THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
Chaos Enhanced Di e en ial E olu ion in he Task
o E olu iona y Con ol o Disc e e Chao ic
Lozi Map
Roman SENKERIK1,2, I an ZELINKA1,3, Michal PLUHACEK2
1Modeling E olu iona y Algo i hms Simula ion and A i icial In elligence, Facul y o Elec ical and Elec onics
Enginee ing, Ton Duc Thang Uni e si y, 19 Nguyen Huu Tho S ee , 700000 Ho Chi Minh Ci y, Vie nam
2Depa men o In o ma ics and A i icial In elligence, Facul y o Applied In o ma ics, Tomas Ba a
Uni e si y in Zlin, T. G. Masa yka 5555, 760 01 Zlin, Czech Republic
3Depa men o Compu e Science, Facul y o Elec ical Enginee ing and Compu e Science,
VSB–Technical Uni e si y o Os a a, 17. lis opadu 15, 708 33 Os a a, Czech Republic
senk[email p o ec ed], i an.zelink[email p o ec ed], [email p o ec ed]
DOI: 10.15598/aeee. 14i3.1166
Abs ac . In his pape , e olu iona y echnique Di -
e en ial E olu ion (DE) is used o he e olu iona y
uning o con olle pa ame e s o he s abiliza ion
o selec ed disc e e chao ic sys em, which is he wo-
dimensional Lozi map. The no el y o he app oach is
ha he selec ed con olled disc e e dissipa i e chao ic
sys em is used wi hin Chaos enhanced heu is ic con-
cep as he chao ic pseudo- andom numbe gene a o
o d i e he mu a ion and c osso e p ocess in he DE.
The idea was o u ilize he hidden chao ic dynamics
in pseudo- andom sequences gi en by chao ic map o
help Di e en ial e olu ion algo i hm in sea ching o
he bes con olle se ings o he same chao ic sys em.
The op imiza ions we e pe o med o h ee di e en
equi ed inal beha io o he chao ic sys em, and wo
ypes o de eloped cos unc ion. To con i m he obus -
ness o p esen ed app oach, compa isons wi h canonical
DE s a egy and PSO algo i hm ha e been pe o med.
Keywo ds
Di e en ial e olu ion, de e minis ic chaos,
chaos con ol, op imiza ion.
1. In oduc ion
In many applica ions, one o he mos challenging asks
is he con olling o highly nonlinea dynamical sys ems
in o de o ei he elimina e o synch onize he chaos.
The i s success ul app oach o con ol chao ic dynam-
ics by means o a simple linea iza ion echnique was
in oduced in he 1990s by O , G ebogy and Yo ke
(i.e. OGY me hod) [1]. La e , apid de elopmen o
me hods o s abilizing chao ic dynamics has a isen,
and mo e ad anced mode n echniques ha e been ap-
plied o chaos con ol and synch oniza ion including
uncon en ional me hods om he so compu ing ield.
The mos cu en in elligen me hods a e mos ly
based on so compu ing, which is a discipline igh ly
bound o compu e s, ep esen ing a se o me hods
including special algo i hms, belonging o he a i i-
cial in elligence pa adigm. The mos popula o hese
me hods a e neu al ne wo ks, E olu iona y Algo i hms
(EA’s) and uzzy logic. Cu en ly, EA’s a e known as a
powe ul se o ools o almos any di icul and com-
plex op imiza ion p oblem.
The in e es abou he connec ion be ween e olu-
iona y echniques and (no only) con ol o chao ic
sys ems is apidly sp eading. The ini ial esea ch was
conduc ed in [2], whe eas [3] and [4] was mo e con-
ce ned wi h he uning o pa ame e s inside he exis -
ing chaos con ol echnique based on he Py agas Ex-
ended Delay Feedback Con ol (ETDAS), [5]. La e
wo ks [6], [7], and [8] in oduce a no el app oach o
gene a ing he en i e con ol law (con ol me hod) o
he pu pose o s abiliza ion o any chao ic sys em.
