en opy
A icle
Highe -O de Hamil onian o Ci cui s wi h
(α,β) Elemen s
Zdenˇek Biolek 1,2 , Dalibo Biolek 1,2 , Vie a Biolko á3,* and Zdenˇek Kolka 3
1Depa men o Mic oelec onics, B no Uni e si y o Technology, 616 00 B no, Czech Republic;
[email p o ec ed] (Z.B.); dalibo [email p o ec ed] (D.B.)
2Depa men o Elec ical Enginee ing, Uni e si y o De ence, 662 10 B no, Czech Republic
3Depa men o Radio Elec onics, B no Uni e si y o Technology, 616 00 B no, Czech Republic;
[email p o ec ed].cz
*Co espondence: [email p o ec ed].cz; Tel.: +420-541-146-584
Recei ed: 22 Feb ua y 2020; Accep ed: 2 Ap il 2020; Published: 5 Ap il 2020


Abs ac :
The pape s udies he cons uc ion o he Hamil onian o ci cui s buil om he (
α
,
β
)
elemen s o Chua’s pe iodic able. I s a s om he Lag ange unc ion, whose exis ence is limi ed
o
Σ
-ci cui s, i.e., ci cui s buil exclusi ely om elemen s loca ed on a common
Σ
-diagonal o he
able. We show ha he Hamil onian can also be cons uc ed ia he gene alized Tellegen’s heo em.
Acco ding o he ideas o p edic i e modeling, he esul ing Hamil onian is made up exclusi ely o
he cons i u i e ela ions o he elemen s in he ci cui . Wi hin he ame o Os og adsky’s o malism,
he simula ion scheme o
Σ
-ci cui s is designed and examined wi h he example o a nonlinea
Pais–Uhlenbeck oscilla o .
Keywo ds:
highe -o de elemen ; cons i u i e ela ion; Hamil onian; Lag angian; Chua’s able;
mem is o ; Eule -Lag ange equa ion
1. In oduc ion
The dynamics o complex nonlinea sys ems a ac s he in e es o esea che s in a ious
b anches o science [
1
–
4
]. One o he app oaches used o s udying sys em dynamics, p edic i e
modeling, conside s he sys em an in e connec ion o one-po de ices, he α,βelemen s, also known
as Highe O de Elemen s (HOEs) [
5
]. Each HOE gua an ees ha i s e minal quan i ies a e coupled
in all ci cums ances ia a cons i u i e ela ion. In elec ical enginee ing, he e minal quan i ies a e
he gene alized ol age
(α)
and cu en i
(β)
, whe e he in ege s
α
and
β
deno e he o de s o he
de i a i e/in eg al wi h espec o ime ( o posi i e/nega i e in ege s). Well-known (
α
,
β
) elemen s
include esis o s (0,0), capaci o s (0,
−
1), and induc o s (
−
1,0), and also he p omising mem is o s
(
−
1,
−
1), memcapaci o s (
−
1,
−
2), and meminduc o s (
−
2,
−
1), equency-dependen esis o s (FDNR,
FDNC), and o he s. HOEs a e usually ep esen ed as poin s wi h in ege (
α
,
β
) coo dina es in Chua’s
pe iodic able (see Figu e 1). The cons i u i e ela ion o he elemen is equen ly w i en in one o he
wo ollowing o ms:
(α)= i(β), o i(β)=g (α)(1)
whe e he unc ion () o g() models he cu en o ol age ep esen a ion o he elemen .
The use o HOEs conside ably inc eases he possibili ies o analyzing and syn hesizing ci cui s
wi h complex dynamic beha io . Thanks o p edic i e modeling, many phenomena can be be e
unde s ood. Fo example, he hypo he ical elemen called a mem is o [
6
] helped o explain he peculia
hys e esis beha io obse ed when measu ing a ious nanoma e ials. When Hewle -Packa d epo ed
he i s TiO
2
-based wo king mem is i e de ice in 2008 [
7
], hey eminded he esea che s o he o iginal
En opy 2020,22, 412; doi:10.3390/e22040412 www.mdpi.com/jou nal/en opy
En opy 2020,22, 412 2 o 20
wo k [
6
] ha explains his beha io as a cha ac e is ic ea u e o he mem is o —a inge p in . Ongoing
esea ch has demons a ed ha a pinched hys e esis loop is a egula phenomenon, which akes e ec
o an a bi a y (α,β) elemen in he space o he ime de i a i es o i s e minal quan i ies [8].
The in e es in sys ems wi h high-o de dynamics has led o an ex ension o Lag ange’s and
Hamil on’s o malisms o sys ems wi h
α
,
β
elemen s. This is because he o iginal Lag ange’s o malism
s a s om he undamen al schema ic o classical mechanics, i.e., om he poin o iew o p edic i e
modeling based on he R(0,0), L(
−
1,0), and C(0,
−
1) elemen s. Mo eo e , he sys em o equa ions
o mo ion can be gene a ed om one scala unc ion, he Lag angian, bu only o conse a i e
sys ems. Recen wo ks [
9
–
11
] ha e ex ended Lag ange’s o malism o ci cui s con aining mem is o s,
memcapaci o s, and meminduc o s. The ollowing wo k [
12
] de ines he po en ial unc ions o gene al
(
α
,
β
) elemen s om which Lag angians and dissipa i e unc ions a e d awn, and ules o gene a ing
he equa ions o mo ions a e es ablished. The po en ial unc ions o a gene al (
α
,
β
) elemen a e de ined
o cu en o ol age ep esen a ion as in eg als:
Sα,β=Z (α)di(β)o ˆ
Sα,β=Zi(β)d (α), (2)
which ep esen a na u al gene aliza ion o he po en ial and kine ic ene gy and he dissipa i e unc ion
o classical mechanics, o he ene gy o capaci o s, induc o s, and he con en o esis o s in classical
elec ical enginee ing. Two di e en ep esen a ions o po en ial unc ions co espond o he concep
o unc ions and co- unc ions in oduced by Milla [13] and Che y [14] in 1951.
En opy 2020, 21, x 2 o 20
ea u e o he mem is o —a inge p in . Ongoing esea ch has demons a ed ha a pinched
hys e esis loop is a egula phenomenon, which akes e ec o an a bi a y (
α
,
β
) elemen in he
space o he ime de i a i es o i s e minal quan i ies [8].
The in e es in sys ems wi h high-o de dynamics has led o an ex ension o Lag ange’s and
Hamil on’s o malisms o sys ems wi h
α
,
β
elemen s. This is because he o iginal Lag ange’s
o malism s a s om he undamen al schema ic o classical mechanics, i.e., om he poin o iew
o p edic i e modeling based on he R (0,0), L (−1,0), and C (0,−1) elemen s. Mo eo e , he sys em o
equa ions o mo ion can be gene a ed om one scala unc ion, he Lag angian, bu only o
conse a i e sys ems. Recen wo ks [9–11] ha e ex ended Lag ange’s o malism o ci cui s
con aining mem is o s, memcapaci o s, and meminduc o s. The ollowing wo k [12] de ines he
po en ial unc ions o gene al (
α
,
β
) elemen s om which Lag angians and dissipa i e unc ions a e
d awn, and ules o gene a ing he equa ions o mo ions a e es ablished. The po en ial unc ions o
a gene al (
α
,
β
) elemen a e de ined o cu en o ol age ep esen a ion as in eg als:
() () () ()
,,
ˆ
o S diSid
α
ββ
α
αβ αβ
==

