a Xi :1811.04862 1 [ma h.DS] 12 No 2018
1
BIFURCATION SET FOR A DISREGARDED
BOGDANOV-TAKENS UNFOLDING. APPLICATION TO 3D
CUBIC MEMRISTOR OSCILLATORS.
ANDR´
ES AMADOR1AND EMILIO FREIRE2AND ENRIQUE PONCE3
Abs ac . We de i e he bi u ca ion se o a no p e iously conside ed h ee-
pa ame ic Bogdano -Takens un olding, showing ha i is possible exp ess i s
ec o ield as wo di e en pe u bed cubic Hamil onians. By using se e al i s -
o de Melniko unc ions, we ob ain o he i s ime analy ical app oxima ions
o he bi u ca ion cu es co esponding o homoclinic and he e oclinic connec-
ions, which along wi h he cu es associa ed o local bi u ca ions o ganize he
pa ame ic egions wi h di e en s uc u es o pe iodic o bi s.
As an applica ion o hese esul s, we s udy a amily o 3D mem is o oscilla-
o s, o which he cha ac e is ic unc ion o he mem is o is a cubic polynomial.
We show ha hese sys ems ha e an in ini y numbe o in a ian mani olds, and
by adding one pa ame e ha s a i ies he 3D dynamics o he amily, i is shown
ha he dynamics in each s a um is opologically equi alen o a ep esen an o
he abo e un olding. Also, based upon he bi u ca ion se ob ained, we show he
exis ence o closed su aces in he 3D s a e space which a e olia ed by pe iodic
o bi s. Finally, we cla i y some misconcep ions ha a ise om he nume ical sim-
ula ions o hese sys ems, emphasizing he impo an ole played by he exis ence
o in a ian mani olds.
Bi u ca ion se , Bogdano -Takens, homoclinic o bi , he e oclinic connec ion, Mel-
niko unc ion, Mem is o oscilla o s
1. In oduc ion
In plana sys ems, he exis ence o some local bi u ca ions may e eal he p esence
o o he bi u ca ions o global cha ac e [13, 14] and he cu es ha de e mine hese
global phenomena a e di icul o de e mine. This is, o ins ance, he case ega ding
he appea ance o homoclinic o he e oclinic connec ions.
A homoclinic connec ion is an o bi o he sys em ha joins a saddle equilib ium
poin o i sel , and gene ally c ea es o des oys pe iodic o bi s (see, o ins ance
[35]). A he e oclinic connec ion joins wo di e en equilib ium poin s o a sys em
and he exis ence o his connec ion can de e mine changes in he basin o a ac ion
o a posi i ely in a ian se .
Following [18], he echniques o s udy homoclinic o bi s in plana ec o ields
we e well de eloped du ing he 1920s in he wo ks o Dulac. The undamen al idea
is ha he ecu en beha io nea a connec ing o bi should be s udied in a ashion
simila o ha used in s udying pe iodic o bi s ia a Poinca ´e e u n map. Bu
he e a e some addi ional complica ions in he s udy o homoclinic o bi s compa ed
o ha o pe iodic o bi s which signi ican ly complica e he analysis.
1Facul ad de Ingenie ´ıa y Ciencias, Depa amen o de Ciencias Na u ales y Ma em´a icas, Pon i-
icia Uni e sidad Ja e iana-Cali, Cali, Colombia.
2,3Depa amen o de Ma em´a ica Aplicada, Escuela T´ecnica Supe io de Ingenie ´ıa, A da. de los
Descub imien os, 41092 Se illa, Spain.
1a amado @ja e ianacali.edu.co, 2e [email protected] 3eponce[email p o ec ed]
1
2 A. AMADOR AND E. FREIRE AND E. PONCE
Usually, he bi u ca ion cu es o homoclinic and he e oclinic connec ions a e
s udied by nume ical con inua ion echniques [17, 18, 9, 36]. On he o he hand,
when a plana sys em can be w i en as a pe u bed Hamil onian sys em, we can
calcula e some Melniko unc ions, in oduced by Melniko in [28], and unde ce ain
hypo heses, he ze os o he associa ed Melniko unc ion de e mine he exis ence
o pe iodic o bi s, homoclinic loops o he e oclinic connec ions, see o ins ance
[8, 20, 3].
As show la e , we will eso o such Melniko unc ions o se ing in o ma ion
on such global bi u ca ions cu es in a wo-pa ame ic plane o a speci ic amily o
di e en ial sys ems.
The no mal o m o he Bogdano -Takens (see [35]) bi u ca ion is gi en by
˙x=y, ˙y=µ1+µ2x+x2±xy.
Following he classi ica ion p oposed in [14], he de o ma ion o codimension h ee
o he p e ious no mal o m is gi en by he un olding
˙x=y,
˙y=µ1+µ2x+αx3+y(µ3+µ4x±x2),
(1)
The p esence o his ype o sys ems has been epo ed in di e en applica ions,
see [16, 2, 24]. On he s udy o bi u ca ion phenomena in hese sys ems many
con ibu ions ha e been made. In [15], he au ho s s udied he global bi u ca ion
diag am o he h ee-pa ame e amily
˙x=y,
˙y=µ1+µ2x−x3+y(µ3−3x2),
and ixing µ3>0, hey ob ained analy ical app oxima ions o he bi u ca ion cu es
o he homoclinic o bi s, by using Melniko unc ions. La e on, he abo e wo k was
quo ed in [23], whe e a nume ical analysis o he same model was pe o med. In
[10, 12, 11], i is conside ed he sys em
˙x=y,
˙y=µ1+µ2x−x3+y(µ3+µ4x−x2),
(2)
and he au ho s showed ha i can be w i en as a pe u bed Hamil onian sys em,
epo ing he maximum numbe o limi cycles. La e , by aking he pa ame e
µ4= 0 in (2), he au ho s in [4, 5, 7, 6] analyzed he sys em as a Li´ena d sys em, i s
local bi u ca ions we e cha ac e ized, and a nume ical s udy o he global bi u ca ions
was done.