O he app oaches u ilizing he EA’s o he s abiliz-
ing o chao ic dynamics ha e mos ly applied he Pa -
icle Swa m Op imiza ion algo i hm (PSO), [9], and
mul i-in e al g adien -me hod [10] o minimum en-
opy con ol echnique [11]. EA’s ha e been also e-
quen ly used in he ask o synch oniza ion o chaos
[12], [13] and [14]. In [15] an EA o op imizing lo-
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 304
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
cal con ol o chaos based on a Lyapuno app oach is
p esen ed. Ano he example o he connec ion be ween
de e minis ic chaos and EA’s ep esen s he embedding
o chao ic dynamics in o he EA’s. Recen esea ch has
p o en ha chao ic app oach is able o bypass local op-
ima s agna ion. A chao ic app oach uses any chao ic
sys em in he place o a pseudo- andom numbe gen-
e a o [16]. This causes he heu is ic o map unique
egions since he chao ic map i e a es o new egions
due o he basic p ope y o de e minis ic chaos, which
is he densi y o pe iodic o bi .
The ini ial concep o embedding chao ic dynamics
in o EA’s is gi en in [17]. La e , he ini ial s udy
[18] was ocused on he simple embedding o chao ic
sys ems in he o m o Chaos Pseudo-Random Num-
be Gene a o (CPRNG) o Di e en ial E olu ion
(DE), [19], and Sel O ganizing Mig a ing Algo i hm
(SOMA), [20], in he ask o op imal PID uning. Also,
he PSO algo i hm wi h elemen s o chaos was in o-
duced as he CPSO [21]. This ield o esea ch was la e
ex ended wi h he success ul expe imen s wi h chaos
d i en DE [22] in eal domain as well as in combina o-
ial p oblems domain [23] and [24].
A he same ime, he chaos embedded PSO wi h in-
e ia weigh s a egy was closely in es iga ed [25], ol-
lowed by he in oduc ion chao ic i e ly algo i hm [26].
The o ganiza ion o his pape is as ollows: i s ly, used
e olu iona y echnique, which is DE, is desc ibed, and
ollowed by he desc ip ion o he ChaosDE concep .
The ea e , he p oblem design and app op ia e co e-
sponding cos unc ions a e in es iga ed and p oposed.
Resul s and conclusion ollow a e wa d.
2. Mo i a ion
This pape ex ends he esea ch o e olu iona y chaos
con ol op imiza ion by means o ChaosDE algo i hm
[27]. In his pape he DE/ and/1/bin s a egy d i en
by di e en chao ic map (sys em) was u ilized o sol e
he issue o e olu iona y op imiza ion o chaos con ol
o he same chao ic sys em used as a CPRNG in he
pa icula case s udy. Thus, he idea was o u ilize he
hidden chao ic dynamics in pseudo- andom sequences
gi en by chao ic map o help Di e en ial e olu ion al-
go i hm in sea ching he bes con olle se ings o he
e y same chao ic sys em. Since he posi i e con ibu-
ion o he chao ic dynamics o he pe o mance o DE
in he ask o e olu iona y chaos con ol op imiza ion
was p o en in compa ison wi h o iginal canonical DE
wi hin he ini ial s udy [28], his pape is no p ima -
ily ocused on he pe o mance compa isons wi h he
di e en heu is ic.
This esea ch ex ends he ini ial wo k wi h he a o e-
men ioned idea and wi h he se e al case s udies com-
bining di e en equi ed s a es o he sys em (i.e. di -
e en Uns able Pe iodic O bi s - UPOs) and di e en
u ilized cos unc ions.
3. Used Heu is ic - Di e en ial
E olu ion
DE is a simple and powe ul popula ion-based op i-
miza ion me hod ha wo ks ei he on eal-numbe -
coded indi iduals o wi h small modi ica ions on dis-
c e e ype indi iduals [19], [29] and [30]. DE is qui e o-
bus , as , and e ec i e, wi h global op imiza ion abil-
i y. This global op imiza ion abili y has been p o en in
many in e disciplina y ypes o esea ch. I wo ks well
e en wi h noisy and ime-dependen objec i e unc-
ions. Recen ly hyb idized DE s a egies ha e been
de eloped [31], [32] and also sel -adap i e DE a ian s
[33], [34] and [35] ha e p o en o be powe ul heu is ics.
Basic canonical p inciple is ollowing.