, (2)
which ep esen a na u al gene aliza ion o he po en ial and kine ic ene gy and he dissipa i e
unc ion o classical mechanics, o he ene gy o capaci o s, induc o s, and he con en o esis o s in
classical elec ical enginee ing. Two di e en ep esen a ions o po en ial unc ions co espond o
he concep o unc ions and co- unc ions in oduced by Milla [13] and Che y [14] in 1951.
0
β
1
-1-2
-1
-2
RL
C
MR
MC
ML FDNC
α
1
Σ=0
Σ=-1
FDNR
(a)
(b)
Figu e 1. Visualiza ion o se e al (α,β) elemen s in Chua’s able. R, L, C = Resis o , Induc o , and
Capaci o ; MR, ML, MC = Mem is o , Meminduc o , and Memcapaci o ; FDNR, FDNC = F equency
Dependen Nega i e Resis o and F equency Dependen Nega i e Conduc o . (a) Lag ange’s
o malism o classical mechanics applies o ci cui s composed exclusi ely o L and C elemen s; he
ene gy, conse ed in he ci cui , co esponds o he diagonal Σ = −1; (b) an example o he applica ion
o a 2nd-o de Lag angian o he desc ip ion o he Pais–Uhlenbeck oscilla o [15] consis ing o
dissipa i e R, FDNR, FDNC elemen s [16]; he powe p ese ed in he ci cui co esponds o he
diagonal Σ = 0.
The classical Lag angian is o he i s o de , so i is a unc ion o gene alized coo dina es and
eloci ies ( i s -o de ime de i a i es o he coo dina es); he esul ing equa ions o mo ion a e o
he second o de . Highe -o de Lag angians we e in oduced by Os og adsky in [17] as ollows:
() ( )
()
1
, ,..., ,
m
LL =xx x (3)
Figu e 1.
Visualiza ion o se e al (
α
,
β
) elemen s in Chua’s able. R,L,C=Resis o , Induc o , and
Capaci o ; MR,ML,MC =Mem is o , Meminduc o , and Memcapaci o ; FDNR,FDNC =F equency
Dependen Nega i e Resis o and F equency Dependen Nega i e Conduc o . (a) Lag ange’s o malism
o classical mechanics applies o ci cui s composed exclusi ely o Land Celemen s; he ene gy,
conse ed in he ci cui , co esponds o he diagonal
Σ
=
−
1; (b) an example o he applica ion o a
2
nd
-o de Lag angian o he desc ip ion o he Pais–Uhlenbeck oscilla o [
15
] consis ing o dissipa i e
R,FDNR,FDNC elemen s [16]; he powe p ese ed in he ci cui co esponds o he diagonal Σ=0.
En opy 2020,22, 412 3 o 20
The classical Lag angian is o he i s o de , so i is a unc ion o gene alized coo dina es and
eloci ies ( i s -o de ime de i a i es o he coo dina es); he esul ing equa ions o mo ion a e o he
second o de . Highe -o de Lag angians we e in oduced by Os og adsky in [17] as ollows:
L=Lx,x(1),. . . ,x(m), (3)
The Lag angian is a unc ion o he 1xn ec o o he gene alized coo dina es
x
=[x
1
.. x
n
]
T
and
hei de i a i es up o he in ege -o de m. The sys em ep esen ed by Lag angian (3) is go e ned by n
equa ions o mo ion:
∂L
∂x−d
d ∂L
∂x(1)!+d2
d 2 ∂L
∂x(2)!−. . . +(−1)mdm
d m ∂L
∂x(m)!=0. (4)
Equa ion (4) is a di ec consequence o he ac ha he sys em ajec o y is ex emal o he ac ion
A=
2
Z
1
Lx,x(1),. . . ,x(m), d , (5)
which is he essence o Hamil on’s a ia ional p inciple [
18
]. The highe -o de Lag angian (3) was
used o he i s ime o desc ibe ci cui s buil om HOEs in [
16
]. This wo k demons a ed ha he
Lag angian, which has he abili y o gene a e equa ions o mo ion, can only be d awn o he
Σ
-ci cui s
ha con ain only elemen s om he common
Σ
-diagonal o Chua’s able, whe e he sum o he indices
αand βis p ese ed; hus, Σ=α+β. The Lag angian hen has he o m
L=X
εi
(−1)iSio ˆ
L=X
εi
(−1)iˆ
Si. (6)
The summa ion is done h ough all
εi
elemen s o he gi en
Σ
-diagonal, whe e iis he posi ion o
he elemen on he diagonal (see Figu e 2). The signs in on o he s a e unc ions in he sums o (6) a e
go e ned by he ype (e en o odd) o he posi ions o he elemen s on he diagonal [
16
]. The di e ence
be ween kine ic and po en ial ene gy, ans e ing be ween he ine ial and he accumula ing elemen s,
which a e he common o m o he Lag angian used in classical mechanics, a e me ely a special
case o (6). The physical dimension o he Lag angian is gi en by he numbe
Σ
o he diagonal:
[Vol
·
Ampe
·
sec
−Σ
]. This is he ene gy [Vol
·
Ampe
·
sec] in LC and he powe [Vol
·
Ampe ] in esis i e
ci cui s. An example o he Lag angian o he Pais–Uhlenbeck oscilla o [
15
], consis ing o h ee
esis i e elemen s o R,FDNR, and FDNC ypes, is gi en in [16].
The ansi ion om he highe -o de Lag angian (3) o he highe -o de Hamil onian is gi en by
he gene alized Legend e ans o ma ion published by Os og adsky in [17]:
Hx,x(1), , x(2m−1)=1pT·x(1)+. . . +mpT·x(m)−Lx,x(1),. . . ,x(m)(7)
whe e 1p o mpa e ec o s o he gene alized momen a
jp=
m
X
k=j
(−1)k−j ∂L
∂x(k)!(k−j)
. (8)
Conside he no a ion
1q=x,2q=x(1),. . . ,mq=x(m−1). (9)
En opy 2020,22, 412 4 o 20
En opy 2020, 21, x 6 o 20
β
ε
0
α
(
α
max
,
β
min
)
ε
1
ε
2
ε
m
(a)
Δ
ε
i
α
ε
i
Δ
ε
i
β
β
ε
m
α
(
α
min
,
β
max
)
ε
1
ε
i
ε
0
(b)
Δ
ε
i
α
ε
2
Δ
ε
i
β
Figu e 2. This ci cui consis s o elemen s o
ε
0 o
ε
m ypes, which a e loca ed on he common
Σ-diagonal. Cha ac e is ic qua e -ci cles a e cons uc ed o he (a) cu en and (b) ol age
ep esen a ion o he ci cui . Since he elemen s a e loca ed on he Σ-diagonal, hei dis ances om
he hidden elemen a e he same in bo h he
α
and
β
di ec ions; hus, Δ
ε
i
α
= Δ
ε
i
β
= i.
Le us ocus on he p oo ha he o mula (18) o he cu en ep esen a ion o elemen s (see
Figu e 2a) is a Hamil onian. The p oo o he o mula (19), which co esponds o he ol age
ep esen a ion, is analogous, so, o he sake o b e i y, i will no be gi en below.
The sum (18) o
ε
∈
ε
0 is equal o
() ()
000
00
xd dx S
εε ε ε ε
εε εε εε
∈∈∈
==