While all he abo e e e ences deal wi h he ocus case, in his wo k we s udy
he saddle case
˙x=y,
˙y=µ1+µ2x+x3+y(µ3−3x2),
(3)
which up o he bes o ou knowledge, seems o be a dis ega ded case wi h a he
in e es ing dynamic beha io .
In ac , ou mo i a ion comes om he analysis o ce ain 3D mem is o oscilla o s
[1, 30, 22], whe e unde speci ic hypo heses on he mem is o cha ac e is ics, such
sys em appea s in a na u al way a e a dimensional educ ion achie ed hanks o
he exis ence o a i s in eg al.
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 3
The pape is o ganized in he ollowing way. Fi s , in sec ion 2 we e iew he
in o ma ion ha can be gained by means o a local analysis o he sys em. Ou
main esul s appea in sec ion 3, whe e we apply Melniko heo y o app oxima e
he homoclinic and he e oclinic cu es on a con enien pa ame e plane. We ob-
ain he global bi u ca ion se o sys em (3), by spli ing such a plane in egions
wi h di e en quali a i e dynamical beha io . Nex , in sec ion 4, We show how he
abo e analysis is use ul o de i ing all he possible esponses o ce ain 3D canonical
mem is o oscilla o s, when he lux-cha ge cha ac e is ics unc ion is a speci ic cubic
polynomial, gene alizing some esul s gi en in [1]. As one o he possible dynamical
beha io s, we ocus o a en ion in showing, hanks o he p e ious analysis, he
exis ence o a opological sphe e in he 3D phase-space comple ely olia ed by pe i-
odic o bi s. Thus, we con i m p e ious nume ical esul s epo ed in [29, 25]. The
necessi y o inco po a ing igo ous echniques in he analysis o mem is o oscilla o s
is emphasized wi h he ma e ial o sec ion 5, whe e ollowing a simila p ocedu e
o he dimensional educ ion o sec ion 4, we can e u e se e al ecen ly published
s udies ha epo he exis ence o an in ini e numbe o hidden a ac o s in a
h ee-dimensional mem is o -based au onomous Du ing oscilla o . Some echnical
esul s a e elega ed o he appendix.
2. Local bi u ca ions
In his sec ion, we s udy he local bi u ca ions ha occu in sys em (3). Fi s , we
no e ha he sys em is in a ian unde he ans o ma ion
(x, y, µ1, µ2, µ3)→(−x, −y, −µ1, µ2, µ3).
The e o e, i is su icien o s udy he bi u ca ion diag am o µ1>0.The equilib-
ium poin s o he sys em a e o he o m (x, y) = (˜x, 0), being ˜xa solu ion o he
cubic µ1+µ2x+x3= 0,and i s Jacobian ma ix o is gi en by
(4) J(x, y) = 0 1
µ2+ 3x2−6yx µ3−3x2.
Rema k 1. No e ha o µ3≤0 he di e gence o sys em (3) does no change sign,
hus om Bendixson’s c i e ion [21], he sys em does no ha e pe iodic solu ions.
Fi s , we p o ide a echnical esul ha p o ides a s udy o he numbe o equi-
lib ia in sys em (3) and hei opological na u e.
Lemma 2. Conside sys em (3), he ollowing s a emen s hold.
(a) I µ2≥0o we ha e µ2<0wi h 27µ2
1+ 4µ3
2>0, hen he sys em has only
one equilib ium poin .
(b) I µ2<0and 27µ2
1+ 4µ3
2= 0 we ha e wo equilib ium poin s.
(c) I µ2<0and 27µ2
1+ 4µ3
2<0, hen he sys em has h ee equilib ium poin s
xi= (si,0) wi h i∈ {L, C, R}such ha
(5) sL<−(−µ2/3)1/2< sC<(−µ2/3)1/2< sR,
and sL+sC+sR= 0.Fu he mo e, xLand xRa e saddles while xCis an
an isaddle (node o ocus).
P oo . We s udy he oo s o he polynomial p(x) = µ1+µ2x+x3.Since p′(x) =
µ2+ 3x2,i µ2≥0 we ob ain p′(x)≥0 and so he polynomial has only one oo .
4 A. AMADOR AND E. FREIRE AND E. PONCE
In he es o he p oo we assume µ2<0. The de i a i e p′(x) anishes a he
poin s x±=±(−µ2/3)1/2being a maximum and minimum local espec i ely, also a
di ec compu a ion gi es p(x±) = µ1∓2(−µ2/3)3/2. When p(x−)<0 o p(x+)>0
he g aph o p(x) only c osses once he x-axis and so hese inequali ies p o ides he
condi ion 27µ2
1+ 4µ3
2>0,and he s a emen (a) ollows. Assuming p(x−) = 0 o
p(x+) = 0, he s a emen (b) ollows.
Finally, i p(x+)<0< p(x−) hen we ha e h ee oo s as indica ed in (5).
Mo eo e , using he ela ion be ween oo s and coe icien s o polynomials, we ge
µ1=−sLsCsR, µ2=sCsL+sCsR+sLsR, sL+sC+sR= 0.