Fo each indi idual ~xi,G in he cu en gene a ion
G, DE gene a es a new ial indi idual ~
x0
i,G by adding
he weigh ed di e ence be ween wo andomly selec ed
indi iduals ~x 1,G and ~x 2,G o a andomly selec ed
hi d indi idual ~x 3,G. The esul ing indi idual ~
x0
i,G
is c ossed-o e wi h he o iginal indi idual ~xi,G. The
i ness o he esul ing indi idual, e e ed o as a pe -
u bed ec o ~ui,G+1, is hen compa ed wi h he i -
ness o ~ui,G. I he i ness o ~ui,G+1 is g ea e han
he i ness o ~xi,G, hen ~xi,G is eplaced wi h ~ui,G+1;
o he wise, ~xi,G emains in he popula ion as ~xi,G+1.
Please e e o Eq. (1) o no a ion o c osso e , and
o [19] o he de ailed desc ip ion o used DERand1Bin
s a egy and all o he DE s a egies:
~ui,G =~x 1,G +F(~x 2,G −~x 3,G).(1)
4. Concep o ChaosDE
This sec ion con ains he desc ip ion o disc e e dissi-
pa i e chao ic map, which can be used as he chao ic
pseudo- andom gene a o s o DE as well as he main
p inciple o he ChaosDE concep . In his esea ch, di-
ec ou pu i e a ions o he chao ic map we e used o
he gene a ion o pseudo andom numbe s. Two ypes
o numbe s a e equi ed: eal numbe s in he p ocess
o c osso e based on he use de ined CR alue and
in ege alues used o selec ion o indi iduals.
The gene al idea o ChaosDE and CPRNG is o e-
place he de aul PRNG wi h he disc e e chao ic map.
Since he disc e e chao ic map is a se o equa ions
wi h a s a ic s a posi ion, a andom s a posi ion
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 305
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
o he map is c ea ed in each un o EA, in o de o
ha e di e en s a posi ion o di e en expe imen s
( uns o EA’s). This andom posi ion is ini ialized wi h
he de aul PRNG, as a one-o andomize . Thus,
he concep o ChaosDE is u ilizing he amous and
well-known “bu e ly e ec ” i.e. ex eme sensi i i y o
chao ic sys em o he ini ial condi ions.
As wo di e en ypes o numbe s a e equi ed in
ChaosDE; eal and in ege s, he use o modulo ope -
a o s is used o ob ain alues be ween he speci ied
anges, as gi en in he ollowing Eq. (2) and Eq. (3):
nd eal = mod (abs ( ndChaos),1.0) ,(2)
ndin = mod (abs ( ndChaos),1.0)×Range+1,(3)
whe e abs e e s o he absolu e po ion o he chao ic
map gene a ed numbe ndChaos, and mod is he
modulo ope a o . Range speci ies he alue (inclusi e),
whe e he numbe is o be scaled.
5. Lozi Map
This sec ion con ains he ma hema ical and g aphical
desc ip ion o he selec ed disc e e dissipa i e sys em,
which se es bo h as o CPRNG and also as he ex-
ample o he sys em o be e olu iona y con olled.
The Lozi map is a simple disc e e wo-dimensional
chao ic map. The map equa ions a e gi en in Eq. (4).
The pa ame e s used in his wo k a e a= 1.7and
b= 0.5as sugges ed in [36]. Fo hese alues, he
sys em exhibi s ypical chao ic beha io and wi h his
pa ame e se ing i is used in he mos esea ch pape s
and o he li e a u e sou ces [37]:
Xn+1 = 1 −a|Xn|+b·Yn
Yn+1 =Xn
.(4)
The x, y plo o he selec ed map is depic ed in Fig. 1.
The chao ic beha io o he chao ic map, ep esen ed
by he example o di ec ou pu i e a ions is depic ed
in Fig. 2, whe eas he Fig. 3 shows he example o
chao ic dynamics ans e ed in o he ange h0,−1i.
Finally, he illus a i e his og am o he dis ibu ion
o eal numbe s ans e ed in o he ange h0,−1igen-
e a ed by means o chao ic Lozi map is shown in Fig. 4.
6. Cos Func ion Design
The idea o he basic cos unc ion (CFSimple), which
could be used p oblem- ee only o he s abiliza ion o
p−1o bi , was o minimize he a ea c ea ed by he di -
e ence be ween he equi ed s a e and he eal sys em
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
x
Fig. 1: x, y plo o he Lozi map.
