Bi u ca ions in non-au onomous scala equa ions
J. A. Langa1, J. C. Robinson2and A. Su´
a ez1
1. Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico
Fac. Ma em´a icas, C/ Ta ia s/n
C.P. 41012, Uni . Se illa, Spain.
2. Ma hema ics Ins i u e,
Uni e si y o Wa wick,
Co en y, CV4 7AL, U. K.
e-mail: [email p o ec ed], [email p o ec ed]a wick.ac.uk, [email p o ec ed]
Running head. Bi u ca ions in non-au onomous scala equa ions
The au ho o whom p oo s should be sen :
James C. Robinson,
Ma hema ics Ins i u e,
Uni e si y o Wa wick,
Co en y, CV4 7AL, U. K.
Tel: 44 24 7652 4657
Fax: 44 24 7652 4182
e-mail: jc @wa wick.ac.uk
1
Abs ac
In a p e ious pape we in oduced a ious de ini ions o s abili y and ins abil-
i y o non-au onomous di e en ial equa ions, and applied hese o in es iga e he
bi u ca ions in some simple models. In his pape we p esen a mo e sys ema ic
heo y o local bi u ca ions in scala non-au onomous equa ions.
Keywo ds: Non-au onomous di e en ial equa ions, bi u ca ion heo y, pullback a -
ac ing se s.
2
1 In oduc ion
In a p e ious pape (Langa, Robinson, & Su´a ez [15]) we in oduced a ious de ini ions o
s abili y and ins abili y ha seemed o be po en ially use ul in discussing he dynamics
o he solu ions o non-au onomous di e en ial equa ions. In pa icula we applied hese
de ini ions o a ious simple model p oblems ha exhibi ed non-au onomous e sions o
s anda d au onomous bi u ca ions: an explici ly sol able pi ch o k bi u ca ion p oblem,
a saddle-node ype bi u ca ion, and a gene al n-dimensional ‘loss o s abili y’.
In his pape we de elop a mo e gene al heo y, concen a ing on he well-known ‘local
bi u ca ions’ om he au onomous heo y, and inding condi ions o simila bi u ca ions
in he scala non-au onomous equa ion
˙x= (x, , λ),
whe e λis a pa ame e . By imposing condi ions on he Taylo coe icien s in he expansion
o nea x=λ= 0 (which educe o he s anda d condi ions in he au onomous case)
we a e able o p o e a ious gene al heo ems gua an eeing ansc i ical, pi ch o k, and
saddle node bi u ca ions. Al hough we equi e a s ong ‘balance hypo hesis’ on he e ms
in he Taylo expansion, we belie e ha hese esul s a e a u he s ep owa ds a gene al
non-au onomous heo y o bi u ca ions. We do no p esen any conc e e examples he e,
ins ead concen a ing on he de elopmen o an abs ac heo y which we belie e should
be applicable o a wide a ie y o pa icula models.
Some pa icula examples ha e been analysed in a ious se ings: using he amewo k
o skew p oduc lows Johnson [8] and Johnson and Yi [9] ha e conside ed a gene alised
no ion o a Hop bi u ca ion; Shen and Yi [19] ea almos pe iodic scala di e en ial
equa ions (bu lea e bi u ca ion phenomena la gely un ouched); mo e ecen ly Kloeden
[12] has analysed ansc i ical and pi ch o k bi u ca ions in an almos pe iodic equa ion;
Johnson, Kloeden, & Pa ani [10] ha e conside ed a non-au onomous ‘ wo s ep bi u ca ion’;
and Kloeden & Siegmund [14] gi e a nice discussion o he gene al p oblem in he con ex
o skew p oduc lows.
In his pape we do no adop he skew p oduc app oach and he es ic ions on he
gene ali y o ha i would en ail, p e e ing o use he language o p ocesses.
3
2 Non-au onomous equa ions as p ocesses
Fo he solu ion o any non-au onomous equa ion
˙x= (x, )x(s) = x0wi h x∈Rm(2.1)
he ini ial ime (s) is as impo an as he inal ime ( ). In o de o ea hese equa ions
as dynamical sys ems we conside a amily o solu ion ope a o s {S( , s)} ≥s( e med a
“p ocess”, see Da e mos [6] o Sell [18]) ha depend on bo h he inal and ini ial imes.
We can hen deno e he solu ion o (2.1) a ime by S( , s)x0. I is su icien ly smoo h
(which i will be in all ha ollows) hen i is clea ha S( , s) : Rm→Rmmus sa is y
a) S( , ) is he iden i y o all ∈R,
b) S( , τ)S(τ, s) = S( , s) o all ,τ, and s∈R, and
c) S( , s)x0is con inuous in ,s, and x0.
The e may in ac be solu ions o (2.1) ha do no exis o all ime, and some es ic ions
o he possible alues o sand may be necessa y, gi ing ise o only a ‘local p ocess’.
Al hough we pass o e hem he e, we will deal wi h such echnicali ies whe e necessa y
in wha ollows.
Since in his pape we will only ea scala equa ions wi h unique solu ions bo h
o wa ds and backwa ds in ime, he esul ing p ocess will be o de -p ese ing, i.e.
xs> ys⇒S( , s)xs> S( , s)ys o all , s ∈R
(allowing S( , s)xso S( , s)ys o be ±∞ i necessa y allows us o ake alues o and s
om all o R).