Fo he Jacobian ma ix gi en in (4), we ge J(si,0) = −(µ2+ 3s2
i) = −p′(si).As
we know ha p′(sL)>0, p′(sC)<0 and p′(sR)>0, he conclusion ollows and he
p oo is comple e. 
In he nex esul , we gi e a cha ac e iza ion o he local bi u ca ions o sys em
(3) on he pa ame ic plane (µ2, µ1),assuming a ixed alue o he pa ame e µ3.
No e ha wee ha e pu he µ1axis in he plane (µ2, µ1) e ically, being he µ2axis
he ho izon al one.
P oposi ion 3. The ollowing s a emen s hold o sys em (3).
(a) Gi en µ3∈R he pa ame e alues in he se
(6) ϕsn ={(µ2, µ1) : 27µ2
1+ 4µ3
2= 0},
co espond wi h saddle-node bi u ca ion poin s o equilib ia. In pa icula ,
he sys em has a cusp bi u ca ion o equilib ia a µ2=µ1= 0.
(b) Gi en µ3>0, he pa ame e alues in he se
(7) ϕH={(µ2, µ1) : µ1=±(µ3/3)3/2∓(µ3/3)1/2µ2, µ2<−µ3},
ep esen And ono -Hop bi u ca ion poin s o codimension one o he cen al
equilib ium poin xC,see Lemma 2(c).
(c) The se de ined in (7) de e mines a symme ic pai o s aigh hal lines
emana ing om wo poin s co esponding o Bogdano -Takens bi u ca ion
poin s, namely
(8) BT±≡−µ3,±2 (µ3/3)3/2.
P oo . S a emen (a) is a di ec consequence o he equa ions
x3+µ2x+µ1= 0, µ2+ 3x2= 0,
o be ul illed o any non-hype bolic equilib ium (x, 0) a a saddle-node bi u ca ion.
Le (˜x, 0) an equilib ium poin o sys em (3). Conside ing he Jacobian ma ix J
gi en in (4), hen J(˜x, 0) has wo pu ely imagina y eigen alues when aking µ3>0,
he alue ˜xsa is ies ˜x=±pµ3/3 wi h µ2<−µ3<0,because hen µ2+ 3˜x2<0.
The las inequali y is ul illed only o he equilib ium poin xC,see Lemma 2(c).
Since (˜x, 0) is an equilib ium poin we ha e
µ1+µ2±pµ3/3+±pµ3/33= 0,
and s a emen (b) ollows. To show s a emen (c) is su icien o conside he equa-
ions ace (J(˜x, 0)) = de (J(˜x, 0)) = 0.
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 5
3. Global bi u ca ions
In his sec ion we will comple e he bi u ca ion analysis o sys em (3). We will
w i e he sys em as a pe u bed Hamil onian o which he Melniko heo y can be
applied. This can be done in di e en ways, as indica ed in he nex esul . The
possibili y o eso ing o one o he wo nex epa ame iza ion o ms will be help ul
la e .
P oposi ion 4. Sys em (3) can be w i en as wo di e en pe u bed Hamil onian
sys ems, as ollows.
(a) Taking
(9) µ1=ε4ν1, µ2=−ε2ν2, µ3=ε2ν3,
he sys em can be ew i en as
˙x=y,
˙y=−ν2x+x3+εν1+ν3y−3x2y,
(10)
which o ε= 0 co esponds o he Hamil onian
(11) H1(x, y) = y2
2+ν2
x2
2−x4
4.
(b) Taking
(12) µ1=ε3ν1, µ2=−ε2ν2, µ3=ε2ν3,
he sys em can be ew i en as
˙x=y,
˙y=ν1−ν2x+x3+ε(ν3y−3x2y),
(13)
which o ε= 0 co esponds o he Hamil onian
(14) H2(x, y) = y2
2−ν1x+ν2
x2
2−x4
4.
P oo . The blow-up ans o ma ion x1= (1/ε)x, y1= (1/ε2)y, and ˜
=ε , allows
o ew i e sys em (3) as
x′
1=y1, y′
1=x3
1+µ2
ε2x1+µ1
ε3+µ3
εy1−3εx2
1y1,
whe e he p ime deno es de i a i es wi h espec o he new ime ˜
. Now, using (9)
and (12), a e some elemen a y algeb a we ob ain sys ems (10) and (13), espec-
i ely. 
The phase po ai o he unpe u bed Hamil onian sys ems (10) and (14) a e
shown in Figu e 1. No e ha when ν1= 0 we ob ain H1(x, y) = H2(x, y),and so in
ha case i is su icien o s udy he p ope ies o he Hamil onian H1.
Now, we will conside he he e oclinic connec ions o unpe u bed Hamil on-
ian sys em (11). The Hamil onian has a pai o he e oclinic connec ions Γ±( ) =
(x( ),±y( )),pa ame e ized by
x( ) = √ν2 anh pν2/2 ,
y( ) = ν2
√2sech2pν2/2 ,
(15)

6 A. AMADOR AND E. FREIRE AND E. PONCE
whe e −∞ < < ∞and ν2>0.In he nex esul , we compu e he Melniko unc-
ion along he he e oclinic connec ion Γ+ o he unpe u bed Hamil onian sys em
(10), and by using (9), we ob ain he app oxima e bi u ca ion cu es o he e oclinic
connec ions o sys em (3).
Figu e 1. (a) Phase po ai o unpe u bed Hamil onian sys em
(10) wi h ν2= 0.2. We show in g een he wo he e oclinic o bi s, while
he non-closing s able and uns able mani olds o he saddle poin s a e
shown in ed. (b) Phase po ai o unpe u bed Hamil onian sys em
(13) wi h ν1= 0.3 and ν2= 1. We d aw in g een he homoclinic o bi ,
he s able and uns able mani olds o he saddle poin s a e shown in
ed.
