Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

NON-HYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS IN PLANAR FILIPPOV SYSTEMS: A CASE STUDY APPROACH MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE Abs ac . Bounda y equilib ium bi u ca ions in piecewise smoo h discon inuous sys ems a e cha ac e ized by he collision o an equi- lib ium poin wi h he discon inui y su ace. Gene ically, hese bi- u ca ions a e o codimension one, bu he e a e scena ios whe e he phenomenon can be o highe codimension. He e, he possible col- lision o a non-hype bolic equilib ium wi h he bounda y in a wo- pa ame e amewo k and he nonlinea phenomena associa ed wi h such collision a e conside ed. By dealing wi h plana discon inuous (Filippo ) sys ems, some o such phenomena a e poin ed ou h ough speci ic ep esen a i e cases. A me hodology o ob aining he co esponding bi-pa ame ic bi u ca ion se s is de eloped. 1. In oduc ion Piecewise smoo h dynamical sys ems a e na u ally used o model a g ea a ie y o enginee ing de ices, which a e discon inuous on a mac oscopic scale. Examples o such sys ems include walking mechanisms (Ga cia, Cha - e jee, Ruina and Coleman, 1998), DC/DC con e e s (Bane jee and Ve gh- ese, 2001) and mechanical sys ems wi h impac s and/o ic ion (B oglia o, 1999; B oglia o, 2000). The co esponding ma hema ical models a e se s o o dina y di e en ial equa ions (ODEs) cha ac e ized by ha ing some discon- inui y ei he in he de i a i es o he ec o ield o in he ec o ield i sel . These discon inui ies appea only in a small subse o he phase space and equen ly such subse is a smoo h mani old, o swi ching mani old, which is gene ically ans e sal o he ec o ield. Discon inuous di e en ial sys ems, also called Filippo sys ems (Filippo , 1988), o m an impo an class which ha e been ecen ly s udied, ocussing on codimension-one bi u ca ions (bo h o equilib ia and closed o bi s) and wo-pa ame e bi u ca ions o limi cycles. Filippo sys ems a e se s o ODEs wi h discon inuous igh -hand side. I has been shown ha hese sys ems can exhibi so-called sliding solu ions. Namely, i is possible o 1 2 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE iden i y egions o he swi ching mani old such ha he ec o ield poin s owa ds he mani old i sel om bo h o i s sides. Then, he sys em ajec- o y is cons ained o e ol e (o slide) on he bounda y i sel un il he ec o ield on one o he o he side o he mani old changes di ec ion. As shown in (Filippo , 1988), and u he de ailed below, i is possible o cons uc an app op ia e ec o ield gene a ing such sliding ajec o ies as a con ex combina ion o he sys em ec o ields on bo h sides o he bounda y. Filippo sys ems ha e been ecen ly shown o exhibi a no el class o bi u ca ions, e med as discon inui y-induced bi u ca ions (DIBs), because o hei discon inuous na u e (di Be na do, Budd, Champneys, Kowalczyk, No dma k, Oli a and Pii oinen, 2006). In pa icula , i was obse ed ha hese sys ems unde go d ama ic ansi ions whene e one o hei in a ian se s (equilib ia, limi cycles e c) in e ac s non- i ially wi h he swi ching mani old in phase space. A s iking example is he g azing-sliding bi u - ca ion which was s udied in (di Be na do, Johansson and Vasca, 2001; di Be na do, Kowalczyk and No dma k, 2002). The e, a limi cycle gains a angen ial in e sec ion wi h he mani old a some pa ame e alue. Then, i can be shown ha in many si ua ions he cycle does no pe sis gi ing ise o mo e complex beha io including chaos. Ano he example is ha o bounda y equilib ium bi u ca ions (BEBs) whe