. (20)
Fo
ε
∈
ε
m, m > 0, epea ed in eg a ion by pa s leads o he ollowing esul :
()
()
()()
()
111
0
11
mm m
mjm
mmjj
j
xd x S
εε ε ε ε
εε εε εε
−−− +
∈∈= ∈

=− +−


 
. (21)
Summing o all he ci cui elemen s and ea anging he e ms yield he conse a i e quan i y (18)
in a compac o m:
()
() ( )
()
1
10
11
ii
mm m
ji
jij
jij i
x
S
εε ε
εε εε
−−
==∈ =∈

=− + −


 

. (22)
The gene alized cu en s x
ε
can be ep esen ed as linea combina ions o gene alized loop cu en s x.
U ilizing incidence ma ices, he addend o he inne summa ion o he i s e m in (22) will assume
he o m
() ( ) () ( )
11
nn
jij j ij
k
kk k
kk
ax x
εε ε
−−
==
=

(23)
whe e k
ε
means ei he ±
ε
o 0 depending on whe he he elemen
ε
is o is no a pa o he k- h loop,
and, possibly, depending on wha i s o ien a ion is wi h ega d o his loop. Subs i u ing (23) in o
(22), expanding he ou e se ies acco ding o index j, and subsequen ly ea anging he summa ion
yield
Figu e 2.
This ci cui consis s o elemen s o
ε0
o
εm
ypes, which a e loca ed on he common
Σ
-diagonal.
Cha ac e is ic qua e -ci cles a e cons uc ed o he (a) cu en and (b) ol age ep esen a ion o he
ci cui . Since he elemen s a e loca ed on he
Σ
-diagonal, hei dis ances om he hidden elemen a e
he same in bo h he αand βdi ec ions; hus, ∆εiα=∆εiβ=i.
Assume ha he Lag angian is egula , i.e.,
de 
∂2L
∂x(m)
i∂x(m)
j
,0, (10)
o i,j=1,...,n[19]. In Equa ion (8), o j=m eads as
mp=∂L
∂x(m). (11)
Then, he unambiguous ela ion,
x(m)=χ1q,. . . ,mq,mp, (12)
can be de i ed om (11), which enables a ansi ion o new coo dina es
1p
,..,
mp
and
1q
,...,
mq
and
allows us o d aw he canonic equa ions,
j.
q=∂H
∂jp,j.
p=−∂H
∂jq. (13)
The Hamil onian His hen a unc ion o he new a iables iq,ip,i=1, ... , m:
H1q,. . . ,mq,1p,. . . ,mp=1pT·2q+. . . +m−1pT·mq+mpT·χ1q,. . . ,mq,mp−L1q,. . . ,mq,mp(14)
whe e
L1q,. . . ,mq,mp=L1q,. . . ,mq,χ1q,. . . ,mq,mp. (15)
The p ocedu e o d awing an al e na i e Hamil onian o cases when condi ion (10) o he
egula i y o he Lag angian is no ul illed is desc ibed, o example, in [
19
]. I espec i e o whe he
he Lag angian is o is no egula , he Hamil onian can be used wi hin Lag ange’s o malism, i.e., in
he sense o gene alized ene gy (7).
En opy 2020,22, 412 5 o 20
This o malism, speci ic o highe -o de Hamil onians o ype (14), is deno ed Os og adsky’s
o malism [
19
]. None o he hi he o published wo ks deal wi h assembling he Hamil onian o sys ems
consis ing o elemen s om Chua’s able. The bene i s o using he Hamil onian ha e been p o en: he
sys em dynamics is gi en by a se o canonic i s -o de di e en ial equa ions, which a e he s anda d
s a ing poin o sol ing complex asks associa ed wi h e ms such as he Lyapuno exponen s, ze o
di e gence, conse a i e phase olumes [
20
], e c. Hamil onians a e an ideal ool when sea ching o
symme ies and associa ed conse a i e quan i ies [
21
]. Today, hey a e well de ined also o sys ems
ha ha e hi he o been conside ed non-conse a i e [22].
The objec i e o his wo k is he e o e o ind he Hamil onian o a
Σ
-sys em consis ing o a bi a y
HOEs wi hin he ame o bo h Lag ange’s o malism (as gene alized ene gy) and Os og adsky’s
o malism. The con en ional p ocedu e o in oducing Os og adsky’s o malism in o physical
heo y in ol es sea ching o he o m o he Lag angian and gene alized momen a, and subsequen ly
ob aining om hem a speci ic o m o he Hamil onian ((7) o (14)). The p ocedu e used in his wo k
will be physically mo e objec i e, s a ing om he gene alized Tellegen’s heo em. In addi ion o i s
cla i y, he eason o selec ing his app oach is p agma ic: Tellegen’s heo em is ex emely gene al and
applicable o ci cui s o a bi a y opologies wi h any elemen s (linea , nonlinea , ime-in a ian , and
ime- a ying), as well as wi h a bi a y elemen s om Chua’s able. Mo eo e , his heo em can also
be o mula ed ia he e minal quan i ies o indi idual elemen s. I is use ul o ci cui s con aining
HOEs, since he sole undamen al cha ac e is ic o he elemen — he cons i u i e ela ion—is a link
only be ween hese e minal quan i ies. Then, he esul ing Hamil onian will na u ally be comp ised
o he cons i u i e ela ions o all HOEs in he ci cui .
This pape has he ollowing s uc u e. Sec ion 2in oduces he gene alized o m o Tellegen’s
heo em applicable o ci cui s wi h HOEs and hus o ci cui s wi h he elemen s de ined ia he
cons i u i e ela ions (1). Sec ion 3is de o ed o d awing a speci ic o m o he Hamil onian o
Σ
-ci cui s in he ame o Lag ange’s o malism and also o Os og adsky’s o malism. Sec ion 4
desc ibes a modeling echnique inspi ed by Os og adsky’s o malism. In he las sec ion, hese new
pieces o knowledge a e applied o a speci ic opology o he Pais–Uhlenbeck oscilla o consis ing
o HOEs.
2. Tellegen‘s Theo em o Ci cui s wi h Highe -O de Elemen s
Conside a ci cui comp ised o a bi a y one-po elemen s
ε
wi h po ol ages
ε
and cu en s
iε. Then, he gene alized Tellegen’s heo em [23] holds o such a ci cui :
X
ε
(α)
εi(β)
ε= (α)
εT
·i(β)
ε=0 (16)
whe e
α
and
β
a e a bi a y in ege s,
(α)
ε
and
i(β)
ε
a e 1xb ec o s o gene alized ol ages and cu en s
o he elemen , and bis he numbe o elemen s in he ci cui . The classical o m o he heo em,
ep esen ing he case
α
=
β
=0, means ha he sum o ins an aneous powe s deli e ed o all elemen s
in he ci cui is ze o. The gene alized heo em (16) eplaces he ins an aneous powe [VA] by a quan i y
[VA]
α+β
. This is because he in eg a ion and di e en ia ion wi h espec o ime belong o Ki chho ’s
ope a o s [
19
], which do no a ec he alidi y o Ki chho ’s laws and he heo ems de i ed om hem.
No e ha he sou ce o he ol age
(α)
o cu en i
(β)
can be subs i u ed by an (
α
,
β
) elemen wi h a
cons an cons i u i e ela ion (1) o a () o g() ype, so heo em (16) also holds o ci cui s wi h ol age
and cu en sou ces.
Conside a ci cui comp ised o gene al HOEs. The elemen
εh
=(
αmax
,
βmin
) is he hidden
elemen o he cu en ep esen a ion o he ci cui [
24
]. Le us in oduce new a iables u=
(αmax)
and
x=i(βmin). Then, Equa ion (16) can be ew i en wi h he aid o Equa ion (1) in he o m
X
ε
(α+αmax−αε)
εx(β+βε−βmin)
ε=0 (17)

