highes a ia ion
occu s o he mul iplici ies o he symbol X). The a e age eloci y (M53,M39) =
(0,0,−1812/14,26/14) ≈(0,0,−129.43,1.86) is qui e di e en om he ins an a-
neous one, which is (M40,M39) = (0,0,780,−767).
4.2 Decay
The adioac i e decay p ocess is such ha he popula ion o a adioac i e iso ope
a ies in ime wi h a ixed a e and usually i is desc ibed by he o dina y di e -
en ial equa ion dX
d =cX. This p ocess is e y simple and can be desc ibed by he