P oposi ion 5. I we conside pe u bed Hamil onian sys em (10) and ν= (ν1, ν2, ν3)
wi h ν2>0and ν2
2/4< ν3
2/27 (see Lemma 2(c)) hen he Melniko unc ion along
o he he e oclinic connec ion Γ±is gi en by
(16) Mh (ν) = 2
15√ν215ν1+ 5√2ν2ν3−3√2ν2
2.
P oo . The sys em can be w i en as
( ˙x, ˙y)T= (x, y) + εg(x, y),
whe e (x, y) = (y, −ν2x+x3)Tand g(x, y) = (0, ν1+ν3y−3x2y)T.Thus, we ha e
∧g=y(ν1+ν3y−3x2y). Acco dingly, he Melniko unc ion is de ined by
Mh (ν) = Z∞
−∞
(x( ),±y( )) ∧g(x( ),±y( ))d =
=Z∞
−∞ ±y( )ν1±(ν3−3x2( ))y( )d ,
whe e x( ) and y( ) a e de ined as in (15). A e a di ec compu a ion we ob ain
(16). 
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 7
By using he Melniko heo y and by ixing one pa ame e o sys em (3), we
can gi e an app oxima ion o he he e oclinic connec ion cu es in he emaining
pa ame e s plane.
P oposi ion 6. Conside sys em (3) wi h µ3>0su icien ly small and he pa ame -
ic plane (µ2, µ1). Then he sys em has a unique hype bolic he e oclinic connec ion
in a neighbo hood o he cu e
(17) ϕh ={(µ2, µ1)∈R2:µ1=±√2
15 µ2(3µ2+ 5µ3), µ26=−5µ3/3}.
P oo . Fixing ν3= 1 in he Melniko unc ion gi en in (16), and imposing he
condi ion Mh (ν1, ν2) = 0,we ob ain
ν1=√2
15 ν2(3ν2−5) .
F om (9) we ge ε=√µ3,µ1=µ2
3ν1,and µ2=−µ3ν2,so ha
µ1=µ2
3ν1=−µ2
3
√2
15
µ2
µ33−µ2
µ3−5,
and he conclusion ollows. 
When µ1= 0 and µ3>0 on he pa ame e plane (µ2, µ1),we ob ain he poin o
double he e oclinic connec ions
(18) DHT ≡(−5µ3/3,0) .
We ecall ha Schec e ’s poin s a e co-dimension wo poin s de ined by he in-
e sec ion o a saddle-node cu e and a homoclinic o he e oclinic cu e, o mo e
de ails see [32]. Taking he in e sec ion poin s o he saddle-node bi u ca ion cu e
and he he e oclinic cu es gi en in (6) and (17) espec i ely, we ob ain a i s -o de
app oxima ion o Schec e ’s poin s o he sys em. Since he sys em is symme ic
wi h espec o he pa ame e µ1, he sys em has ou Schec e ’s poin s (see Figu e
3), hese poin s a e
S±
1≡ρ1(5/27),∓(5√10/729) p18µ3+ 5 + √5,
S±
2≡ρ2(5/27),±(5√10/729) p18µ3+ 5 −√5,
(19)
whe e
ρ1=9µ3+ 5 −√5p18µ3+ 5, ρ2=√5p18µ3+ 5 −9µ3−5.
Now, by using he homoclinic connec ion o Hamil onian sys em (14), we compu e
he associa ed Melniko unc ion o sys em (3) when ν3= 1.
P oposi ion 7. I we conside sys em (13) and ν= (ν1, ν2, ν3)wi h ν1>0,ν2>0
and ν3= 1, hen he Melniko unc ion associa ed o he homoclinic o bi wi h
connec ion poin (0, sR), i is gi en by
(20) M(ν) = √2cosh2(θ)
cosh2(θ) + 2 (F1(θ) + ν2F2(θ)) ,
8 A. AMADOR AND E. FREIRE AND E. PONCE
whe e
F1(θ) =720θ−320 sinh θ+ 240θcosh3θ−320 cosh2θsinh θ−
−80 cosh4θsinh θ+ 480θcosh θ,
F2(θ) =1440θcosh θ−768 sinh θ−cosh3θ−1344 cosh2θsinh θ−
−48 cosh4θsinh θ
(21)
and 0< θ < ∞, wi h
cosh θ=2s
ω, ω2= 2(ν2−s2
R)>0, ν1=ν2sR−s3
R,
being sR he bigges posi i e oo o he equa ion ν1−ν2x+x3= 0, see Figu e 2.
P oo . We conside he unpe u bed Hamil onian sys em gi en in (13) wi h ν2>0.
F om Lemma 2(c), he sys em has 3 equilib ium poin s xi= (si,0), whe e xLand
xRa e saddle poin s and xCis a ocus o node and
sL< sC< sR, sL+sC+sR= 0, sLsCsR=−ν1.
We s udy only he case ν1>0, o he case ν1<0 is analogous.
Sys em (13) can w i en as
( ˙x, ˙y)T= (x, y) + εg(x, y).