e, unde pa ame e a ia- ion, he collision is obse ed o an equilib ium poin wi h he discon inui y su ace. This class o bi u ca ions was ecen ly s udied in (di Be na do, No dma k and Oli a , 2007), whe e a en ion is ocussed on one-pa ame e BEBs. Many enginee ing sys ems a e cha ac e ized by mo e han one pa ame- e . A basic mechanical sys em, o example, has a leas h ee pa ame e s (mass, ic ion coe icien and damping). The e o e, i is becoming inc eas- ingly appa en ha highe -codimension DIBs need o be in es iga ed o gain a deepe unde s anding o he bi u ca ion beha io de ec ed expe imen ally in a g owing numbe o eal-wo ld applica ions. A emp s ha e been made in he li e a u e o classi y and s udy some wo-pa ame e DIBs, see o ex- ample he esul s in (Cunha, Pagano and Mo eno, 2003) and (Kowalczyk, di Be na do, Champneys, Hogan, Home , Kuzne so , No dma k and Pi- i oinen, 2006). While a en ion has mos ly ocussed o DIBs o limi cycles, li le is known on wo-pa ame e bounda y equilib ium bi u ca ions. The main aim o his pape is o s a illing in his gap in he li e a u e by illus a ing some o he possible ea u es o wo-pa ame e BEBs. The idea is o analyze in de ail a numbe o simple case s udies unde going a wo- pa ame e BEB. In pa icula , we s udy a se o cases whe e he equilib ium colliding wi h he discon inui y su ace is non-hype bolic. We s udy he case o such equilib ium ha ing a null eal eigen alue a he BEB (bounda y- old NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 3 bi u ca ion) o a pai o complex conjuga e eigen alues (bounda y-cen e and bounda y-Hop ). We p esen he un olding o he wo-pa ame e BEB o each o he plana case s udies p esen ed in he pape , showing ha he local bi u ca ion dia- g am is cha ac e ized by he p esence o bi u ca ion cu es co esponding o bo h smoo h and discon inui y-induced bi u ca ions. In so doing, we p esen a new in e p e a ion o he condi ions o classi y one-pa ame e BEB in Fil- ippo sys ems. Also, we illus a e a simple me hodology o cons uc he local bi u ca ion diag ams in he neighbo hood o a wo-pa ame e BEB. I is wo h men ioning he e ha such me hodology can be easily ex ended o cope wi h highe dimensional p oblems. The es o he pape can be ou lined as ollows. In Sec. 2 we gi e a b ie o e iew o he sys ems o ou conce n and de ine he di e en ypes o equilib ia which a e deal wi h in he pape . Then, in Sec. 3, we in o- duce a classi ica ion s a egy o one-pa ame e BEBs which is hen applied o each o he case s udies discussed in he pape . The me hodology is de- sc ibed in de ail in Sec. 4. Then, we conside h ee case s udies in Sec. 5. The bounda y- old bi u ca ion o a non-hype bolic equilib ium is s udied in Sec. 5.1. Namely, we assume ha a he bounda y equilib ium bi u ca ion, he non-hype bolic equilib ium also unde goes a smoo h old and p esen he un olding o he co esponding wo-pa ame e BEB. The case o a lin- ea cen e colliding wi h he bounda y is discussed in Sec. 5.2, whe e he bounda y-cen e bi u ca ion is un olded. Finally, he simul aneous occu - ence o a Hop bi u ca ion and a BEB (bounda y-Hop ) is discussed in Sec. 5.3. Conclusions a e d awn in Sec. 6. 