3 S abili y & ins abili y in non-au onomous sys ems
We now ecall some o he de ini ions om Langa e al. [15] which we will use in ou
bi u ca ion analysis. The simple no ion o a comple e ajec o y will be cen al:
De ini ion 1 The con inuous map x:R→Rmis a comple e ajec o y i
S( , s)x(s) = x( ) o all , s ∈R.
4
We will in es iga e he appea ance and disappea ance o comple e ajec o ies ha a e
‘s able’ o ‘uns able’ in ce ain senses ha appea o be app op ia e o non-au onomous
sys ems. No e ha comple e ajec o ies a e me ely pa icula examples o in a ian se s
in non-au onomous sys ems:
De ini ion 2 A ime- a ying amily o se s {Σ( )} ∈Ris in a ian (we say “Σ(·)is in-
a ian ”) i
S( , s)Σ(s) = Σ( ) o all , s ∈R.
In wha ollows we make cons an use o he Hausdo semidis ance be ween wo se s
Aand B, dis [A, B], which is de ined as
dis [A, B] = sup
a∈A
in
b∈Bd(a, b) :
no e ha his only measu es how a Ais om B(dis [A, B] = 0 only implies ha A⊆B).
We also use he no a ion N(X, ²) o deno e he closed ²-neighbou hood o a se X:
N(X, ²) = {y:y=x+z, x ∈X, z ∈Rmwi h |z| ≤ ²}.
3.1 No ions o a ac ion
Fi s we de ine o mally he amilia no ion o a se ha is a ac ing o wa ds in ime,
wi h a speci ied domain o a ac ion D. Fo any choice o Dwe say ha Σ(·)⊂Di
Σ( )⊂D o e e y ∈R.
De ini ion 3 An in a ian se Σ(·)is o wa ds a ac ing wi hin Di Σ(·)⊂Dand o
each s∈R
lim
→∞ dis [S( , s)K, Σ( )] = 0
o all compac subse s1Ko D.
In a non-au onomous sys em he no ion o being ‘locally o wa ds a ac ing’ is a li le
mo e sub le; we allow he neighbou hood o Σ ha is a ac ed o depend on he ini ial
ime. I is clea ha i Σ(·) is o wa ds a ac ing wi hin D hen i is also locally o wa ds
a ac ing wi hin D.
1No e ha he de ini ion implies a ac ion o e e y ini ial condi ion in Ka a uni o m a e. Ou
de ini ion in Langa e al. [15] only equi ed con e gence o each ixed ini ial condi ion. Con a y o he
s a emen in he oo no e in ha pape , he wo de ini ions a e mos ce ainly no equi alen , e en o
ini e-dimensional sys ems.
5

De ini ion 4 An in a ian se Σ(·)is locally o wa ds a ac ing wi hin Di Σ(·)⊂D
and o each s∈R he e exis s a δ(s)such ha
lim
→∞ dis [S( , s)K, Σ( )] = 0
o all compac K⊂N(Σ(s), δ(s)) ∩D.
We now in oduce he no ion o pullback a ac ion
De ini ion 5 An in a ian se Σ(·)is pullback a ac ing wi hin Di Σ(·)⊂Dand o
e e y ∈Rand e e y compac se K⊂D,
lim
s→−∞ dis [S( , s)K, Σ( )] = 0.
Σ(·)is globally pullback a ac ing i we can ake D=Rm.
Fo a se Σ(·) o be locally pullback a ac ing, he neighbou hood o Σ(·) ha is
a ac ed can depend only on he inal ime. No e ha he de ini ion allows a di e en
collec ion o compac se s K(·) o be a ac ed o Σ( ) o each ixed ∈R.
De ini ion 6 We say ha Σ(·)is locally pullback a ac ing wi hin Di Σ(·)⊂Dand
o e e y ∈R he e exis s a δ( )>0such ha i K(·)⊂Dis compac and
lim
s→−∞ dis [K(s),Σ(s)] < δ( )
hen
lim
s→−∞ dis [S( , s)K(s),Σ( )] = 0.(3.1)
I Dis bounded i is once again clea ha any se ha is pullback a ac ing wi hin
Dis locally pullback a ac ing wi hin D. Howe e , i is an uncom o able consequence
o ou de ini ions ha a se can be globally pullback a ac ing bu no locally pullback
a ac ing i Dis unbounded. Ne e heless, his canno occu i he se is ‘bounded in
he pas ’, as shown by he ollowing lemma.
Lemma 1 I an in a ian se Σ(·)is pullback a ac ing wi hin Dand bounded ‘in he
pas ’, i.e.
[
<T
Σ( )
is bounded o some T, hen Σ(·)is locally pullback a ac ing.
6
P oo . We show ha Σ(·) is locally pullback a ac ing o any choice o cons an δ
( his was called ‘uni o mly pullback a ac ing’ in Langa e al. [15]). I
lim
s→−∞ dis [K(s),Σ(s)] < δ
hen o some τ, which we choose o be less han T, we mus ha e dis [K(s),Σ(s)] <2δ
o all s < τ. Since Σ(s) is bounded o s < T , all such K(s) a e con ained in a bounded
se Xδ.