Fig. 2: I e a ions o he uncon olled Lozi map ( a iable x).
Fig. 3: Example o he chao ic dynamics: ange h0,−1igene -
a ed by means o he Lozi map.
ou pu in he whole simula ion in e al −τiEq. (5).
This CF design is e y con enien o he e olu iona y
sea ching p ocess due o he ela i ely a o able CF
su ace. The disad an age o he app oach is ha CF
alue is in luenced by chao ic ansien beha io o he
non-s abilized sys em. As a esul o his, he small
change in con ol me hod se ing o ex emely sensi-
i e chao ic sys em (gi en by he e y small change o
CF alue), can be supp essed by he abo e-men ioned
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 306
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
0.2
0.4
0.6
0.8
1.0
Value
100
200
300
400
Fig. 4: His og am o he dis ibu ion o eal numbe s ans-
e ed in o he ange h0,−1igene a ed by he Lozi map
(5000 samples).
including o ini ial chao ic ansien :
CFSimple =
τi
X
=0
|T S −AS |,(5)
whe e T S is a ge s a e and AS is ac ual s a e.
Di e en ype o uni e sal cos unc ion is pu ely
based on sea ching o he desi ed s abilized pe iodic
o bi and he ea e calcula ion o he di e ence be-
ween desi ed and ac ual pe iodic o bi in he sho
ime in e al −τs(20 i e a ions) om he poin , whe e
he i s minimal alue o di e ence be ween desi ed
and ac ual sys em ou pu is ound (i.e. loa ing win-
dow o minimiza ion, Fig. 5).
Fig. 5: Floa ing window o he op imiza ion.
Such a design o uni e sal CF should secu e he suc-
cess ul s abiliza ion o ei he p−1o bi (s able s a e) o
any highe pe iodic o bi anywise phase shi ed. Fu -
he mo e, due o CF alues con e ging owa ds ze o,
his CF also allows using decision ules and a oiding
e y ime demanding simula ions. This ule s ops EA
immedia ely, when he i s indi idual wi h good pa-
ame e s uc u e is eached, since he alue o CF
is lowe han he accep able one (CFacc). Based on
he nume ous expe imen s, ypically CFacc = 0.001 a
ime in e al τs= 20 i e a ions, hus he di e ence be-
ween desi ed and ac ual ou pu has he alue o 0.0005
pe i e a ion – i.e. success ul s abiliza ion o he used
con ol echnique. The CFUNI has he o m desc ibed
in Eq. (6):
CFUNI =pen1+
τ2
X
=τ1
|T S −AS |,(6)
whe e τ1is he i s min alue o di e ence be ween T S
and AS,τ2is he end o op imiza ion in e al (τ1+τs),
pen1= 0 i τi−τ2≥τsand pen1= 10 ·(τi−τ2)i
τi−τ2< τs(i.e. la e s abiliza ion).
7. Expe imen Design
This esea ch encompasses six case s udies. Th ee di -
e en equi ed beha io o he chao ic sys em and wo
di e en cos unc ions we e combined in he ollowing
o m:
•Case s udy 1: p−1UPO, Lozi map as
CPRNG/Con olled sys em wi h CFSimple.
•Case s udy 2: p−1UPO, Lozi map as
CPRNG/Con olled sys em wi h CFUNI.
•Case s udy 3: p−2UPO, Lozi map as
CPRNG/Con olled sys em wi h CFSimple.
•Case s udy 4: p−2UPO, Lozi map as
CPRNG/Con olled sys em wi h CFUNI.
•Case s udy 5: highe o de p−4UPO, Lozi map
as CPRNG/Con olled sys em wi h CFSimple.
•Case s udy 6: highe o de p−4UPO, Lozi map
as CPRNG/Con olled sys em wi h CFUNI.