En opy 2020,22, 412 6 o 20
whe e
ε
and x
ε
a e he cons i u i e ela ion (1) and he x a iable o he elemen
ε
. Since Equa ion (16)
mus hold o a bi a y in ege s
α
and
β
, i also holds o
α
=0 and
β
=1+
βmin −βε
. Subs i u ing
hese alues in o mula (17), which is iden ically equal o ze o, and in eg a ing i wi h espec o ime,
he esul ing quan i y mus be cons an in all ci cums ances:
H=ZX
ε
(∆εα)
ε
.
xεd =cons (18)
whe e
∆εα
=
αmax −αε
is he dis ance o he
ε
elemen om he hidden elemen in he
α
di ec ion
(see Figu e 2a).
A simila p ocedu e leads o he dual o m o Equa ion (18) o ol age ep esen a ion o he ci cui :
H∗=ZX
ε
g(∆εβ)
ε
.
uεd =cons ∗(19)
whe e
∆εβ
=
βε−βmin
is he dis ance o he
ε
elemen om he hidden elemen
εh
=(
αmin
,
βmax
) in he
βdi ec ion (see Figu e 2b).
I will be shown in he ollowing sec ion ha he quan i y (18) o (19) is, o he case o
Σ
-ci cui s,
a Hamil onian o ype (7) and ha i becomes a Hamil onian o ype (14) a e ansi ioning o
coo dina es ha co espond o Os og adsky’s o malism.
3. F om Tellegen‘s Theo em o he Hamil onian
Le all he elemen s o he ci cui be loca ed on a common
Σ
-diagonal acco ding o Figu e 2.
Deno e such a ci cui as he
Σ
-ci cui . All ypes o elemen s will be speci ied as
ε0
o
εm
, whe e mis
he dis ance o he
εm
elemen om he hidden elemen in he
α
o
β
di ec ion (since he elemen is
loca ed on he
Σ
-diagonal, hese dis ances a e he same, hus
∆εα
=
∆εβ
=
∆ε
). The me hod o indexing
depends on he posi ion o he hidden elemen , and his posi ion depends on he choice be ween he
cu en , (), and he ol age, g(), ep esen a ion (1) o he elemen . The ela ion
ε∈εi
,i=0,...,mwill hus
speci y ha he gi en εelemen is only o he εi ype.
Le us ocus on he p oo ha he o mula (18) o he cu en ep esen a ion o elemen s
(
see Figu e 2a
) is a Hamil onian. The p oo o he o mula (19), which co esponds o he ol age
ep esen a ion, is analogous, so, o he sake o b e i y, i will no be gi en below.
The sum (18) o ε∈ε0is equal o
X
ε∈ε0Z (0)
ε
.
xεd =X
ε∈ε0Z (0)
εdxε=X
ε∈ε0
Sε. (20)
Fo ε∈εm,m>0, epea ed in eg a ion by pa s leads o he ollowing esul :
X
ε∈εmZ (m)
ε
.
xεd =X
ε∈εm
m−1
X
j=0
(−1)j (m−1−j)
εx(j+1)
ε
+X
ε∈εm
(−1)mSε. (21)
Summing o all he ci cui elemen s and ea anging he e ms yield he conse a i e quan i y
(18) in a compac o m:
H=
m
X
j=1
(−1)j−1
m
X
i=jX
ε∈εi
x(j)
ε (i−j)
ε
+
m
X
i=0X
ε∈εi
(−1)iSε. (22)
En opy 2020,22, 412 7 o 20
The gene alized cu en s x
ε
can be ep esen ed as linea combina ions o gene alized loop
cu en s x. U ilizing incidence ma ices, he addend o he inne summa ion o he i s e m in (22)
will assume he o m n
X
k=1
aεkx(j)
k (i−j)
ε=
n
X
k=1
x(j)
k
k (i−j)
ε(23)
whe e
k
ε
means ei he
±
ε
o 0 depending on whe he he elemen
ε
is o is no a pa o he k- h loop,
and, possibly, depending on wha i s o ien a ion is wi h ega d o his loop. Subs i u ing (23) in o (22),
expanding he ou e se ies acco ding o index j, and subsequen ly ea anging he summa ion yield
H=n
P
k=1

x(1)
k
m
X
i=1X
ε∈εik (i−1)
ε
| {z }
1pk
+x(2)
k
m
X
i=2X
ε∈εi−k (i−2)
ε
| {z }
2pk
+. . . +x(m)
kX
ε∈εm(−1)m−1k (0)
ε
| {z }
mpk