Now, assuming ν1>0,by G een’s Theo em, he homoclinic Melniko unc ion o
he sys em can ew i en as
Mh(ν) = Z ZD(ν1,ν2)−∂g(x, y)
∂y dA,
whe e Dis he egion bounded by he homoclinic o bi which joins he equilib ium
poin (sR,0) o i sel . By ixing ν3= 1 ( ha is µ3>0), and aking p(x) =
ν1−ν2x+x3,we ge p(sR) = ν1−ν2sR+s3
R= 0, ha is
(22) ν1=sR(ν2−s2
R),
and so ν2−s2
R>0.Taking he auxilia y unc ion
q(x) = Zx
0
p(x)dx =ν1x−ν2
x2
2+x4
4,
and using (14), he homoclinic loop is gi en by he poin s (x, y±
s(x)) whe e x≤x≤
sR,
y±
s(x) = ±√2pq(x)−q(sR),
and y±
s(x) = y±
s(sR) = 0,see Figu e 2. Now, he Melniko unc ion is hanks o he
symme y o he loop
Mh(ν) = 2 ZsR
x
(3x2−1)dx Zy+
s(x)
0
dy =
=√2Zs
x
(3x2−1)(sR−x)q(x+sR)2−2(ν2−s2
R)dx =
=√2Zs
x
(3x2−1)(sR−x)p(x+sR)2−ω2dx,
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 9
whe e om (22) ω2= 2(ν2−s2
R),and we ha e used ha
q(x)−q(sR) = 1
4(x−sR)2(x+sR)2−ω2.
Taking he change o a iable
x+sR=ωcosh θ,
and no ing ha q(sR)−q(x) = 0 we see ha x+sR=ω, which co esponds o θ= 0,
while o x=sR he co esponding alues o θ=θsRsa is y cosh θsR= 2sR/ω, o
also ω2cosh2θsR= 4s2
R, ha is (s2
R+ν2) cosh2θsR= 2s2
R,and so we ge
s2
R=cosh2θsR
2 + cosh2θsR
ν2.
Now we a i ed o
Mh(ν) = √2ω2ZθsR
0
(1 −3(ωcosh θ−sR)2)(2sR−ωcosh θ) sinh2θdθ,
and a e some compu a ions, we ob ain (20) and (21), whe e θsRhas been simpli ied
o θ.
0
x
0
y
y+
s(x)
y−
s(x)
sC
sLxsR
Figu e 2. Homoclinic o bi which joins he saddle equilib ium poin
(sR,0) o i sel .
As a di ec consequence o he abo e esul , we gi e an analy ical app oxima ion
o he bi u ca ion cu es o homoclinic connec ions o sys em (3).
P oposi ion 8. Conside sys em (3) wi h µ3>0su icien ly small and he pa ame -
ic plane (µ2, µ1).Then he sys em has a unique homoclinic o bi in a neighbo hood
o he cu e
(23) ϕh={(µ2, µ1)∈R2:µ2=−µ3ν2(θ), µ1=±µ3/2
3ν1(θ),0< θ < ∞},
whe e
ν2(θ) = 10(cosh 2θ+ 5)(9 sinh θ+ sinh 3θ−12θcosh θ)
3(370 sinh θ+ 115 sinh 3θ+ sinh 5θ−60θ(11 cosh θ+ cosh 3θ)),
ν1(θ) = ν2(θ)s−s3, s2=cosh2θ
2 + cosh2θν2(θ).
(24)
16 A. AMADOR AND E. FREIRE AND E. PONCE
sys em (46). This esul gua an ees he exis ence o a opological sphe e in he 3D
phase-space comple ely olia ed by pe iodic o bi s.
P oposi ion 11. Conside sys em (30) wi h β, ξ > 0,, he unc ion qde ined as in
(31),a2−3b < 0and
(37) a2−3b+ 3β > 0
su icien ly small. Addi ionally, suppose ha he ollowing inequali ies hold
0<3b−a2<3ξ/β,
β(3b−a2)−3ξ < (5/2) 3b−a2−3ββ < 0.
(38)
Then o all h∈Rwi h
−A
27 < h < B
27,
whe e
A=4a2β+ 3β2−12bβ + 9ξpa2−3b+ 3β+ 9aξ + 2a3β−9abβ,
B=4a2β+ 3β2−12bβ + 9ξpa2−3b+ 3β−9aξ −2a3β+ 9abβ,
(39)
he sys em has a s able pe iodic o bi . Mo eo e , he e exis a opological sphe e Ω
(see Figu e 10) olia ed by such pe iodic o bi s.
P oo . F om Rema k 10, and a e subs i u ing he alues o µ1, µ2and µ3gi en
in (35), we ob ain he inequali ies (37)-(38). Now, om (36) we ob ain |µ1|<
(1/3) (µ3/3)1/2(µ3−3µ2),so ha µ3−3µ2>0,since om hypo heses we ha e
µ2<−(5/2)µ3<0.Now a e some algeb a we ob ain
|27h+ 9aξ + 2a3β−9abβ|<4a2β+ 3β2−12bβ + 9ξpa2−3b+ 3β.
Taking in o accoun he absolu e alue, and g ouping e ms, we ob ain he alues o
Aand Bde ined in (39). Finally, om Rema k 10 sys em (30) has a s able pe iodic
o bi on each Shde ined in (32), so a ying he pa ame e h, we ob ain a sphe e
olia ed by such pe iodic o bi s. 
5. False Hidden A ac o s in Mem is o -Based Au onomous
Du ing Oscilla o s
An a ac o is called a hidden a ac o i i s basin o a ac ion does no in e sec
any neighbo hood o equilib ia; o he wise, i is called a sel -exci ed a ac o , o
mo e de ails see [26, 27]. Recen ly in [33, 34, 19] i was epo ed he exis ence o
an in ini e numbe o hidden a ac o s in a mem is o -based au onomous Du ing
oscilla o s, whose mem is ance unc ion is a cubic polynomial. He e, by using a
simila app oach o he ollowed in he p e ious sec ion, we will show ha such
hidden a ac o s a e no possible, so ha he nume ical simula ions included in
[33, 34, 19] a e misleading.