Since Σ is globally pullback a ac ing, his bounded se is (pullback) a ac ed o Σ:
he e exis s a σsuch ha
dis [S( , s)Xδ,Σ( )] < ² o all s≤σ.
Since K(s)⊂Xδ o all s < T, i ollows ha
dis [S( , s)K(s),Σ( )] < ² o all s≤σ,
and so Σ is locally pullback a ac ing. ¤
3.2 S abili y
We now gi e a de ini ion o ‘s abili y’ in he pullback sense.
De ini ion 7 Σ(·)is pullback Lyapuno s able i o e e y ∈Rand ² > 0 he e exis s
aδ( )>0such ha o any s < ,xs∈N(Σ(s), δ( )) implies ha S( , s)xs∈N(Σ( ), ²).
The ollowing esul , analogous o he ac ha a ac ion implies s abili y o s a-
iona y poin s o scala au onomous sys ems, means ha in wha ollows we need no
be conce ned wi h Lyapuno s abili y p ope ies o comple e ajec o ies, bu only hei
a ac ion p ope ies.
Lemma 2 Le x∗(·)be a comple e ajec o y in a non-au onomous scala ODE ha is
locally pullback a ac ing; hen his ajec o y is also pullback Lyapuno s able.
P oo . Fix ∈R. Gi en an ² > 0, we can gua an ee ha i x±(s) = x∗(s)±1
2δ( ) hen
lim
s→−∞ |S( , s)x±(s)−x∗( )|= 0,
and so in pa icula he e exis s a σsuch ha
|S( , s)x±(s)−x∗( )|< ² o all s≤σ.
7
Since he sys em is o de p ese ing
|xs−x∗(s)|<δ( )
2⇒ |S( , s)xs−x∗( )|< ² o all s≤σ.
Now we can use he con inuous dependence on ini ial condi ions o s∈[σ, ], along wi h
he in a iance o x∗(·), o gua an ee ha o δσ< δ( ) and su icien ly small
|xs−x∗(s)|< δσ⇒ |S( , s)xs−x∗( )|< ² o all σ≤s≤ .
Thus x∗(·) is pullback Lyapuno s able. ¤
We no e he e ha Kloeden [11] has shown ha one can gene alise he classical no ion
o a Lyapuno unc ion o co e many non-au onomous sys ems in such a way ha he e is
a Lyapuno unc ion associa ed wi h any pullback a ac ing se . In pa icula his esul s
imply he exis ence o a Lyapuno unc ion o a bounded locally pullback a ac ing
ajec o y o he equa ion ˙x= (x, ) p o ided ha (x, ) is locally Lipschi z in x.
3.3 No ions o ins abili y
In Langa e al. [15] we in oduced wo no ions o ins abili y. One is simply he con e se
o Lyapuno s abili y, while he o he , s onge , p ope y appea s o be mo e use ul.
De ini ion 8 We say ha Σ(·)is pullback uns able i i is no pullback Lyapuno s able,
i.e. i he e exis s a ∈Rand an ² > 0such ha , o each δ > 0, he e exis s an s <
and an x0∈N(Σ(s), δ)such ha
dis [S( , s)x0,Σ( )] > ².
We say ha Σ(·) is ‘asymp o ically uns able’ i i s uns able se UΣ(·), de ined below
(c . C auel [4]), is non- i ial (i.e. i UΣ( )6= Σ( )).
De ini ion 9 I Σ(·)is an in a ian se hen he uns able se o Σ,UΣ(·), is de ined as
UΣ(s) = {x0: lim
→−∞ dis [S( , s)x0,Σ( )] = 0}.
We say ha Σ(·)is asymp o ically uns able i o some we ha e
UΣ( )6= Σ( ).(3.2)
The powe o his de ini ion comes om he ollowing simple esul (see Langa e
al. [15] o he p oo ).
8
P oposi ion 3 I Σ(·)is asymp o ically uns able hen i is also pullback uns able and
canno be locally pullback a ac ing.
Mos no ions o ins abili y a e ela ed o he beha iou o solu ions x( ) as → −∞;
he no ion o ‘asymp o ic ins abili y’ de ined abo e is essen ially a ime- e e sed no ion o
‘ o wa ds a ac ion’. I should he e o e be unsu p ising ha i is possible o de ine an
al e na i e no ion o ins abili y based on a ime- e e sed e sion o pullback a ac ion:
De ini ion 10 An in a ian se Σ(·)is (locally) pullback epelling wi hin Di i is (lo-
cally) pullback a ac ing wi hin D o he ime- e e sed sys em, i.e. i Σ(·)⊂Dand o
any compac se K⊂Dand o each ∈R,
lim
s→+∞dis [S( , s)K, Σ( )] = 0.
3.4 An aside: linea s abili y in non-au onomous sys ems
We men ion he e ha we make li le use o linea no ions o s abili y in his pape . The e
appea o be majo p oblems wi h deducing any hing om such ‘in ini esimal’ beha iou
wi hou u he cons ain s. As an example, conside he equa ion
˙x=x−e−
1 + 2x2,
whose solu ion can be gi en explici ly as
x( , s;xs) = e
esx−1
s+ an−1( )− an−1(s).
I is clea ha i xsis ixed hen as s→ −∞
x( , s;xs)→x∗( ) = e
an−1( ) + π/2.