This wo k is ocused on he u iliza ion o he chaos
d i en DE o uning o pa ame e s o ETDAS con ol
me hod o s abilize desi ed Uns able Pe iodic O bi s
(UPO). In he desc ibed esea ch, desi ed UPO was p−
1(s able s a e). The o iginal con ol me hod, ETDAS,
in he disc e e o m sui able o disc e e chao ic maps
has he o m Eq. (7) and Eq. (8):
Fn=K[(1 −R)Sn−m−xn],(7)
sn=xn·R·Sn−m,(8)
whe e Kand Ra e adjus able cons an s, Fis he
pe u ba ion; Sis gi en by a delay equa ion u ilizing
p e ious s a es o he sys em, mis he pe iod o m-
pe iodic o bi o be s abilized. The pe u ba ion Fn
in Eq. (8) may ha e a bi a ily la ge alue, which can
cause di e ging o he sys em. The e o e, Fnshould all
be ween −Fmax,Fmax. The anges o all e olu iona y
es ima ed pa ame e s a e gi en in Tab. 1.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 307
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
Tab. 1: Es ima ed pa ame e s.
Pa ame e Min Max
K-2 2
R0 0.99
Fmax 0 0.9
Tab. 2: DE e sions se ings.
Pa ame e Value
PopSize 25
FCanonicalDE 0.5
C Canonical DE 0.9
FChaosDE 0.5
C ChaosDE 0.6
Gene a ions 300
Max. CF E alua ions (CFE) 7500
Wi hin he esea ch, a o al numbe o 50 simula-
ions o each case s udy and each DE e sion we e
ca ied ou . The pa ame e se ings o bo h ChaosDE
and Canonical DE we e gi en ollowing way (Tab. 2).
Besides he pa ame e s de ining he size o popula ion
and numbe o gene a ions, o he wo in e nal un-
ing pa ame e s o DE, which a e mu a ion cons an F
and c osso e pa ame e C , a e e y impo an in he
DE pe o mance issue. Canonical DE u ilizes he well-
p o en and ecommended se ings [38] Recen esea ch
in he chaos and complex dynamics d i en heu is ics
shows ha DE equi es lowe alues o a o emen ioned
in e nal pa ame e s [30]. The e o e, simple uning wi h
he inc emen al s ep o 0.1 has been conduc ed o
ChaosDE (Tab. 3).
Expe imen s we e pe o med in an en i onmen o
Wol am Ma hema ica, PRNG ope a ions. The e o e,
used he buil -in Ma hema ica so wa e pseudo- andom
numbe gene a o . All expe imen s used di e en ini-
ializa ion, i.e. di e en ini ial popula ion was gene -
a ed in each un o Canonical/ChaosDE.
Tab. 4: The alues o desi edUPOs.
UPO Values o UPO o unpe u bed sys em
p−1xF= 0.454545
p−2x1=−0.382166;x2= 0.700637
p−4x1=−0.691899;x2= 0.334059;
x3= 0.086151;x4= 1.020573
8. Resul s
All simula ions we e success ul and ga e new op imal
se ings o ETDAS con ol me hod secu ing he as
s abiliza ion o he chao ic sys em a equi ed beha -
io s, which we e p−1UPO (s able s a e), p−2UPO
(oscilla ion be ween wo alues) and inally p−4UPO.
Pe o mances o bo h s udied DE s a egies a e com-
pa ed wi h he ep esen a i e o swa m algo i hm,
which is Pa icle Swa m Op imize (PSO). The canon-
ical e sion wi h ine ia weigh s a egy [9] has been
u ilized. The maximal cos unc ion e alua ion alue,
popula ion size and he numbe o i e a ions we e se
iden ically as o he bo h DE s a egies.
The o ganiza ion o he esul s is ollowing. Table 5,
Tab. 6, Tab. 8, Tab. 9, Tab. 11, and Tab. 12 a e ocused
on he pe o mance compa isons be ween canonical
DE, ChaosDE d i en by Lozi map; and swa m based
PSO algo i hm. These ables con ain simple s a is i-
cal o e iew o e olu iona y op imiza ion/simula ion
esul s i.e. a e age, median maximum, minimum ( he
bes solu ion), s d. de . alues o he pa icula cos
unc ion and o all 50 uns o bo h compa ed heu is-
ics. I alic numbe s ep esen he bes esul .
Tab. 3: Resul s o uning o in e nal pa ame e s Fand C o ChaosDE om he in e al h0.1,0.9iand inc emen al s ep o 0.1
– A e age cos unc ion esul s o 30 uns o ChaosDE op imizing he highe nonlinea ask: Case s udy 2.