The quo ed mem is o based au onomous Du ing oscilla o is modeled by he
dynamical sys em
˙x=y,
˙y=z,
˙z=−αz −M(x)y,
(40)

BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 17
Figu e 10. Using (62) on each in a ian mani old Shde ined in (32),
some slices o he su ace Ω gi en by P oposi ion 11 o sys em (30)
wi h pa ame e s a= 1, b = 4.8, β = 5 and ξ= 80 a e shown. Fo
his se o pa ame e s we ge µ3= 0.106 >0, µ2=−2.3<−(5/2)µ3,
A= 1180.1 and B= 152.2.
whe e he mem is ance unc ion M(possibly discon inuous) is de ined as
(41) M(x) = dφ(x)
dx
and φis a con inuous unc ion. Sys em (40) has a con inuum o equilib ia, since any
poin o he x-axis is an equilib ium poin . In he nex esul , we show ha e en
sys em (40) does no belong o he amily (46) o he appendix, he sys em also has
he p ope y o possessing an in ini e numbe o in a ian mani olds.
P oposi ion 12. Conside sys em (40) wi h he unc ion Mde ined as in (41). Fo
any h∈R he se
(42) Sh={(x, y, z)∈R3:H(x, y, z) = h}
is an in a ian mani old o he sys em, whe e we ha e in oduced he con inuous
unc ion
(43) H(x, y, z) = φ(x) + αy +z.
The e o e, he sys em has an in ini e numbe o in a ian mani olds olia ing he
whole R3, and so he dynamics is essen ially wo-dimensional.
P oo . Taking Has in (43), de ine o any solu ion (x(τ), y(τ), z(τ)) o (40) he
auxilia y con inuous unc ion
h(τ) = H(x(τ), y(τ), z(τ))
Now, a di ec compu a ion gi es, excep ing he poin s o possible non-di e en iabili y,
h′(τ) = dφ(x)
dx ˙x+α˙y+ ˙z=M(x)y+αz −αz −M(x)y= 0.
Then his piecewise cons an along he o bi s o (40), bu as his con inuous by
de ini ion, i should be globally cons an . In sho , he le el se s o Ha e in a ian
o he low. 
18 A. AMADOR AND E. FREIRE AND E. PONCE
Now, by using he abo e esul , we educe he s udy o he dynamical beha io
o he sys em, o he s udy o a plana sys em.
P oposi ion 13. Conside sys em (40) wi h he unc ion Mde ined as in (41).
Then on each in a ian se Shde ined in (42) he sys em is opologically equi alen
o he plana sys em
˙x=y,
˙y=−φ(x)−αy +h.
(44)
Mo eo e , (x(τ), y (τ)) ∈R2is a solu ion o he abo e sys em i and only i Eh(x(τ), y (τ))
is a solu ion o sys em (40), whe e
(45) Eh(X(τ), Y (τ)) = 
x(τ)
y(τ)
h−φ(x(τ)) −αy(τ)


P oo . F om P oposi ion 12 we can sol e o zin he equa ion H(x, y, z) = h, and
w i e
z=h−φ(x)−αy.
Replacing his exp ession in o he i s and second equa ion o (40) we ob ain sys em
(44). Suppose ha (x(τ), y (τ)) ∈R2is a solu ion o sys em (44). Taking
z(τ) = h−αy(τ)−φ(x(τ))
we ob ain
˙z(τ) = −α˙y(τ)−dφ(x(τ))
dx ˙x(τ) = −α(h−φ(x(τ)) −αy(τ)) −M(x(τ))y(τ) =
=−α(z(τ)) −M(x(τ))y(τ).
and he p oposi ion ollows. 
In he ollowing esul , we show ha o α6= 0, he sys em does no ha e pe iodic
solu ions.
P oposi ion 14. Conside sys em (44). The ollowing s a emen s hold.
(a) Fo α= 0 he sys em is Hamil onian.
(b) Fo α6= 0 he sys em does no ha e pe iodic solu ions.
P oo . The di e gence o he sys em is ∆ = −α. Then, when α= 0 he sys em
co esponds o he Hamil onian
H(x, y) = y2
2+φ′(x).
Fo α6= 0 he di e gence o sys em (44) does no change sign, hus om Bendixson’s
c i e ion [21], sys em (40) does no ha e pe iodic solu ions. 
Rema k 15. No e ha as a consequence o p oposi ion 13 and 14, he 3Dsys em
(40) sys em canno ha e pe iodic o bi s o any con inuous unc ion φand α6= 0.
Howe e , when α= 0 he sys em could ha e an in ini e numbe o pe iodic o bi s on
each in a ian se Shde ined in (42).
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 19
Rega ding [33, 34, 19] au ho s conside sys em (40) wi h he unc ion φ(x) =
ωx +βx3and he se o pa ame e s α= 0.0001, ω = 0.35, β = 0.85.In bo h quo ed
e e ences, au ho s epo ed he exis ence o an in ini e numbe o s able pe iodic
o bi s coexis ing wi h an in ini e numbe o s able equilib ia, by aking in o accoun
se e al nume ical simula ions, so concluding he exis ence o hidden a ac o s.
F om P oposi ions 12 and 13, we ob ain he in a ian mani olds
Sh={(x, y, z)∈R3:ωx +βx3+αy +z=h},
and he plana sys em uling he dynamics on each Shgi en by
˙x=y, ˙y=−ωx −βx3−αy +h.