−
m
X
i=0X
ε∈εi
(−1)iSε
| {z }
L
(24)
Since i holds ha ∂L
∂xk
=(−1)k+1k ε, (25)
compa ing (24) and (7) e eals ha he unc ion
H
om Equa ion (24) is he gene alized ene gy (7) o
he cu en ep esen a ion o a ci cui wi h HOEs, and he gene alized momen a a e
jpk=(−1)j−1
m
X
i=jX
ε∈εik (i−j)
ε. (26)
The one-dimensional case o n=1 signi ies ha all he elemen s a e in se ies, wi h he common
gene alized cu en x. The se ies connec ion o he elemen s o he same ype can, he e o e, be ega ded
as one elemen o he same ype, wi h he cons i u i e ela ion gi en as he sum o cons i u i e ela ions
o he indi idual elemen s. Equa ion (26) can, he e o e, be ew i en in he simpli ied o m
jp=(−1)j−1
m
X
i=j
(i−j)
ix(i)(27)
whe e
i
() deno es he cons i u i e ela ion o an elemen o he
εi
ype. The gene alized momen a a e
hen gi en as summa ions o he ime de i a i es o he cons i u i e ela ions
1p= (0)
1+ (1)
2+ + (m−1)
m
1p= (0)
1+ (1)
2+ + (m−1)
m
.
.
.
mp=(−1)m−1 (0)
m.
(28)
This one-dimensional case clea ly illus a es he ansi ion om he o mula ion o he
Hamil onian (7) o he o mula ion (14) by changing he coo dina es (x
(0)
,..., x
(m)
) o Lag ange’s
o malism o new coo dina es (
1
p,...,
m
p,
1
q,...,
m
q) o Os og adsky’s o malism. The key ela ion
χ
()
om (12) can be ob ained ia a simple in e sion o he cons i u i e ela ion o he elemen o he
εm ype:
mp=(−1)m−1 mx(m)⇒x(m)=gm(−1)m−1mp(29)
whe e gm() is he cons i u i e ela ion o he ol age ep esen a ion o he elemen o he εm ype.
En opy 2020,22, 412 8 o 20
4. Modeling o he Σ-Ci cui s
Os og adsky’s o malism leads o a sys em o canonic i s -o de equa ions (13). This ac
acili a es he modeling p ocess. I ollows om (28) ha he momen a o a one-dimensional sys em a e
1.
p=− 01q
2.
p=−1p+ 12q
3.
p=2p+ 23q
.
.
.
m.
p=(−1)m−1m−1p+ m−1(mq).
(30)
Simila ly, he gene alized coo dina es a e
1.
q=2q
2.
q=3q
.
.
.
m−1.
q=mq
m.
q=gm(−1)mmp.
(31)
The sys ems o Equa ions (30) and (31) lead o he elegan p og amming diag am in Figu e 3,
consis ing o 2xmin eg a o s o compu ing mgene alized momen a and mgene alized coo dina es, as
well as m+1 unc ion blocks o modeling he cons i u i e ela ions o he indi idual elemen s.
En opy 2020, 21, x 8 o 20
()
32 3
2
pp q=+


()
()
11
1
1m
mmm
m
pp q
−−
−
=− +
.
Simila ly, he gene alized coo dina es a e
12
qq=

23
qq=


1mm
qq
−=

()
()
1m
mm
m
qg p=−
.
(31)
The sys ems o Equa ions (30) and (31) lead o he elegan p og amming diag am in Figu e 3,
consis ing o 2xm in eg a o s o compu ing m gene alized momen a and m gene alized coo dina es,
as well as m+1 unc ion blocks o modeling he cons i u i e ela ions o he indi idual elemen s.
     
m-1
(⋅)
2
(⋅)
1
(⋅)
0
(⋅)
g
m
(⋅)
… …
++ +
Figu e 3. Model o he Σ-sys em acco ding o Os og adsky’s o malism. The colo -codes ep esen
he m in eg a o s o compu ing he iq coo dina es and he m in eg a o s o compu ing he ip
momen a. The unc ion blocks model he cons i u i e ela ions o indi idual elemen s. The gm() block
is o modeling he in e se unc ion o m() acco ding o (29), which enables he ansi ion om
Lag ange’s o Os og adsky’s o malism.
5. Applica ion: Non-linea Pais–Uhlenbeck Oscilla o
The use ulness o he Hamil onian o he analysis o HOE ci cui s can be demons a ed by he
example o he Pais–Uhlenbeck (PU) oscilla o [15]. The PU oscilla o is equen ly employed o es
new physical heo ies. I gene a es a signal wi h wo ha monic componen s whose ampli udes and
ini ial phases a e gi en by he ini ial condi ions. Bo h equencies a e coupled ia he o mula [25]
114
2
ωω
±−
=
 (32)
whe e ϵ is a eal numbe , 0 ≤ ϵ < ¼. Fo ϵ → 0, he PU oscilla o changes o a classical sinusoidal
oscilla o wi h an oscilla ion equency o
ω
.
The PU oscilla o is go e ned by he di e en ial equa ion
2
20xx x
ω
ω
++ =
 
. (33)
Figu e 3.
Model o he
Σ
-sys em acco ding o Os og adsky’s o malism. The colo -codes ep esen he
min eg a o s o compu ing he
i
qcoo dina es and he min eg a o s o compu ing he
i
pmomen a.
The unc ion blocks model he cons i u i e ela ions o indi idual elemen s. The g
m
() block is o
modeling he in e se unc ion o
m
() acco ding o (29), which enables he ansi ion om Lag ange’s o
Os og adsky’s o malism.
5. Applica ion: Non-Linea Pais–Uhlenbeck Oscilla o
The use ulness o he Hamil onian o he analysis o HOE ci cui s can be demons a ed by he
example o he Pais–Uhlenbeck (PU) oscilla o [
15
]. The PU oscilla o is equen ly employed o es
new physical heo ies. I gene a es a signal wi h wo ha monic componen s whose ampli udes and
ini ial phases a e gi en by he ini ial condi ions. Bo h equencies a e coupled ia he o mula [25]
ω±=ωs1∓√1−4
2(32)
En opy 2020,22, 412 9 o 20
whe e

is a eal numbe , 0
≤
<
1
4
. Fo
→
0, he PU oscilla o changes o a classical sinusoidal
oscilla o wi h an oscilla ion equency o ω.
The PU oscilla o is go e ned by he di e en ial equa ion