The ajec o y x∗( ) is globally pullback a ac ing, and also, since i is bounded as
→ −∞, locally pullback a ac ing (Lemma 1). Since we a e ea ing a scala equa ion,
he ajec o y is also pullback Lyapuno s able (Lemma 2). Howe e , suppose ha we
linea ise abou x∗( ), and ob ain
˙
X=·1−2e−
1 + 2x∗( )¸X
=·1−2
(1 + 2)( an−1( ) + π/2)¸X
9
and once again he igh -hand side is bounded below by mλ/2, his ime using (5.17).
Thus o each ixed he e exis s a σ such ha i s≤σ and |xs|is su icien ly small
he solu ion exis s on [s, ] and hence he o igin is locally pullback a ac ing.
When λ= 0.When λ= 0 he explici solu ion is
x( ) = 1
x−1
s+R
sg( ) d ,(5.19)
and o xs>0 i ollows om he asymp o ic posi i i y o gand a simpli ied e sion o he
abo e a gumen ha he o igin is locally pullback a ac ing in R+; and ha o xs<0
bu su icien ly small (depending on s), x( , s;xs)→0 as → −∞, and so he o igin is
asymp o ically uns able.
When λ > 0.This case was ea ed be o e he o mal s a emen o he p oposi ion. Only
he asymp o ic ins abili y o he o igin and he con e gence o xλ o ze o emain.
We deal i s wi h he asymp o ic ins abili y o he o igin. Since x( )≡0 and xλ(·) a e
solu ions and he equa ion is o de -p ese ing, any solu ion wi h 0 < xs< xλ(s) exis s
o all ≤s. Since 0 <Rs
−∞ eλF( )g( ) d < +∞and eλF( )→0 as → −∞ i ollows
ha o such a solu ion x( , s;xs)→0 as → −∞.
To show ha xλ( )→0 as λ→0, ix and ² > 0. Choose Tsuch ha
Z
T
g( ) d > 2eλF ( )/²
(which is possible since gis asymp o ically posi i e). Then
Z
−∞
eλF( )g( ) d =ZT
−∞
eλF( )g( ) d +Z
T
eλF( )g( ) d > Z
T
eλF( )g( ) d .
Now, choose λsu icien ly small ha
sup
∈[T, ]|eλF ( )−1|<eλF ( )
²R
T|g( )|d ,
and hen Z
−∞
eλF( )g( ) d > eλF ( )/²,
which implies ha xλ( )< ².
Including he ex a ‘ o wa ds’ condi ions in (5.13) and (5.12), when λ < 0 he o igin is
locally o wa ds a ac ing when xsis su icien ly small, since (5.12) gua an ees ha
in
≥sZ
s
eλF( )g( ) d > −∞.
16

When λ= 0 he o igin becomes locally o wa ds a ac ing. When λ > 0 he ajec o y
xλ(·) is now locally o wa ds a ac ing: o show his we can ea ange he explici solu ion
in o he al e na i e o m
µ1
x( )−1
xλ( )¶= eλ(F(s)−F( )) µ1
xs−1
xλ(s)¶.(5.20)
The e o e
|x( )−xλ( )|=xλ( )x( )
eλF( )
eλF(s)
xλ(s)xs|xλ(s)−xs|.(5.21)
The balance condi ion in (5.10) implies ha any solu ion wi h xs>0 is bounded as
→+∞. To see his, conside
eλF( )
x−1
seλF(s)+R
seλF( )g( ) d ≤MλR
−∞ eλF( )g( ) d
x−1
seλF(s)+R
seλF( )g( ) d
=MλRs
−∞ eλF( )g( ) d +R
seλF( )g( ) d
x−1
seλF(s)+R
seλF( )g( ) d .
Condi ion (5.12) gua an ees ha he second e ms in he nume a o and denomina o a e
posi i e o su icien ly la ge, and so
lim sup
→∞
x( )≤Mλmax µ1,xs
xλ(s)¶.
I he e o e ollows om (5.21) ha xλ(·) is o wa ds a ac ing while solu ions exis .
To show ha solu ions do no blow up o xs<(1 + αs)xλ(s), obse e ha
x−1
seλF(s)+Z
s
eλF( )g( ) d > 1
1 + αsZs
−∞
eλF( )g( ) d +Z
s
eλF( )g( ) d
=Z
−∞
eλF( )g( ) d −αs
1 + αsZs
−∞
eλF( )g( ) d .
Using he asymp o ic posi i i y o g his exp ession is posi i e o αssu icien ly small.
This implies ha xλ(·) is locally o wa ds a ac ing.
Unde he inal condi ion he esul s ollow by making he ans o ma ions
λ7→ −λ, x 7→ −x, and 7→ − .
¤
We no e he e ha an al e na i e o equi ing s onge condi ions a in ini y (such as
he asymp o ic posi i i y o g) migh be o make assump ions on in eg als o and g ha
a e uni o m in ime, e.g.
Z +T
g(s) ds≥γ > 0 and Z +T
|g(s)|ds≤Γ<+∞ o all ∈R.
17
Since (see Boh [1]) almos pe iodic unc ions ϕ(·) ha e ime a e ages ha con e ge
uni o mly,
sup
∈R¯¯¯¯Z +T
ϕ(s) ds−¯ϕ¯¯¯¯→0
as T→ ∞ (he e ¯ϕis he ime a e age o ϕ), such condi ions would na u ally include his
impo an class o speci ic examples.