2. Filippo sys ems, o dina y equilib ia, pseudo-equilib ia and bounda y equilib ium poin s To s a wi h, le us e iew some elemen a y ac s abou Filippo sys- ems. We will adop he simples se ing which is gene al enough o appli- ca ions, wi hou in ending o cope wi h all possible si ua ions. Ou main e e ence will be he wo k o Kuzne so e al. (Kuzne so , Rinaldi and G agnani, 2003) and we es ic ou sel es o conside only au onomous ec- o ields, wi h pa icula a en ion o equilib ia and hei possible bi u ca- ions. Suppose ha we a e gi en wo smoo h ec o ields (i):Rn→Rn, i= 1,2,and a smoo h non-cons an scala unc ion h:Rn→Rsuch ha he discon inui y mani old Σ = {x∈Rn:h(x) = 0} is su icien ly egula and gene ically ans e sal o bo h ec o ields. Namely, he scala p oduc s ∇h, (i), o i= 1,2,gi ing each no mal 4 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE componen o he ec o ields wi h espec o he mani old, a e no iden- ically ze o on Σ (e en i hey could anish in some poin s o he man- i old). Le us de ine he wo open egions S1={x∈Rn:h(x)<0}and S2={x∈Rn:h(x)>0}, and he Filippo ec o ield (1) F:Rn Σ→Rn,such ha F(x) =  (1)(x),i x∈S1, (2)(x),i x∈S2. Then, in gene al he esul ing ec o ield canno be con inuously ex ended o he mani old Σ and some p ocedu e mus be designed in o de o de ine app op ia ely he solu ions o he di e en ial sys em (2) ˙x =F(x). Take a poin ˆx ∈Σ and assume ha ∇h(ˆx), (1)(ˆx)>0.Then, such a poin is eachable o some o bi o he sys em ˙x = (1)(x) wi h ini- ial condi ion wi hin S1. I u he mo e i u ns ou ha we also ha e ∇h(ˆx), (2)(ˆx)>0,i seems na u al o conca ena e he o bi o ˙x= (1)(x) inishing a ˆx wi h he o bi o ˙x = (2)(x) s a ing a he same poin . We can adop he same easoning in he analogous case, when bo h ∇h(ˆx), (2)(ˆx)<0 and ∇h(ˆx), (1)(ˆx)<0, and de ine he c ossing se (some imes called he sewing pa o Σ) as Σc=nx∈Σ : D∇h(x), (1)(x)E·D∇h(x), (2)(x)E>0o⊂Σ. In poin s belonging o Σc, s anda d solu ions o he wo sys ems can be joined o o m a solu ion whose o bi c osses he discon inui y mani old. No e ha , apa om he abo e geome ical in e p e a ion on he poin ing di ec ions o he ec o ields in poin s o he discon inui y su ace, one can also hink o he ins an aneous change a e o he alue o halong a solu ion x( ), and conclude ha d d [h(x( ))] = ∇h(x( )) ·d d x( ) = D∇h(x( )), (i)(x( ))E, wi h i= 1 o 2, depending on he co esponding ec o ield associa ed o he solu ion conside ed. Acco ding o Kuzne so e al. (Kuzne so e al., 2003), we can de ine he non-c ossing pa o Σ,o sliding pa , as he complemen o Σcin Σ, ha is, (3) Σs=nx∈Σ : D∇h(x), (1)(x)E·D∇h(x), (2)(x)E≤0o. NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 5 Thus, while he c ossing se is an open se , he sliding se should be he union o closed subse s (some o hem possibly consis ing o isola ed sliding poin s)1. Wi hin he sliding se , he Filippo me hod can be used o cons uc solu ions, o be conside ed as ex ensions o solu ions o (2). Such a me hod consis s in de ining a new ec o ield compu ed om an adequa e con ex combina ion o he wo o iginal ec o ields, namely (4) Fs(x) = (1 −λ) (1)(x) + λ (2)(x), whe e o each x∈Σs he alue o λis selec ed such ha h∇h(x), Fs(x)i= 0. A simple compu a ion shows ha (5) λ=λ(x) = ∇h(x), (1)(x) ∇h(x), (1)(x)− (2)(x), p o ided he abo e denomina o does no anish and hen, by using he de ini ion o Σs, one concludes ha 0≤λ(x)≤1, o all x∈Σs. The e o e, we ha e a explici de ini ion o he sliding ec o ield, namely (6) Fs(x) = ∇h(x), (2)(x) (1)(x)−∇h(x), (1)(x) (2)(x) ∇h(x), (2)(x)− (1)(x). I ∇h(x), (2)(x)− (1)(x)= 0 o some x∈Σs, hen we say ha such xis a singula sliding poin . This can happen in h ee cases: bo h ec o ields a e angen o Σ; one is angen and he o he anishes; bo h ec o ields anish a he poin . In all hese singula cases, i we exclude in ini ely degene a e cases and he sliding poin is non-isola ed, i is possible o de ine he ec o ield Fsby con inui y a gumen s. Fo isola ed singula