ω2
....
x+..
x+ω2x=0. (33)
In gene al, he signal xgene a ed by he oscilla o is non-pe iodical. Pe iodici y is achie ed in
special cases when he a io o bo h equencies is a a ional numbe . The co esponding alues o he
cons an a e as ollows:
=1
41− 1−k2
1+k2!2,k=ω+
ω−=
s<1 (34)
whe e ,sa e in ege s. Only in hese cases will he s a e-space ajec o ies be closed cu es.
I is shown in [
16
] ha he PU oscilla o , modeled by Equa ion (33), can be implemen ed as a
se ies o pa allel
Σ
-ci cui consis ing o an a bi a y iad o HOEs, which a e immedia e neighbo s
on an a bi a y
Σ
-diagonal. One possible combina ion is shown in he inse in Figu e 4a, namely, he
se ies connec ion o he FDNC,R, and FDNR elemen s wi h he cons i u i e ela ions
0
(),
1
(), and
2
().
In gene al, hese ela ions can be nonlinea . The indices 0 o 2 co espond o he dis ance o he elemen
om he hidden elemen in he
α
o
β
di ec ion. The hidden elemen is he FDNC. The gene alized
cu en xis go e ned by he di e en ial equa ion
..
2..
x+.
1.
x+ 0(x)=0 (35)
I all h ee elemen s ha e linea cons i u i e ela ions,
0(x)=ω2x, 1.
x=.
x, 2..
x=
ω2
..
x, (36)
hen he gene alized cu en will be modeled ia he linea di e en ial Equa ion (33).
The subsequen analysis is made o linea FDNC and Rand nonlinea FDNR elemen s wi h he
ollowing cons i u i e ela ions:
0(x)=k0x, 1.
x=k1
.
x, 2..
x=k2a c an..
x. (37)
The Lag angian o he ci cui is
Lx,.
x,..
x=−1
2k0x2+1
2k1
.
x2−1
2k22..
xa c an..
x−ln1+..
x2. (38)
The equa ion o mo ion, gene a ed by he Lag angian acco ding o (4), akes he o m
k0x+k1
..
x+k2
....
x1+..
x−...
x2
1+..
x2=0. (39)
Acco ding o (28), he gene alized momen a a e
1p=k1
.
x+k2
...
x
1+..
x2
2p=−k2a c an..
x.(40)
Fo Lag ange’s o malism, he Hamil onian (7) will be in he o m
Hx,.
x,..
x,...
x=1
2k0x2+1
2k1
.
x2+k2
.
x...
x
1+..
x2−1
2k2ln1+..
x2. (41)
En opy 2020,22, 412 16 o 20
En opy 2020, 21, x 16 o 20
0 20 40 60 80 100
-1.0
-0.5
0.0
0.5
1.0
1.5
Hamil onian = cons
Lag angian
Time [s]
L, H
[Js]
165 nJs
Figu e 11. Lag angian and Hamil onian s. ime.
-15m -10m -5m 0m 5m 10m 15m
-5m
0m
5m
-5m 0m 5m
-4m
0m
4m
-4m 0m 4m
-4m
-2m
0m
2m
4m
x
(2)
[As]
x
(1)
[As
2
]
x
(0)
[As
3
]
x
(3)
[A]
x
(2)
[As]
x
(1)
[As
2
]
Figu e 11. Lag angian and Hamil onian s. ime.
En opy 2020, 21, x 16 o 20
0 20 40 60 80 100
-1.0
-0.5
0.0
0.5
1.0
1.5
Hamil onian = cons
Lag angian
Time [s]
L, H
[Js]
165 nJs
Figu e 11. Lag angian and Hamil onian s. ime.
-15m -10m -5m 0m 5m 10m 15m
-5m
0m
5m
-5m 0m 5m
-4m
0m
4m
-4m 0m 4m
-4m
-2m
0m
2m
4m
x
(2)
[As]
x
(1)
[As
2
]
x
(0)
[As
3
]
x
(3)
[A]
x
(2)
[As]
x
(1)
[As
2
]
Figu e 12.
Phase ajec o ies o he PU oscilla o wi h he HP mem is o in he coo dina es acco ding o
Lag ange’s o malism.

