Bifurcations in non-autonomous scalar equations

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. Re e ences [1] H. Boh . Almos pe iodic unc ions. Chelsea Publishing Company, New Yo k, 1947. [2] D.N. Cheban, P.E. Kloeden, & B. Schmal uß. The ela ionship be ween pullback, o wa ds and global a ac o s o nonau onomous dynamical sys ems. 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