sliding poin s i will be aken Fs(x) = 0. Clea ly, he bounda y o he sliding se Σs(as a subse o Σ) is associa ed o each one o he wo equali ies ∇h(x), (1)(x)= 0, ∇h(x), (2)(x)= 0; ha is, λ(x) = 0 o λ(x) = 1 wi h x∈Σ. As usual in he analysis o dynamical sys ems, we mus look o he simples solu ions o ganizing he dynamics, namely he cons an solu ions associa ed o es poin s no mally called equilib ia. O cou se, he Filippo 1We mus men ion ha some au ho s p e e o quali y as sliding only he case when ∇h(x), (2)(x)<0<∇h(x), (1)(x) ( o be called he e a ac i e sliding case) and, o he wise, ha is i ∇h(x), (1)(x)≤0≤∇h(x), (2)(x), hen hey speak o he escaping pa o Σ. 6 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE sys em (2) inhe i he equilib ia o each ec o ield i(x), bu we mus be cau ious and dis inguish be ween eal o i ual equilib ia. In pa icula , we call •admissible o eal equilib ium poin s o all he solu ions o (1)(x) = 0 ha belong o S1and he solu ions o (2)(x) = 0 ha belong o S2, while • i ual equilib ium poin s a e he solu ions o (1)(x) = 0 ha belong o S2, and he solu ions o (2)(x) = 0 ha belong o S1. Al hough i ual equilib ia a e no equilib ia o F(x), hey can s ill o ga- nize he dynamics in he co esponding egion. Rega ding now he induced dynamical sys em ˙x =Fs(x) on he sliding se , we no e ha apa om iso- la ed singula sliding poin s, he sliding ec o ield Fscan anish in o he poin s which beha e like eal equilib ia o such dynamical sys em i we es ic ou a en ion o he se Σs. They also a e, in some sense, equilib ia o sys em (2) and will be called pseudo-equilib ium poin s. Fo ins ance, when bo h ec o s (i)a e ans e sal o Σ in a ce ain poin o his su ace and u he mo e hey a e an i-collinea , ha is, he e exis λ1, λ2>0, such ha (7) λ1 (1)(x) + λ2 (2)(x) = 0, hen a simple a gumen shows ha he poin is necessa ily in Σs, since he alue o ∇h(x), (i)(x),i= 1,2 is non-ze o and wi h di e en sign. In ac , i is immedia e o conclude ha a such poin one has Fs(x) = 0 and so i is a pseudo-equilib ium o (2). Recip ocally, i xis a poin in Σswi h Fs(x) = 0 and i is no a angency poin , we easily conclude ha bo h ec o ields a e an i-collinea a he poin . I o ins ance a such poin we assume ∇h(x), (2)(x)<0<∇h(x), (1)(x),we conclude ha wo o bi s, each one co esponding o a ec o ield, collide wi h opposi e di ec ions and de e mine a es poin o he global ec o ield. This es poin , di e en ly om wha obse ed o admissible equilib ia, can be eached in ini e ime om poin s belonging o hese wo o bi s. A gene al and uni ied necessa y condi ion o equilib ia, o be called Gene alized Condi ion o Equilib ia (GCE), can be es ablished om Eq. (6). E ec i ely, he condi ion Fs(x) = 0 ansla es o (8) D∇h(x), (2)(x)E (1)(x)−D∇h(x), (1)(x)E (2)(x) = 0 and i is ob ious ha his condi ion is also sa is ied by i ual and admissible equilib ia. To end his sec ion, an impo an ema k is ha Fs(x) = 0 when any o he ec o ields anishes a a poin o Σ. Then we de ine such a poin as a bounda y equilib ium poin . Thus, a poin xis a bounda y equilib ium poin NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 7 i (1)(x) = 0 o (2)(x) = 0 and h(x) = 0. On he one hand, a bounda y equilib ium poin can be seen as he c i ical case be ween admissible and i ual equilib ia. On he o he hand, i we exclude he excep ional case o being a singula sliding poin , we deduce om (5) ha ei he λ(x) = 0 o λ(x) = 1, ha is, he poin is also a he bounda y o he sliding se Σs. 3. Bounda y equilib ium bi u ca ions We will now discuss he occu ence o bounda y equilib