F om Rema k 15 we no e ha , he sys em canno ha e pe iodic o bi s, and so,
he s a emen made in he quo ed pape s is clea ly w ong, p obably a e gi ing
oo much c edi o nume ical simula ions. This emphasizes he ele ance o he
app oach ollowed in his wo k which allows o a oid misconcep ions coming jus
om nume ical simula ions.
6. Conclusions
Mo i a ed by he dynamical analysis o 3D mem is o oscilla o s whose nonlin-
ea cha ac e is ics is a cubic polynomial, and a e showing ha hei dynamics
is essen ially wo-dimensional, he need o conside a dis ega ded un olding o he
Bogdano -Takens singula i y na u ally a ose. The co esponding bi u ca ion se ,
including bo h local and global bi u ca ions has been desc ibed. While local bi u -
ca ions can be easily de ec ed, he cha ac e iza ion o global bi u ca ions pa ame e s
cu es is much mo e in ol ed; only by eso ing o Melniko ’s heo y i was possible
o ob ain such cu es p o iding a comple e desc ip ion o he bi u ca ion se .
Rega ding he conside ed 3D mem is o oscilla o s, and by wo king wi hin some
pa ame e s egions o he abo e bi u ca ion se , i has been possible o show igo -
ously he exis ence o mul iple pe iodic o bi s leading o a opological sphe e.
When he same app oach is applied o a di e en amily o 3D mem is o oscilla-
o s, i has been shown ha he oscilla ions a e no possible, con a ily o wha had
been ecen ly claimed.
Acknowledgemen s
The i s au ho is suppo ed by Pon i icia Uni e sidad Ja e iana Cali-Colombia.
E. F ei e and E. Ponce a e pa ially suppo ed by MINECO/FEDER g an MTM2015-
65608-P and by he Conseje ´ıa de Econom´ıa, Inno aci´on, Ciencia y Empleo de la
Jun a de Andaluc´ıa unde g an P12-FQM-1658.
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Appendix: Dimensional educ ion in 3D mem is o oscilla o s
We conside a amily o h ee-dimensional sys ems, which is gene al enough o
cap u e all he ma hema ical models o mem is o oscilla o s gi en in (30). Such
amily was been s udied in [1] and [30], whe e he au ho s showed ha he dynamics
o such a amily o h ee-dimensional sys ems is essen ially uled by a one pa ame e
se o wo-dimensional sys ems. We conside he sys em
(46)
˙x=a11W(z)x+a12y,
˙y=a21x+a22y,
˙z=x,
whe e he cons an s a11, a12, a21, a22 ∈Rand he unc ion Wallows o de ine a
con inuous unc ion
(47) q(z) = Zz
0
W(s)ds.
The nex esul gua an ees ha he dynamics o sys em (46) is essen ially wo-
dimensional, see [1] o a p oo .
P oposi ion 16. Conside sys em (46) whe e he unc ions Wand qa e ela ed as
in (47). Fo any h∈R, he se
(48) Sh={(x, y, z)∈R3:−a22x+a12y−a12a21z+a11a22q(z) = h}
is an in a ian mani old o he sys em. The e o e, he sys em has an in ini e amily
o in a ian mani olds olia ing he whole R3, and so he dynamics is essen ially
wo-dimensional.
In he ollowing esul we show ha on each in a ian se Shgi en in (48), and o
any con inuous unc ion qde ined as in (47), he dynamics is opologically equi alen
o a Li´ena d sys em. Fu he mo e, we gi e o any solu ion o he Li´ena d sys em
wi h a gi en alue o h, he co esponding solu ion o he 3D canonical model (46).

22 A. AMADOR AND E. FREIRE AND E. PONCE
This esul is a gene aliza ion o Theo em 3 gi en in [1], whe e he unc ion qwas
conside ed o be a con inuous piecewise linea unc ion.
P oposi ion 17. Conside sys em (46) wi h he unc ion qde ined as in (47). I
a12 6= 0, hen on each in a ian se Shgi en in (48), he dynamics is opologically
equi alen o he Li´ena d sys em
(49) ˙
X=Y−F(X),˙
Y=−g(X) + h,
whe e Fand ga e gi en by
(50) F(X) = −a11q(X)−a22X, g(X) = a11a22q(X)−a12a21X
Mo eo e , (X(τ), Y (τ)) ∈R2is a solu ion o he Li´ena d sys em (49) o a gi en
h∈R, i and only i Eh(X(τ), Y (τ)) ∈R3is a solu ion o sys em (46) on Sh,
whe e
(51) Eh(X(τ), Y (τ)) = 

Y(τ)−F(X(τ))
1
a12 [(a2
22 +a12a21)Y(τ)−a22Y(τ) + h]
X(τ)

.
P oo . Fi s , wi h a12 6= 0 he change o a iables
(52) x=x, y =a22x−a12y, z =z
ans o ms sys em (46) in o he sys em
˙
x= 1(z)x−y,(53)
˙
y= 2(z)x,
˙
z=x,
whe e he unc ions 1and 2a e de ined as
(54) 1(z) = a11W(z) + a22, 2(z) = a22a11W(z)−a12a21.
F om P oposi ion 16, he in a ian mani olds (48) o sys em (53)-(54) can be w i en
in he new a iables as
(55) e
Sh={(x, y, z)∈R3:−y+g(z) = h}.
Now, eplacing he condi ion gi en in (55) in he i s equa ion o (53) and emo ing
he unnecessa y second equa ion, we ob ain he sys em
(56) ˙
x= 1(z)x−g(z) + h,
˙
z=x.
whe e he unc ion gis de ined by
(57) g(u) = a11a22q(u)−a12a21u.