5.2 Condi ions o localised bi u ca ing solu ions
We now gi e s onge , bu pe haps mo e na u al, condi ions on ( ) and g( ) ha ensu e
ha he balance condi ions (5.10) and (5.11) hold.
Lemma 6 Suppose ha
lim in
→−∞ g( )>0 (5.22)
and ha
0< m = lim in
→−∞
( )
g( )≤lim sup
→−∞
( )
g( )=M < +∞.(5.23)
Then o λ > 0
λm ≤lim in
→−∞ xλ( )≤lim sup
→−∞
xλ( )≤λM, (5.24)
while o λ < 0we ha e
lim in
s→−∞
eλF(s)
R
seλF( )g( ) d ≥ −mλ, (5.25)
whe e Fis an an ide i a i e o .
P oo . Fo any K > M he e exis s a Tsuch ha o all ≤Twe ha e g( )>0 and
( )
g( )≤K.
Fo such i ollows ha
Z
−∞
eλF(s)g(s) ds≥1
KZ
−∞
eλF(s) (s) ds
≥1
K·eλF(s)
λ¸
s=−∞
=1
λK eλF( ),
18
since F( )→ −∞ as → −∞ by (5.22) and (5.23). The e o e
xλ( ) = eλF( )
R
−∞ eλF(s)g(s) ds≤Kλ o all ≤T,
and hence
lim sup
→−∞
xλ( )≤Mλ.
Fo he lowe bound he p oo is simila , bu now using he ac ha o any k < m he e
exis s a Tsuch ha ( )
g( )> k o all ≤T.
The p oo o (5.25) ollows he same lines. ¤
5.3 The gene al case
We will now conside he gene al equa ion ˙x=G( , x, λ), and p o e a bi u ca ion heo em
based on assump ions on he Taylo coe icien s o G. Since we will impose condi ions
on hese coe icien s simila o hose in Lemma 6, we will be able o show ha he
sys em unde goes a ansc i ical bi u ca ion ha is a li le mo e akin o i s au onomous
coun e pa han ha in P oposi ion 5.
We now gi e ou o mal de ini ion o a ‘ ansc i ical bi u ca ion’ in a non-au onomous
sys em. No e ha we insis in he de ini ion ha he non-ze o ajec o y is in some sense
‘localised’ nea he o igin, and ha he equi ed beha iou depends only on he sys em in
he pas (pullback a ac ion and asymp o ic ins abili y). In ou esul s we will be able
o deduce u he de ails o he beha iou o solu ions by making addi ional assump ions
on he sys em in he u u e.
De ini ion 12 The sys em ˙x= (x, , λ)unde goes a local ansc i ical bi u ca ion a
x= 0,λ= 0 i he e exis s a λ0>0and an ² > 0such ha
(i) o all −λ0< λ < 0 he ze o solu ion is locally pullback a ac ing wi hin (−², 0] and
pullback a ac ing wi hin [0, ²); and he e is ano he nega i e comple e ajec o y
xλ( )wi hin (−², 0) ha is asymp o ically uns able and sa is ies
xλ( )→0as λ→0; (5.26)
(ii) o λ= 0 he ze o solu ion is asymp o ically uns able bu s ill pullback a ac ing
wi hin [0, ²); and
19
(iii) o 0<λ<λ0 he ze o solu ion is asymp o ically uns able, and he e is ano he
posi i e comple e ajec o y xλ( )wi hin (0, ²) ha sa is ies
xλ( )→0as λ→0 (5.27)
and is pullback a ac ing wi hin (0, ²).
While we only equi e poin wise con e gence in (5.26) and (5.27), we will in ac
ob ain uni o m con e gence in Theo em 7, which ea s he equa ion ˙x=G( , x, λ) whose
igh -hand side has he Taylo expansion
G( , x, λ) = G+Gxx+Gλλ+1
2Gxxx2+Gxλxλ +1
2Gλλλ2
+1
6Gxxxx3+1
3Gxxλx2λ+1
3Gxλλxλ2+1
6Gλλλλ3+. . .
(all exp essions in ol ing Gand i s de i a i es on he igh -hand side a e e alua ed a
( , 0,0)). We assume ha G( , 0, λ) = 0 o all and λ, and u he mo e ha Gx( , 0,0) =
0. This implies ha ∂kG/∂λk( , 0,0) = 0 o all and k∈Z+.
We he e o e ha e
G( , x, λ) = λ£Gxλ +1
3Gxλλλ+. . .¤x+£1
2Gxx +1
6Gxxxx+1
3Gxxλλ+. . .¤x2
and his mo i a es he ollowing heo em.
Theo em 7 Conside
˙x=G( , x, λ),
and assume ha
G( , 0, λ) = 0 o all λ∈Rand Gx( , 0,0) = 0.