En opy 2020,22, 412 17 o 20
En opy 2020, 21, x 17 o 20
Figu e 12. Phase ajec o ies o he PU oscilla o wi h he HP mem is o in he coo dina es acco ding
o Lag ange’s o malism.
-15m -10m -5m 0m 5m 10m 15m
-120
0
120
-5m -3m 0m 2m 5m
-150
0
150
-120 -60 0 60 120
-150
0
150
1
p [V]
2
q [As
2
]
1
q [As
3
]
1
p
[V]
2
p
[Vs]
2
p
[Vs]
Figu e 13. Phase ajec o ies o he PU oscilla o wi h he HP mem is o in he coo dina es acco ding
o Os og adsky’s o malism.
The phase ajec o ies in Figu e 12 and 13 ep esen he con ou s o he gene alized ene gy (47)
and Hamil onian (48). They can be discussed he same way as he ajec o ies in Figs. 8 and 9.
6. Discussion
Hamil on’s a ia ional p inciple, i.e., ha he ajec o y is he ex emal o Lag ange’s unc ion,
holds o he Σ-ci cui s [16]. This wo k o e s new knowledge, namely ha a Hamil onian can be
cons uc ed o Σ-ci cui s and ha his Hamil onian p ese es i s ixed alue du ing mo ion. I s
physical dimension is gi en by he numbe Σ o he diagonal, and i s uni is [VAs—Σ]. The
Hamil onian o a ci cui consis ing o gene ally nonlinea induc o s and capaci o s, he e o e,
ep esen s he ene gy because bo h he induc o and capaci o a e loca ed on he diagonal wi h he
numbe Σ = −1. The physical cha ac e o he Hamil onian does no change e en i o he ypes o
elemen s om his diagonal appea in he ci cui .
The wo k in [9] was he beginning o his ype o esea ch. This s udy in oduced he s a e
unc ions o mem is o s, memcapaci o s, and meminduc o s and cons uc ed a Lag angian om
hose unc ions. The esea ch in [11,12,16], which appea ed la e , in oduced he condi ions o
alida ing Hamil on’s a ia ional p inciple, building up Hamil on’s o malism, and gene alizing all
hese undamen al pieces o knowledge o gene al highe -o de elemen s. I was demons a ed in
[16] ha he highe -o de Lag angian is “na i e” o gene al HOEs. Ou wo k o e s ano he insigh :
Fo his Lag angian we cons uc he co esponding Hamil onian, which is “na i e” o ci cui s wi h
HOEs.
In o de o de i e bene i s om Hamil on’s o malism in ci cui s wi h HOEs, we should use
Os og adsky’s o malism because only he highe -o de Hamil onian is na i e o such ci cui s. One
Figu e 13.
Phase ajec o ies o he PU oscilla o wi h he HP mem is o in he coo dina es acco ding o
Os og adsky’s o malism.
6. Discussion
Hamil on’s a ia ional p inciple, i.e., ha he ajec o y is he ex emal o Lag ange’s unc ion,
holds o he
Σ
-ci cui s [
16
]. This wo k o e s new knowledge, namely ha a Hamil onian can be
cons uc ed o
Σ
-ci cui s and ha his Hamil onian p ese es i s ixed alue du ing mo ion. I s
physical dimension is gi en by he numbe
Σ
o he diagonal, and i s uni is [VAs
—Σ
]. The Hamil onian
o a ci cui consis ing o gene ally nonlinea induc o s and capaci o s, he e o e, ep esen s he
ene gy because bo h he induc o and capaci o a e loca ed on he diagonal wi h he numbe
Σ
=
−
1.
The physical cha ac e o he Hamil onian does no change e en i o he ypes o elemen s om his
diagonal appea in he ci cui .
The wo k in [
9
] was he beginning o his ype o esea ch. This s udy in oduced he s a e unc ions
o mem is o s, memcapaci o s, and meminduc o s and cons uc ed a Lag angian om hose unc ions.
The esea ch in [
11
,
12
,
16
], which appea ed la e , in oduced he condi ions o alida ing Hamil on’s
a ia ional p inciple, building up Hamil on’s o malism, and gene alizing all hese undamen al pieces
o knowledge o gene al highe -o de elemen s. I was demons a ed in [
16
] ha he highe -o de
Lag angian is “na i e” o gene al HOEs. Ou wo k o e s ano he insigh : Fo his Lag angian we
cons uc he co esponding Hamil onian, which is “na i e” o ci cui s wi h HOEs.
In o de o de i e bene i s om Hamil on’s o malism in ci cui s wi h HOEs, we should use
Os og adsky’s o malism because only he highe -o de Hamil onian is na i e o such ci cui s. One o
i s ad an ages is he abili y o cons uc com o able analyses o he p ese ed quan i ies when inding
pe iodical s eady s a es o he condi ions o he occu ence o sel -oscilla ion, as demons a ed by he
example o he PU oscilla o . The syn hesis o his ci cui ia nonlinea HOEs leads o a nonlinea
oscilla o , which has no been hi he o desc ibed in he li e a u e. The Hamil onian o he PU oscilla o
consis ing o he FDNR,R, and FDNC elemen s, acco ding o Figu e 4a, ep esen s he powe because
En opy 2020,22, 412 18 o 20
hese elemen s a e loca ed on he diagonal wi h
Σ
=0. The Hamil onian o he PU oscilla o consis ing
o MR,linea FDNC, and linea FDPC elemen s, acco ding o Figu e 4b, ep esen s he ac ion (in eg al o
ene gy) because hese elemen s a e loca ed on he diagonal wi h
Σ
=
−
2. This quan i y is also p ese ed
in he sys em in spi e o he ac ha he sys em is dissipa i e. Quod no a, he pa adox is only illuso y
because wo o he h ee elemen s o hese speci ic ci cui s a e, in p inciple, ac i e elemen s, namely he
FDNR and FDNC o he linea FDNC and linea FDPC elemen s.
The Hamil onian o he ci cui , con aining only he HOEs om one diagonal, can be cons uc ed
ia Equa ion (24). Equa ion (24) ep esen s he known s uc u e o a highe -o de Hamil onian,
in which he gene alized coo dina es, gene alized momen a, and highe -o de Lag angian appea .
The Lag angian is se up ia he cons i u i e ela ions o he indi idual elemen s. The indi idual
momen a a e a anged acco ding o (24) as sums o he gene alized ol ages o he co esponding
o de s ac oss selec ed elemen s o he ci cui . The Lag angian is gi en as a sum o he s a e unc ions
o he indi idual elemen s p o ided wi h he app op ia e sign. The s a e unc ion o he elemen
co esponds o he a ea below i s cons i u i e ela ion.
I has been newly ound ha he s a e unc ion o he well-known HP mem is o wi h he Jogleka
window unc ion o p=1 is a composi ion o he diloga i hm and he exponen ial unc ion.
The modeling diag am o he one-dimensional
Σ
-ci cui om Figu e 3can be gene alized in a
classical way o a mul i-dimensional sys em (e.g., a ci cui wi h se e al loops), which can be used o
implemen a ion in he co esponding simula ion p og am. Fo simula ion, i is desi able o selec a
me hod o nume ical in eg a ion ha is sui able o he analysis o he se s o Hamil on’s di e en ial
equa ions [
27
]. The wa e o m o he Hamil onian mus exhibi negligible de lec ions om he cons an
alue. This c i e ion p o ides a good eedback when seeking o an op imum con igu a ion o he
simula ion ask and pa ame e s, including he ime s ep.
The heo e ical appa a us buil in his wo k can also be used o ci cui s conis ing only o wo ypes
o HOEs immedia ely neighbo ing each o he on he common
Σ
-diagonal. Howe e , he Hamil onian
o such a ci cui is o a classical no a highe -o de ype. A ypical example is a ne wo k o mu ually
in e connec ed memcapaci o s and meminduc o s.
Fo esea che s dealing wi h conc e e mem is i e de ices, o example he TiO
x
, TaO