A e he change o a iables
(58) X=z,
Y=−˜
F(z) + x,
whe e Fis
(59) ˜
F(z) = a11q(z) + a22z,
we ob ain ˙
X=˙
z=x=Y+˜
F(X) = Y−(−˜
F(X)),
BIFURCATION SET FOR A DISREGARDED BOGDANOV-TAKENS UNFOLDING 23
so ha
˙
Y=−˜
F′(z)˙
z+˙
x=−(a11q′(z) + a22) + ( 1(z)x−g(z) + h) =
=− 1(z)x+ 1(z)x−g(z) + h=−g(z) + h,
and aking F(X) = −˜
F(X) we ob ain sys em (49)-(50).
I (X(τ), Y (τ)) ∈R2is a solu ion o sys em (49)-(50) o a gi en h∈R, we ha e
om (58) ha x(τ)
z(τ)=Y(τ)−F(z(τ))
X(τ)
is a solu ion o sys em (56). F om (55), we ob ain on e
Sh ha y=g(z)−h, wi h g
as in (57). Thus, 
x(τ)
y(τ)
z(τ)

=
Y(τ)−F(X(τ))
g(X(τ)) −h
X(τ)

,
is a solu ion o sys em (53) on Sh. Finally, om (52) we ob ain o sys em (46) he
solu ion x(τ) = x(τ),
y(τ) = 1
a12
[a22x(τ)−y(τ)] = 1
a12
[a22Y(τ)−a22F(X(τ)) −g(X(τ)) + h]
=1
a12 ha22Y(τ) + a22 ˜
F(X(τ)) −g(X(τ)) + hi,
and z(τ) = z(τ). The conclusion ollows om he ac ha o all Xwe ha e
a22 ˜
F(X)−g(X) = (a2
22 +a12a21)X.
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In o de o apply he analysis pe o med o sys em (3), in wha ollows we conside
he unc ion qde ined by a cubic polynomial, ha is, we assume
(60) W(z) = 3cz2+ 2az +b, q(z) = cz3+az2+bz,
wi h c6= 0. As a di ec consequence o P oposi ions 16 and 17, we ob ain he nex
esul .
Co olla y 18. Conside sys em (46) wi h he unc ions qand Wde ined as in (60).
I a12 6= 0, hen on each in a ian se Shgi en by
Sh={(x, y, z)∈R3:−a22x+a12y+a11a22cz3+aa11a22z2+(ba11a22 −a12a21)z=h}
he dynamics is opologically equi alen o he Li´ena d sys em
˙x=y+ca11x3+aa11x2+ (ba11 +a22)x,
˙y=−a11a22cx3−a11a22ax2+ (a12a21 −a11a22b)x+h.
(61)
Mo eo e , (x(τ), y (τ)) ∈R2is a solu ion o he Li´ena d sys em (61) o a gi en
h∈R, i and only i Eh(x(τ), y (τ)) ∈R3is a solu ion o sys em (46) on Sh,whe e
(62) Eh(x(τ), y (τ)) = 

y(τ) + ca3
11x(τ)3+aa2
11x(τ)2+ (ba11 +a22)x(τ)2
1
a12 [(a2
22 +a12a21)y(τ)−a22y(τ) + h]
x(τ)

.
In he nex P oposi ion, we show ha sys em (61) can be w i en in o he o m
(1).
24 A. AMADOR AND E. FREIRE AND E. PONCE
P oposi ion 19. The ollowing s a emen s hold o sys em (61).
(a) I a22 6= 0 and a11a22 <0 hen he sys em can be w i en in o he o m
(63) ˙x=y, ˙y=µ1+µ2x+cx3+µ3y+ 3ca11x2y.
whe e he new pa ame e s µ1, µ2and µ3a e gi en by
µ1=27ch +a11a22a(9cb −2a2)−9caa12a21
27c2(−a11a22)5/2,
µ2=a11a22(a2−3cb) + 3ca12a21
3c(a11a22)2, µ3=a11(a2−3cb)−3ca22
3ca11a22
.
(64)
(b) I a22 = 0 hen he sys em can be w i en in o he o m
(65) ˙x=y, ˙y=µ1+µ2x+µ3y+ 3ca11x2y,
whe e he new pa ame e s µ1, µ2and µ3a e de ined by
(66) µ1=h−aa12a21
3c, µ2=a12a21, µ3=ba11 −a2a11
3c.
P oo . Fi s , he change o a iables
u=x+a
3c, =y+2
27
a3
c2a11 −1
3
a
ca22 −1
3ab
ca11,
ans o ms sys em (61) in o
˙u= +ca11u3+λ1u,
˙ =−ca11a22u3+λ2u+λ3,
(67)
whe e he new pa ame e s a e
λ1=a22 +ba11 −1
3
a2
ca11, λ2=a12a21 −ba11a22 +1
3
a2
ca11a22,
λ3=h+1
3ab
ca11a22 −1
3
a
ca12a21 −2
27
a3
c2a11a22.
(68)
I a11a22 <0, he change o a iable
x=1
(−a11a22)1/2u, y = , τ =1
−a11a22
,
ans o ms sys em (67)-(68) in o
˙x=1
(−a11a22)3/2y+ca11x3−λ1
a11a22
x,
˙y=λ3
−a11a22
+λ2
(−a11a22)1/2x+c(−a11a22)3/2x3
and aking in o accoun ha
¨x=1
(−a11a22)3/2˙y+ 3ca11x2˙x−λ1
a11a22
˙x,
and a e some algeb a, s a emen (a) ollows.
I a22 = 0, hen om sys em (67), we ob ain s a emen (b) a e a di ec compu a ion.
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