Se ( ) = Gxλ( , 0,0) and g( ) = −1
2Gxx( , 0,0), and ew i e he equa ion as
˙x=λ[ ( ) + λφ( , λ)]x−[g( ) + γ( , x, λ)]x2,
whe e
φ( , 0) = 1
3Gxλλ( , 0,0) and γ( , 0,0) = 0.(5.28)
Assume ha
lim in
→±∞ g( )>0,(5.29)
20
ha
0< m = lim in
→±∞
( )
g( )≤lim sup
→±∞
( )
g( )=M < +∞,(5.30)
and ha
|φ( , λ)| ≤ h( ),|γλ( , x, λ)| ≤ h( ),and |γx( , x, λ)| ≤ h( ),(5.31)
whe e
lim sup
→±∞
h( )
g( )≤K.
Then he e is a local ansc i ical bi u ca ion as λpasses h ough ze o. Fu he mo e when
λ < 0 he ‘uns able’ ajec o y is pullback epelling in (−², 0); when λ= 0 he o igin is
locally o wa ds a ac ing in R+; and when λ > 0 he pullback a ac ing ajec o y xλ(·)
is o wa ds a ac ing in (0, ²).
No e ha he s anda d condi ions o a ansc i ical bi u ca ion in he au onomous equa-
ion ˙x= (x, λ) a e (see Glendinning, [7]):
(0, λ) = 0, x(0,0) = 0, xλ(0,0) >0,and xx(0,0) <0.
I G( , x, λ) = (x, λ) hen we eco e hese condi ions in ou heo em.
P oo . We assume h oughou ha |λ| ≤ ², whe e ²will be chosen ‘su icien ly small’.
No e ha i ollows om (5.28) and (5.31) ha
|γ( , x, λ)| ≤ h( )[|x|+|λ|].(5.32)
The o igin is locally pullback a ac ing in (−², ²) o λ < 0.While 0 < x( , s;xs)≤²we
ha e 0 ≤x( , s;xs)≤ ( , s;xs) whe e ( ) sol es
˙ =λ[ ( ) + ²h( )] −[g( )−2²h( )] 2wi h (s) = xs.
The e exis s a Tsuch ha i s≤ ≤T hen we can neglec he second e m; changing he
de ini ion o Ti necessa y, we can use he bound |h( )| ≤ K0 ( )/m ( o some K0> K)
o deduce ha
˙ ≤λ(1 −(²K0/m)) ( ) ,
om which i ollows ha
( )≤e(1−(²K0/m))λ(F( )−F( 0)) ( 0).
21

Once mo e dec easing Ti necessa y, so ha ( )>0 o all ≤T, i ollows ha
o s≤ ≤Twe ha e ( , s;xs)≤²p o ided ha 0 < xs≤²and hence, since he
compa ison x( , s;xs)≤ ( , s;xs) emains alid, i ollows ha
lim
s→−∞ S( , s)xs= 0 o all ≤T.
Since S(τ, ) is con inuous and ze o is in a ian we ha e
lim
s→−∞ S(τ, s)xs=S(τ, )·lim
s→−∞ S( , s)xs¸=S(τ, )0 = 0 o all τ∈R,
and he o igin is pullback a ac ing wi hin [0, ²).
While −²≤x( , s;xs)≤0 we ha e u( , s;xs)≤x( , s;xs)≤0 whe e u( ) sol es
˙u=λ[ ( )−²h( )]u−[g( )+2²h( )]u2wi h u(s) = xs;
he e o e while u≥ −²
˙u≥λ ( )[(1 −²K/m)u]−(1 + 2²K)u2/m.
Fo Tchosen such ha ( )>0 o all ≤T, and o 0 ≥xs≥ −λ(m−²K)/m(1 + 2²K)
i ollows ha
lim
s→−∞ S( , s)xs= 0 o all ≤T,
and a guing as abo e he o igin is locally pullback a ac ing wi hin (−², 0].
When λ= 0.While |x| ≤ ²we ha e
˙x≤ −[g( )−2²h( )]x2,
which immedia ely gi es he pullback a ac ion o he ze o solu ion wi hin [0, ²), and he
asymp o ic ins abili y o ze o, since o xs<0 we ha e x( , s;xs)→0 as → −∞.
The e is a posi i e ajec o y ha is pullback a ac ing in [0,∞)when λ > 0.While
|x( , s;xs)|,|λ|< ² we ha e
u( , s;xs)≤x( , s;xs)≤ ( , s;xs),(5.33)
whe e u( , s;xs) and ( , s;xs) a e he solu ions o
˙u=λ[ ( )−²h( )]
| {z }
−( )
u−[g( )+2²h( )]
| {z }
g+( )
u2wi h u(s) = xs
22
and
˙ =λ[ ( ) + ²h( )]
| {z }
+( )
−[g( )−2²h( )]
| {z }
g−( )
2wi h (s) = xs.(5.34)
In pa icula , we ha e an explici o m o he solu ion o (5.34), namely
( ) = eλF+( )
x−1
seλF+(s)+R
seλF+( )g−( ) d .
Using he balance condi ion (5.23) i ollows ha o λand xssu icien ly small, ( )≤²
o all ≤0. In his case he compa ison (5.33) emains alid o all such .
Due o he wo-sided balance and he balance be ween hand gi ollows ha we can
de ine he uppe and lowe solu ions
x+( ) = eλF+( )
R
−∞ eλF+( )g−( ) d
and
x−( ) = eλF−( )
R
−∞ eλF−( )g+( ) d ,
he pullback a ac o s o he uppe and lowe equa ions. We hen ha e
x−( )≤lim in
s→−∞ x( , s;xs)≤lim sup
s→−∞
x( , s;xs)≤x+( ).