ium bi u ca ions in Filippo sys ems. We suppose ha he wo de ining ec o ields and he mani old de e mining unc ion hdepend on some eal pa ame e µ∈R. Also, we assume ha o a c i ical alue o he pa ame e , say µ= 0, he e appea s a bounda y equilib ium poin associa ed o he i s ec o ield (1). Wi hou loss o gene ali y, we assume ha he poin is a he o igin, ha is, (x, µ) = (0,0) is a solu ion o he equa ions (1)(x, µ) = 0,(9) h(x, µ) = 0, and he co esponding solu ion b anch ¯x(µ),wi h ¯x(0) = 0, is egula , ha is, A=Dx (1)(0,0) is in e ible. I we also assume ha he b anch is ans e sal o Σ,namely, d dµh(¯x(µ), µ)µ=0 6= 0, a na u al ques ion a ises. How does he si ua ion change in passing he pa ame e h ough he c i ical alue µ= 0? Le us i s w i e in a explici way he ans e sali y condi ion. Fo ha , we apply he implici unc ion heo em (I.F.T. o sho , in he sequel) o he i s equa ion in (9) by di e en ia ing wi h espec o µ, o ge Dx (1)(¯x(µ), µ)d dµ ¯x(µ) + Dµ (1)(¯x(µ), µ) = 0, and, by conside ing µ= 0,a e de ining M=Dµ (1)(0,0),we ob ain ¯x′(0) = d dµ ¯x(µ)µ=0 =−A−1M. Then, now d dµh(¯x(µ), µ) = Dxh(¯x(µ), µ)d dµ ¯x(µ) + Dµ (1)(¯x(µ), µ), 8 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE and de ining CT=Dxh(0,0), N=Dµh(0,0), he ans e sali y condi ion educes o (10) d dµh(¯x(µ), µ)µ=0 =N−CTA−1M6= 0. I should be no iced ha o µ= 0, he o igin is also an equilib ium poin o he sliding ec o ield (4) co esponding o he alue λ= 0, ha is, i is in one o he wo bounda ies o he sliding se Σs. Thus, coming back o he pending ques ion, such equilib ium could be also a poin o a solu ion b anch ˜x(µ),˜ λ(µ), wi h ˜x(0),˜ λ(0)= (0,0), o he sys em o equa ions (1 −λ) (1)(x, µ) + λ (2)(x, µ) = 0, h(x, µ) = 0. Such a solu ion b anch o he sliding ec o ield will be locally egula when he hypo heses o I.F.T. a e ul illed in o de o sol e he abo e sys em o (x, λ) in e ms o µ, nea he poin (x, λ, µ) = (0,0,0). I hen su ices ha in he ollowing sys em o equa ions, ob ained by implici di e en ia ion wi h espec o µ, namely (1−λ)Dx (1)(x, µ)+λDx (2)(x, µ) (2)(x, µ)− (1)(x, µ) Dxh(x, µ) 0  x′(µ) λ′(µ)+ +(1 −λ)Dµ (1)(x, µ) + λDµ (2)(x, µ) Dµh(x, µ)=0 0 (11) he ma ix o coe icien s be non-singula a (x, λ, µ) = (0,0,0).Taking in o accoun ha (1)(0,0) = 0 by ou ini ial hypo hesis, and in oducing he ec o B= (2)(0,0), he condi ion can be w i en as de A B CT06= 0. Using he equali y A−10 CTA−1−1 A B CT0=I A−1B 0CTA−1B, we see ha de A B CT0=−CTA−1B·de A, and ou conclusion is ha i CTA−1B6= 0, he e exis s locally a unique equilib ium b anch ˜x(µ),˜ λ(µ), wi h ˜x(0),˜ λ(0)= (0,0) o he sliding NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 9 ec o ield which will be admissible whene e 0 ≤˜ λ(µ)≤1. Mo eo e , om (11), using he I.F.T. we ha e A˜x′(0) + B˜ λ′(0) + M= 0, CT˜x′(0) + N= 0. A e some s anda d manipula ion and ecalling (10), we hen ha e ˜ λ′(0) = N−CTA−1M CTA−1B=1 CTA−1B·d dµh(¯x(µ), µ)µ=0 . Hence, o he wo colliding equilib ium b anches, we conclude ha (12) d dµ ˜ λ(µ)µ=0 =1 CTA−1B·d dµh(¯x(µ), µ)µ=0 , so ha we can easily p o e he ollowing esul . No e ha he heo em below al eady appea ed in (di Be na do e al., 2007), bu he e we p o ide a mo e de ailed de i a ion by eso ing o he I.F.T. Theo em 1. Assume ha o µ= 0 an equilib ium b anch o he ec o ield (1)(x, µ)c osses he discon inui y mani old h(x, µ) = 0 a he o i- gin, being his poin egula , ha is, A=Dx (1)(0,0) is a non-singula ma ix. Fu he mo e, suppose ha he c ossing is ans e sal, ha is, i M=Dµ (1)(0,0),CT=Dxh(0,0), and N=Dµh(0,0) hen d dµh(¯x(µ), µ)µ=0 =N−CTA−1M6= 