C =0.1 C =0.2 C =0.3 C =0.4 C =0.5
F=0.1 4.39444 ·10−15 4.14344 ·10−15 4.16289 ·10−15 4.37956 ·10−15 4.14627 ·10−15
F=0.2 4.37891 ·10−15 4.09678 ·10−15 3.88405 ·10−15 3.95291 ·10−15 3.92013 ·10−15
F=0.3 4.35117 ·10−15 3.96566 ·10−15 3.84068 ·10−15 3.82903 ·10−15 3.90125 ·10−15
F=0.4 4.47881 ·10−15 3.98678 ·10−15 3.85901 ·10−15 3.81791 ·10−15 3.82793 ·10−15
F=0.5 4.43944 ·10−15 4.00844 ·10−15 3.85903 ·10−15 3.81515 ·10−15 3.78464 ·10−15
F=0.6 4.44328 ·10−15 4.01397 ·10−15 3.92736 ·10−15 3.82460 ·10−15 3.81850 ·10−15
F=0.7 4.43664 ·10−15 4.00505 ·10−15 3.87791 ·10−15 3.80962 ·10−15 3.82570 ·10−15
F=0.8 4.45275 ·10−15 4.04009 ·10−15 3.90736 ·10−15 3.83625 ·10−15 3.83625 ·10−15
F=0.9 4.44226 ·10−15 4.13174 ·10−15 3.91846 ·10−15 3.82625 ·10−15 3.83793 ·10−15
C =0.6 C =0.7 C =0.8 C =0.9
F=0.1 5.83432 ·10−73.52617 ·10−10 0.0000271935 0.172951
F=0.2 3.99240 ·10−15 4.15958 ·10−15 3.16606 ·10−12 0.0000531224
F=0.3 3.86901 ·10−15 0.0000501611 3.96350 ·10−15 4.16068 ·10−15
F=0.4 3.79960 ·10−15 3.82458 ·10−15 3.85238 ·10−15 3.90238 ·10−15
F=0.5 3.75962 ·10−15 3.77407 ·10−15 3.79903 ·10−15 3.87791 ·10−15
F=0.6 3.79684 ·10−15 3.79572 ·10−15 3.79625 ·10−15 3.85182 ·10−15
F=0.7 3.79680 ·10−15 3.76240 ·10−15 3.81350 ·10−15 3.88070 ·10−15
F=0.8 3.79125 ·10−15 3.76964 ·10−15 3.80127 ·10−15 3.86570 ·10−15
F=0.9 3.78905 ·10−15 3.79184 ·10−15 3.78738 ·10−15 3.85848 ·10−15
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 308

THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
Tab. 5: Compa ison o ChaosDE, canonical DE and PSO case s udy 1.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 0.520639 0.520639 0.530679
Max 0.522148 0.527132 0.573742
A e age 0.520696 0.520769 0.548688
Median 0.520639 0.520639 0.549588
S d. De . 2.78·10−49.18·10−41.07·10−2
Tab. 6: Compa ison o ChaosDE, canonical DE and PSO case s udy 2.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 3.53307 ·10−15 3.55511 ·10−15 4.41022 ·10−15
Max 4.05511 ·10−15 3.91062 ·10−15 5.21022 ·10−15
A e age 3.75362 ·10−15 3.7514 ·10−15 4.79336 ·10−15
Median 3.75511 ·10−15 3.75511 ·10−15 4.78267 ·10−15
S d. De . 9.96 ·10−17 7.42 ·10−17 2.02977 ·10−16
Tab. 7: Bes solu ions – Joined case s udies 1 and 2, p−1UPO.
Pa ame e
Case s udy 1,
CNSimple,
ChaosDE
Case s udy 2,
CNUNI,
ChaosDE
K−1.11259 −0.859989
Fmax 0.9 0.65695
R0.289232 0.065673
CF Value 0.520639 3.53307 ·10−15
Is ab. Value 21 9
A g. e o pe i e a ion 7.21 ·10−15 2.07 ·10−15
Tab. 8: Compa ison o ChaosDE, canonical DE and PSO case s udy 3.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 7.04967 6.99829 7.29818
Max 7.54409 7.33379 8.05143
A e age 7.30428 7.2827 7.67487
Median 7.33379 7.33379 7.6944
S d. De . 7.77 ·10−29.29 ·10−20.216771
Tab. 9: Compa ison o ChaosDE, canonical DE and PSO case s udy 4.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 1.62665 ·10−91.68548 ·10−91.81061 ·10−6
Max 1.28207 ·10−21.36095 ·10−20.136879
A e age 7.57338 ·10−46.40434 ·10−40.008665
Median 1.42814 ·10−41.35227 ·10−40.001515
S d. De . 1.86 ·10−31.96 ·10−30.023862
Tab. 10: Bes solu ions – Joined case s udies 3 and 4, p−2UPO.