The e o e he e exis s a pullback a ac o A( ) wi hin he phase space consis ing o he
in e al (0, ²). Since he sys em is o de -p ese ing, he e a e wo solu ions x1( ) and x2( )
such ha A( ) = [x1( ), x2( )], and so we ha e x−( )≤xj( )≤x+( ) o j= 1,2.
I we se z( ) = x1( )−x2( ) hen
dz
d ≤λ[ ( ) + ²h( )]z−g( )(x1+x2)z−[γ( , x1, λ)x2
1−γ( , x2, λ)x2
2]
≤λ +( )z−g( )(x1+x2)z−γ( , x1, λ)(x2
1−x2
2)
+[γ( , x1, λ)−γ( , x2, λ)]x2
2
≤λ +( )z−g( )(x1+x2)z+ 2²h( )[x1+x2]z+²x1h( )z
≤[λ +( )−(2g( )−5²h( ))x−( )]z.
Since
2g( )−5²h( )≥2−5²K
1 + ²K g+( )
his gi es
dz
d ≤"λ +( )−2−5²K
1+2²K
g+( )eλF−( )
R
−∞ eλF−( )g+( ) d #z.
23
We ha e
z( )≤z( 0)eI( , 0),
whe e
I( , 0) := Z
0
λ +(s)−2−5²K
1+2²K
g+(s)eλF−(s)
R
−∞ eλF−( )g+( ) d ds
=λ(F+( )−F+( 0)) −2−5²K
1+2²K ·ln Zs
−∞
eλF−( )g+( ) d ¸
s= 0
.
Now, −
M≤g+≤1+2²K
m−²K −,
and so
·ln Zs
−∞
eλF−( )g+( ) d ¸
s= 0≥ln µ1
λM eλF−( )¶−ln µ1+2²K
λ(m−²K)eλF−( 0)¶
=λ(F−( )−F−( 0)) + ln m−²K
M(1 + 2²K).
The e o e
I≤λ(F+( )−F+( 0)) −2−5²K
1+2²K λ(F−( )−F−( 0)) + C²,
whe e
C²=−2−5²K
1+2²K ln m−²K
M(1 + 2²K)>0.
Since
−≥1−(²K/m)
1+(²K/m) +
we also ha e
F−( )−F−( 0)≥1−(²K/m)
1+(²K/m)[F+( )−F+( 0)],
and so
I≤λ(F+( )−F+( 0)) ·1−2(1 −5
2²K)(1 −²K/m)
(1 + 2²K)(1 + ²K/m)¸+C².
I ollows ha o ²su icien ly small we can gua an ee ha z( ) = 0, and hence ha
he e is a single pullback a ac ing posi i e ajec o y x∗(·).
Now no e ha he abo e a gumen is in ac alid o any wo ajec o ies x1(·) and
x2(·) ha a e bounded below by x−( ). Now also no e ha any ajec o y x( , s;xs) wi h
xs>0 has x( , s;xs)>3
4x−( ) o la ge enough (c . a gumen ollowing (5.20) in he
p oo o P oposi ion 5); his is also enough o apply he abo e a gumen , and so x∗(·) is
a ac ing in (0, ²) as →+∞.
24
The o igin is uns able ‘downwa ds’ when λ > 0.We ha e 0 ≥x( )≥u( ) whe e u( )
sol es
˙u=λ[ ( )−²h( )]u.
As → −∞ we he e o e ha e u( )→0, and so we ha e x( )→0 oo.
The uns able ajec o y when λ < 0.The ans o ma ion x7→ −x, 7→ − , gi es he
exis ence o a candida e o he nega i e uns able ajec o y; i s ins abili y ollows om
he ac ha x∗(·) is a ac ing ‘ om abo e’ as →+∞.¤
6 Non-au onomous ‘simple pi ch o k’ bi u ca ion
The canonical au onomous example o an equa ion exhibi ing a pi ch o k bi u ca ion is
˙y=µy −y3.(6.1)
Fo µ < 0 he only ixed poin is he o igin, which is s able; while o µ > 0 he o igin is
uns able and he e a e wo new ixed poin s a ±√µwhich a e s able.
We now gi e a o mal de ini ion o wha we unde s and by a ‘pi ch o k bi u ca ion’ o
a non-au onomous sys em. No e ha as be o e all he beha iou in he de ini ion only
elies on he p ope ies o he equa ion ‘in he pas ’.
De ini ion 13 The sys em ˙x= (x, , λ)unde goes a localised pi ch o k bi u ca ion a
x= 0,λ= 0 i he e exis s a λ0>0and an ² > 0such ha
(i) o all −λ0< λ ≤0 he ze o solu ion is pullback a ac ing wi hin (−², ²);
(ii) when 0< λ < λ0 he ze o solu ion is asymp o ically uns able, and he e exis bounded
ajec o ies x+
λ( )and x−
λ( ) ha a e pullback a ac ing in (0, ²)and (−², 0) espec-
i ely, and sa is y
x±
λ( )→0 as λ↓0
uni o mly on compac subse s o R.