0, and he non-degene acy condi ion CTA−1B6= 0,whe e B= (2)(0,0), is sa is ied. The ollowing s a emen s hold. (a): (Pe sis ence o one equilib ium o he Filippo sys em) I CTA−1B > 0 he equilib ium b anch o (1)(x, µ)and he equilib- ium b anch o he sliding ec o ield Fsdo no coexis o µ6= 0 and small. Mo e p ecisely, when µ a ies in a neighbo hood o 0one admissible equilib ium o (1) becomes non-admissible o µ= 0 and i is ans o med in o an admissible pseudo-equilib ium, ha is, in o an admissible equilib ium o he sliding ec o ield (o ice e sa). (b): (Non-smoo h old o annihila ion o equilib ia o he Filippo sys em) When he condi ion CTA−1B < 0holds, he equilib ium b anch o (1)(x, µ)and he equilib ium b anch o he sliding ec- o ield Fscoexis o small alues de µ, bu only ei he o µ < 0 o o µ > 0. Tha is, when µ a ies in a neighbo hood o 0one ad- missible equilib ium o (1) collides o µ= 0 wi h one admissible pseudo-equilib ium and bo h become non-admissible. 16 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE he le , namely (2)(x, y, ν) = −1 0, which ob iously canno gi e ise o any admissible equilib ia. No e ha his choice also implies ha any sliding egion will be a ac i e, and so we will no deal wi h escaping egions. The second s ep in ou analysis is o compu e he sliding se Σs. Acco d- ing o (15), we ha e he se (x, y)∈R2:x=ρ, (−1) ν−x2=x2−ν≤0 ha is Σs= Σs(ρ, ν) = (x, y)∈R2:x=ρi ν≥ρ2, ∅i ν < ρ2, so ha , when i is non-emp y he e is no es ic ion o yand we ge Σs= Σ, ha is he ull e ical line x=ρ. Assuming ν≥ρ2, and using (16) we de ec a pseudo-equilib ium poin in Σswhen ν−x2·0−(−y)(−1) = 0, ha is y= 0.No e ha , since D∇h(x), (2)(x)− (1)(x)E= [1,0]Tx2−ν−1 y=x2−ν−1, acco ding o (6) we ge (17) Fs(x)= 1 ρ2−ν−1(−1) ν−ρ2 −y−ν−ρ2−1 0=1 ρ2−ν−10 y. Thus, he dynamics on he sliding se Σsis go e ned by he di e en ial equa ion ˙y=y ρ2−ν−1, and as always ρ2−ν−1<0, he pseudo-equilib ium poin (ρ, 0) is a ac i e, beha ing as a pseudo-node whene e i exis s. The las s ep is o analyze he possible bounda y equilib ium bi u ca ion when he pa ame e ρis a ied. Looking a he admissible equilib ia o (1), namely (−√ν, 0) and (√ν, 0) o ν≥0, i is immedia e o conclude ha we ha e collisions be ween hese equilib ia and he discon inui y bounda y o ρ=±√ν, o equi alen ly o he poin s o he pa abola ν=ρ2in he pa ame e plane. To apply Theo em 1, we need o check he ans e sali y and he non-degene acy condi ion, o which we compu e NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 17 A=Dx (1)(±√ν, 0) = ∓2√ν0 0−1, B= (2)(±√ν, 0) = −1 0, CT=Dxh(±√ν, 0) = 1 0  along wi h M=Dρ (1)(±√ν, 0) = 0 0,and N=Dρh(±√ν, 0) = −1. We mus exclude he case ν= 0, he o igin o he pa ame e plane, whe e A u ns ou o be singula . Fo ν6= 0, we see ha N−CTA−1M=−1 and he ans e sali y equi emen holds. Rega ding he non-degene acy condi ion, we ob ain CTA−1B=1 0 ∓1 2√ν0 0−1 −1 0=±1 2√ν, so ha we will ha e pe sis ence a he collision o (√ν, 0) and non-smoo h old a (−√ν, 0). The bi u ca ion se compu ed abo e is depic ed in Fig. 1 whe e, apa om he wo b anches o he pa abola co esponding o he wo di e en bounda y equilib ium bi u ca ions, we see he hal -s aigh line ν= 0, wi h ρ > 0,whe e a s anda d saddle-node bi u ca ion akes place. These lines, o ganized a ound he o igin which beha es like a codimension wo bi u - ca ion poin , de e mine ou open egions in he pa ame e plane. Wi hin each pa ame e egion he phase planes a e quali a i e simila . The phase space diag ams co esponding o he a ious egions labelled in Fig.1 a e shown in Fig. 2. S a ing om egion 1 