Pa ame e
Case s udy 3,
CNSimple,
ChaosDE
Case s udy 4,
CNUNI,
ChaosDE
K0.574025 −0.614527
Fmax 0.430788 0.508694
R0.445453 0.528986
CF Value 6.99829 1.62665 ·10−9
Is ab. Value 22 18
A g. e o pe i e a ion 2.98 ·10−81.99 ·10−11
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 309
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
Tab. 11: Compa ison o ChaosDE, canonical DE and PSO case s udy 5.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 13.8025 13.7305 14.9151
Max 19.2155 51.9391 29.6333
A e age 15.4354 21.7940 14.5647
Median 14.2014 14.2548 22.4428
S d. De . 1.1312 5.4219 3.5719
Tab. 12: Compa ison o ChaosDE, canonical DE and PSO case s udy 6.
S a is ical
da a
ChaosDE Canonical DE PSO
CF Value CF Value CF Value
Min 1.39076 ·10−31.45946 ·10−32.04937 ·10−3
Max 1.4131 ·10−21.28539 ·10−21.14558
A e age 2.79004 ·10−32.09308 ·10−30.15435
Median 1.60824 ·10−31.62911 ·10−30.11778
S d. De . 3.49 ·10−32.19 ·10−30.25369
Tab. 13: Bes solu ions – Joined case s udies 5 and 6, p−4UPO.
Pa ame e
Case s udy 5,
CNSimple,
ChaosDE
Case s udy 6,
CNUNI,
ChaosDE
K−0.869336 −0.935913
Fmax 0.255559 0.623172
CF Value 13.7305 1.39076 ·10−3
Is ab. Value 46 39
A g. e o pe i e a ion 2.66 ·10−41.39 ·10−6
Resul s gi en in Tab. 7, Tab. 10 and Tab. 13 ep e-
sen he di ec compa ison o chaos s abiliza ion p op-
e ies o he joined case s udies ela ed o he iden ical
UPO o be con olled. Tables show he bes ounded
indi idual solu ions o pa ame e s se up o ETDAS
con ol me hod, co esponding inal CF alue. Also
hese ables show he Is ab. alue ep esen ing he
numbe o i e a ions equi ed o s abiliza ion on he
desi ed UPO and u he he a e age e o be ween de-
si ed ou pu alue and eal sys em ou pu om he las
20 i e a ions.
G aphical simula ion ou pu s o he bes indi id-
ual solu ions o pa icula case s udies a e depic ed
in Fig. 6, Fig. 7, Fig. 10, Fig. 11, Fig. 14 and Fig. 15
Fig. 6: Simula ion o he bes indi idual solu ion – ChaosDE
and Lozi map: Case s udy 1, CNSimple,p−1UPO.
whe eas he Fig. 8, Fig. 9, Fig. 12, Fig. 13, Fig. 16 and
Fig. 17 show he simula ion ou pu o all 50 uns o
ChaosDE con i ming he obus ness o his app oach.
Fo he illus a i e pu poses, all g aphical simula ions
ou pu s a e depic ed only o he a iable xo he
chao ic sys ems.
The alues o desi ed UPOs o unpe u bed chao ic
Lozi map based on he ma hema ical analysis o he
sys ems a e gi en in Tab. 4.
F om he esul s p esen ed in he Tab. 5, Tab. 6,
Tab. 7, Tab. 8, Tab. 9, Tab. 10, Tab. 11, Tab. 12 and
Tab. 13 i is clea ha he CFSimple is e y con e-
nien o he e olu iona y p ocess, which means ha
epea ed uns o EA a e p o iding iden ical op imal
Fig. 7: Simula ion o he bes indi idual solu ion – ChaosDE
and Lozi map: Case s udy 2, CNUNI,p−1UPO.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 310
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
esul s (i.e. e y close o he possible global ex eme).