x
, SiO
x
,
H O
x
and o he ypes, i will be use ul o de e mine whe he he p esen ed Hamil on’s o malism o
highe -o de elemen s can also be applied o hese de ices.
I ollows om he essence o he gi en o malism ha his o malism can be used only o ci cui s
ha a e made up o wo- e minal HOEs, whe eas all he HOEs om he ci cui mus be loca ed on he
common
Σ
-diagonal o Chua’s able. The ques ion is whe he he models o hese mem is i e sys ems
can be buil om he abo e HOEs and whe he all he o he elemen s om he ci cui a e also loca ed
on he common diagonal.
His o ically, he i s model o he HP mem is o wi h a simple window unc ion [
7
,
28
] o modeling
he nonlinea dopan d i can be classi ied as an ideal gene ic mem is o . Since his mem is o is
equi alen o an ideal mem is o , i can be conside ed an HOE o he (
−
1,
−
1) ype, which complies
wi h Hamil on’s o malism. Howe e , he co esponding applica ion ci cui mus con ain, in addi ion
o his mem is o , only he o he app op ia e HOEs om he
Σ
=
−
2 diagonal— o example, (
−
2, 0),
(
0, −2
) and o he elemen s. I some o hese elemen s a e linea , hen hey can be mo ed in Chua’s
able along he co esponding
∆
-diagonals [
31
]. Then, he se o hese admissible linea elemen s will
g ow, as is illus a ed in Sec ion 5.
Howe e , he complex physical models o he abo e men ioned mem is i e de ices, such as he
Tunneling ba ie model, TEAM, and o he s [
32
], a e classi ied as ex ended mem is o s, which canno be
gene ally buil om wo- e minal HOEs. Thus, a emp s o u ilize he p esen ed Hamil on’s o malism
o such sys ems in e e e wi h he p incipal limi s. When looking o a Hamil on’s o malism sui able
o hese elemen s, i will be necessa y o use o he me hods, such as hose men ioned, o example,
in [21,22].
En opy 2020,22, 412 19 o 20
Au ho Con ibu ions:
Concep ualiza ion, Z.B., D.B., V.B., and Z.K.; me hodology, D.B.; w i ing—o iginal d a
p epa a ion, Z.B.; w i ing— e iew and edi ing, Z.B., D.B., V.B., and Z.K.; p ojec adminis a ion and unding
acquisi ion, V.B. and Z.K. All au ho s ha e ead and app o ed he inal manusc ip .
Funding:
This wo k was suppo ed by he Czech Science Founda ion unde g an no. 18-21608S. Fo he esea ch,
he in as uc u e o K217 Depa men , UD B no, was also used. The APC was unded by he Open Access Fund
o B no Uni e si y o Technology.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1. Fuchs, A. Nonlinea Dynamics in Complex Sys ems; Sp inge : Be lin/Heidelbe g, Ge many, 2013. [C ossRe ]
2.
Chua, L.O. CNN: A Pa adigm o Complexi y; Wo ld Scien i ic Se ies on Nonlinea Science Se ies A;
Wo ld Scien i ic: Singapo e, 1998; Volume 31.
3. Ca a elli, F.; T a e sa, F.L.; Di Ven a, M. Complex dynamics o mem is i e ci cui s: Analy ical esul s and
uni e sal slow elaxa ion. Phys. Re . 2017,E95, 022140. [C ossRe ]
4.
Di Ven a, M.; T a e sa, F.L. Absence o chaos in digi al memcompu ing machines wi h solu ions.
Phys. Le . A
2017,381, 3255–3257. [C ossRe ]
5.
Chua, L.O. De ice modeling ia nonlinea ci cui elemen s. IEEE T ans. Ci cui s Sys .
1980
,27, 1014–1044.
[C ossRe ]
6.
Chua, L.O. Mem is o —The missing ci cui elemen . IEEE T ans. Ci cui Theo y
1971
,18, 507–519. [C ossRe ]
7.
S uko , D.B.; Snide , G.S.; S ewa , D.R.; Williams, R.S. The missing mem is o ound. Na u e
2008
,453,
80–83. [C ossRe ] [PubMed]
8.
Biolek, D.; Biolek, Z.; Biolko
á
, V. E e y nonlinea elemen om Chua’s able can gene a e pinched hys e esis
loops: Gene alised homo he y heo em. Elec on. Le . 2016,52, 1744–1746. [C ossRe ]
9.
Jel sema, D. Memo y Elemen s: A Pa adigm Shi in Lag angian Modeling o Elec ical Ci cui s.
IFAC P oc. Vol. 2012,45, 445–450. [C ossRe ]
10.
Cohen, G.Z.; Pe shin, Y.V.; Di Ven a, M. Lag ange Fo malism o Memo y Ci cui Elemen s: Classical and
Quan um Fo mula ions. Phys. Re . B 2012,85, 165428–165430. [C ossRe ]
11.
Biolek, Z.; Biolek, D.; Biolko
á
, V. U iliza ion o Eule -Lag ange Equa ions in Ci cui s wi h Memo y Elemen s.
Radioenginee ing 2016,25, 783–789. [C ossRe ]
12.
Biolek, Z.; Biolek, D.; Biolko
á
, V. Eule -Lag ange Equa ions o Ne wo ks wi h Highe -O de Elemen s.
Radioenginee ing 2017,26, 397–405. [C ossRe ]
13.
Milla , W. Some gene al heo ems o nonlinea sys ems possessing esis ance. Philos. Mag.
1951
,42,
1150–1160. [C ossRe ]
14.
Che y, E.C. Some gene al heo ems o nonlinea sys ems possessing eac ance. Philos. Mag.
1951
,42,
1161–1177. [C ossRe ]
15.
Pais, A.; Uhlenbeck, G.E. On Field Theo ies wi h Non-Localized Ac ion. Phys. Re .
1950
,79, 145–165.
[C ossRe ]
16.
Biolek, D.; Biolek, Z.; Biolko
á
, V. Lag angian o Ci cui s wi h Highe -O de Elemen s. En opy
2019
,21,
1059. [C ossRe ]
17.
Os og adsky, M. M
é
moi e su les de
é
en ielles ela i es aux p oblemes des isop
é
ime es. Mem. Acad. Sci.
S . Pe e sb. 1850,6, 385–517.
18.
Lanczos, C. The Va ia ional P inciples o Mechanics, 4 h ed.; Do e Books on Physics (Book 4); Do e Publica ions:
Mineola, NY, USA, 1986.
19.
Kne e , C.G. E ec i e Lag angians wi h highe de i a i es and equa ions o mo ion. Phys. Re . D
1994
,49,
6709. [C ossRe ]
20.
Deng, Y.; Li, Y. A mem is i e conse a i e chao ic ci cui consis ing o a mem is o and capaci o . Chaos
2020
,
30, 013120. [C ossRe ]
21. I oh, M.; Chua, L.O. Mem is o Hamil onian Ci cui s. In . J. Bi . Chaos 2011,21, 2395–24255. [C ossRe ]
22.
I oh, M.; Chua, L.O. Dynamics o Hamil onian Sys ems and Mem is o Ci cui s. In . J. Bi . Chaos
2017
,27,
1730005. [C ossRe ]
23.
Pen ield, P.; Spence, R.; Duinke , S. A gene alized o m o Tellegen’s heo em. IEEE T ans. Ci cui Theo y
1970,17, 302–305. [C ossRe ]
En opy 2020,22, 412 20 o 20
24.
Biolek, Z.; Biolek, D.; Biolko
á
, V.; Kolka, Z. Taxicab Geome y in Table o Highe -O de Elemen s.
Nonlinea Dyn. 2019,98, 623–636. [C ossRe ]
25. Wooda d, R.P. The Theo em o Os og adsky. a Xi 2015, a Xi :1506.02210.
26.
Soliman, A.M. Realiza ions o ideal FDNC and FDNR elemen s using new ypes o mu a o s. In . J. Elec on.
1978,44, 317–323. [C ossRe ]
27. Ba bu, A.; Zhu, S.-C. Mon e Ca lo Me hods, 4 h ed.; Sp inge : Singapo e, 2020; p. 422. [C ossRe ]
28.
Jogleka , Y.N.; Wol , S.J. The elusi e mem is o : P ope ies o basic elec ical ci cui s. Eu . J. Phys.
2009
,30,
661–675. [C ossRe ]
29.
Biolek, Z.; Biolek, D. Fou h Fundamen al Ci cui Elemen : SPICE Modeling and Simula ion. In Mem is o
and Mem is i e Sys ems; Sp inge : New Yo k, NY, USA, 2014; pp. 105–162. [C ossRe ]
30. Lewin, L. Diloga i hms and Associa ed Func ions, 1s ed.; MacDonald: London, UK, 1958; p. 353.
31.
Biolek, D.; Biolek, Z.; Biolko
á
, V. Duali y o Complex Sys ems Buil om Highe -O de Elemen s. Complexi y
2018,2018, 15. [C ossRe ]
32.
Ascoli, A.; Co in o, F.; Senge , V.; Te zla , R. Mem is o Model Compa ison. IEEE Ci cui s Sys . Mag.
2013
,
13, 89–105. [C ossRe ]
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