Since equa ion (6.1) is in a ian unde he ans o ma ion y7→ −yi is con enien o
conside he new a iable x= 2y2, which sa is ies he equa ion
˙x= 2µx −x2.
25
i ollows ha any ajec o y wi h
x−:= −δ m−²K
1 + ²K < xs≤²
has |x( , s;xs)| ≤ ² o all ≥s, and hence ha
δ m−²K
1 + ²K ≤lim
s→−∞ x( , s;xs)≤².
Thus he pullback a ac o in (x−, ²] consis s o he in e al [x1( ), x2( )]. Conside ing
he di e ence z=x1−x2 his sa is ies
dz
d =λ[φ( , x1, λ)−φ( , x2, λ)] −(x1+x2)g( )z−x2
1ψ( , x1) + x2
2ψ( , x2)
≤δ2h( )z−(x1+x2)g( )z+ (x2
2−x2
1)ψ( , x1)+[ψ( , x2)−ψ( , x1)]x2
1
≤δ2h( )z−(x1+x2)g( )z+ [²+δ2]h( )(x1+x2)z+h( )z²2
≤C[²2h( )−²g( )]z
≤ −C(1 −K²)²g( )z
as ²→0. I ollows ha o ²chosen su icien ly small, z( ) = 0.
Using he same a gumen o he ime- e e sed sys ems gi es a saddle-node bi u ca ion.
¤
8 The balance hypo hesis: examples
In his inal sec ion we gi e some examples demons a ing ha wi hou some kind o
‘balance’ be ween successi e e ms in he Taylo se ies we canno expec he ype o bi-
u ca ion esul s abo e. No e ha while all hese examples a e asymp o ically au onomous
(as → ∞), he beha iou o he non-au onomous equa ion is di e en om ha o i s
au onomous limi .
Ou simples example is
˙x=λx −e− x2wi h x(s) = xs≥0,
whe e he exponen ial e m p oduces e y s ong dissipa i i y as → −∞. F om he
explici solu ion
x( , s;xs) = eλ
x−1
seλs + (λ−1)−1(e(λ−1) −e(λ−1)s)
32

i is clea ha while o λ < 0 he o igin is pullback a ac ing in R+, his is also he
case when 0 <λ<1. Thus he ‘one-sided pi ch o k’ ype bi u ca ion ha we migh
expec is supp essed. [No e, howe e , ha he comple e (bu unbounded) ajec o y
x∗( ) = (λ−1)e is o wa ds a ac ing o all λ > 0.]
In he p e ious example we made one o he e ms o he Taylo expansion ha plays
a p ime ˆole in he bi u ca ion blow up as → −∞. Howe e , we can also shi his
beha iou o he highe -o de e ms and un in o simila p oblems. Fo he equa ion
˙x=λx −x2−e− x3wi h x(s) = xs≥0
i is clea ha o λ < 0 he o igin is globally pullback (and o wa ds) a ac ing; while
o λ > 0 we ha e
λx −x2−e− x3≤λx −e− x3,
so ha he con inued pullback a ac ion o he ze o solu ion ollows he p e ious example
a e se ing y=x2and µ= 2λ(see Sec ion 6).
A simila example, bu one in which he highe -o de e ms p oduce ins abili y ( a he
han enhance he s abili y), is
˙x=λx −2x2+ e− x3.
Gi en an ini ial condi ion xs, wha e e he alue o λwe can choose Tsu icien ly la ge
and nega i e ha
λx −2x2+ e− x3≥1
2e− x3 o all ≤T.
I ollows ha o any xs,
lim
s→−∞ x( , s;xs) = +∞,
and he e is ne e a pullback a ac ing ajec o y.
9 Conclusion
We ha e ied o de elop a gene al heo y o bi u ca ions in non-au onomous scala
sys ems, in pa icula gi ing a se o possible de ini ions o ansc i ical, pi ch o k, and
saddle-node bi u ca ions ha depend only on p ope ies o he sys em in he pas .
The e a e, o cou se, many ways in which hese esul s could be imp o ed. The main
p oblem is he es ic i e na u e o some o he condi ions ha we ha e equi ed on he
33
e ms in ou Taylo expansion. As we ema ked a he end o Sec ion 5.1, i should be
possible o p o e simila esul s eplacing assump ions such as he asymp o ic posi i i y
o e ms in he equa ion by ime in eg a ed (o pe haps ime-a e aged) condi ions such
as Z +T
g(s) ds≥γ > 0 o all ∈R.
The e a e highe -dimensional bi u ca ion esul s o ce ain sys ems, in pa icula in
he almos pe iodic case (see Kloeden [12], o example). We hope o ex end he esul s
he e o gene al highe dimensional sys ems, by conside ing he scala sys ems ob ained
by es ic ing a en ion o an app op ia e cen e mani old, as is done in he au onomous
case.
Acknowledgemen s.
We a e all mos g a e ul o he Royal Socie y o hei suppo : ou collabo a ion has
been unded by a Royal Socie y Join P ojec g an . JCR is cu en ly a Royal Socie y
Uni e si y Resea ch Fellow, JAL and ASF ha e been suppo ed in pa by M.E.C. (Spain,
Fede ), P oyec os BFM2002-03068 and BFM2003-06446, espec i ely. We would like o
hank wo anonymous e e ees o hei help ul and de ailed sugges ions.
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