This is g aphically con i med in he Fig. 8, Fig. 12 and
Fig. 16, which show all 50 simula ions. All he uns
a e me ged in o he one line.
Fig. 8: All 50 uns o EA – ChaosDE and Lozi map: Case s udy
1, CNSimple,p−1UPO.
Fig. 9: All 50 uns o EA – ChaosDE and Lozi map: Case s udy
2, CNUNI,p−1UPO.
Fig. 10: Simula ion o he bes indi idual solu ion – Canonical
DE and Lozi map: Case s udy 3, CNSimple,p−2
UPO.
Fig. 11: Simula ion o he bes indi idual solu ion – ChaosDE
and Lozi map: Case s udy 4, CNUNI,p−2UPO.
Fig. 12: All 50 uns o EA– Canonical DE and Lozi map: Case
s udy 3, CNSimple,p−2UPO.
Fig. 13: All 50 uns o EA – ChaosDE and Lozi map, Case
s udy 4, CNUNI,p−2UPO.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 311
THEORETICAL COMPUTER SCIENCE VOLUME: 14 |NUMBER: 3 |2016 |SEPTEMBER
Fig. 14: Simula ion o he bes indi idual solu ion – Canonical
DE and Lozi map: Case s udy 5, CNSimple,p−4
UPO.
Fig. 15: Simula ion o he bes indi idual solu ion – Canonical
DE and Lozi map: Case s udy 5, CNSimple,p−4
UPO.
On he o he hand he disad an age o including o
ini ial chao ic ansien beha io o no s abilized sys-
em in o he cos unc ion alue and esul ing e y iny
change o con ol me hod se ing o he ex emely sen-
si i e chao ic sys em is causing supp ession o s abiliza-
ion speed and nume ical p ecision.
Resul s ob ained in he cases u ilizing he CNUNI
lend weigh o he a gumen ha he echnique o pu e
sea ching o pe iodic o bi s is ad an ageous o as e
and mo e p ecise s abiliza ion o he chao ic sys em.
The g aphical compa isons o he pe o mance anal-
ysis o ChaosDE, Canonical DE and PSO wi hin all 6
case s udies a e gi en in complex Fig. 18 and Fig. 19.
The i s one ep esen s he ime e olu ion o he cos
unc ion alue o he bes indi idual solu ion, which
a e gi en in Tab. 10, Tab. 11 and Tab. 13. Mo eo e ,
hese solu ions we e also used o he g aphical sim-
ula ions in Fig. 6, Fig. 7, Fig. 10, Fig. 11, Fig. 14
and Fig. 15. Figu e 19 shows he compa isons o
ime e olu ion o a e age CF alues o all 50 uns o
ChaosDE/Canonical DE/PSO, con i ming he obus -
ness o bo h used DE s a egies wi hin many epea ed
Fig. 16: All 50 uns o EA – Canonical DE and Lozi map: Case
s udy 5 - CNSimple,p−4UPO.
Fig. 17: All 50 uns o EA – ChaosDE and Lozi map: Case
s udy 6, CNUNI,p−4UPO.
uns. Mo e abou he indings is gi en in he conclusion
sec ion o his pape .
9. Conclusion
In his pape , e olu iona y algo i hm Di e en ial E o-
lu ion was used o he e olu iona y uning o con-
olle pa ame e s o he s abiliza ion o selec ed dis-
c e e chao ic sys em, which was he wo-dimensional
Lozi map. The o iginali y o he p esen ed app oach is
ha he selec ed con olled disc e e dissipa i e chao ic
sys em is used also wi hin chaos enhanced heu is ic
concep as he chao ic pseudo- andom numbe gene a-
o o d i e he mu a ion and c osso e p ocess in he
Di e en ial E olu ion. The idea was o u ilize he hid-
den chao ic dynamics in he pseudo- andom sequences
gi en by chao ic map o help Di e en ial E olu ion al-
go i hm in sea ching o he bes con olle se ings o
he iden ical chao ic sys em.
The indings o he wo di e en cos unc ion de-
signs can be summa ized as ollows:
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 312