in Fig. 2(1), we see ha no equilib ium exis . Then, unde pa ame e a ia ion, he bi u ca ion cu e co esponding o a smoo h old bi u ca ion is c ossed as we mo e in o egion 2. As shown in Fig. 2(2), his co esponds o he c ea ion o wo admissible equilib ia. Fu he pa ame e a ia ions, cause one o his equilib ia o unde go a BEB associa ed o pe sis ence (as shown analy ically abo e). Hence, in egion 3, we see he coexis ence o one admissible equilib ium and a pseudo equilib ium (see Fig. 2(3)) which a e hen annihila ed h ough ano he BEB along he cu e labelled as EAin Fig. 1. 5.2. Case S udy 2: Bounda y-cen e bi u ca ion. As a second exam- ple o a possible non-hype bolic poin which collides wi h he discon inui y su ace, le us conside as be o e ha he mani old Σ is gi en by he e ical 18 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE line x=ρ, ha is h(x, y) = x−ρ, so ha o x < ρ we ha e he ec o ield (1)(x, y) = νx −y x. Thus, he e is only a possible equilib ium ( o be admissible o i ual de- pending on ρ) a (¯x, ¯y) = (0,0) ha becomes non-hype bolic o ν= 0 and will c oss Σ o ρ= 0.Now he non-hype bolici y comes om he ac ha he eigen alues a e on he imagina y axis o he complex plane o c i ical alues o pa ame e s. Thus, he equilib ium is a linea ocus as long as 0<|ν|<2,being a linea cen e o ν= 0. As he second ec o ield in ol ed, le us conside again he case o a cons an ho izon al ec o ield pushing o he le , say (2)(x, y) = −1 0. We mus an icipa e ha o ρ > 0, he passage h ough he line ν= 0 cons i u es a bi u ca ion, since he o igin changes i s abili y becoming a uns able ocus o ν > 0.When ν= 0,we ha e a linea cen e con igu a ion bounded by Σ. To compu e he sliding se Σs, we p oceed as be o e by ecalling (15), ge ing he condi ions x=ρ, y −νx ≤0,and so we a i e a Σs= Σs(ρ, ν) = (x, y)∈R2:x=ρ, y ≤νρ, which ep esen s a hal pa o he s aigh line Σ.No e ha he bounda y o his se is he poin (ρ, νρ),whe e he o bi s o (1) ha e a angency wi h Σ, isible o ρ > 0 and in isible when ρ < 0.Using GCE condi ion (16), which educes he e o x= 0, we de ec pseudo-equilib ium poin s in Σsonly i ρ= 0. Then he sys em has a con inuum o pseudo-equilib ia a (0, y) wi h y≤0, independen ly on he alue o ν, which leads o a e ical line in he bi u ca ion se , namely he ν-axis o he pa ame e plane. Since now D∇h(x), (2)(x)− (1)(x)E= [1,0]Ty−νx −1 x=y−νx −1, om (6) we ge Fs(x) = 1 y−νx −1(−1) νx −y x−(νx −y)−1 0= =1 y−νx −10 −x. Thus, he dynamics on he sliding se Σsis go e ned by he nonlinea di - e en ial equa ion ˙y=ρ 1 + νρ −y, NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 19 so ha , as y≤νρ, he sign o ˙ycoincides wi h he sign o ρ. This dynamics is esponsible o c ea ing one s able limi cycle including some uppe pa o he sliding se , su ounding he uns able ocus ha exis s o 0 < ν, < 2 when ρ > 0.The bi u ca ion unde gone by he sys em o ρ > 0 a ν= 0, is eminiscen o he ocus-cen e -limi cycle bi u ca ion epo ed in (F ei e- Ponce-Ros 1999), in he sense ha he phenomenon is simila , o he limi cycle appea s suddenly wi h signi ican size, ‘coming’ om he ou e mos pe iodic o bi o he cen e . I we wan o analyze he bounda y equilib ium bi u ca ion o he o igin a ρ= 0 acco ding o Theo em 1, we mus i s o check i s hypo heses. He e A=Dx (1)(0,0) = ν−1 1 0 , while he o he elemen s a e as be o e, namely B= (2)(0,0) = −1 0, CT=Dxh(0,0) = 1 0 , M=Dρ (1)(0,0) = 0 0,and N=Dρh(0,0) = −1. We ge N−CTA−1M=−1,so ha he collision is ans e sal. Bu , ega ding he non-degene acy condi ion, we now ob ain CTA−1B=1 0 0 1 −1ν −1 0= 0, elling us ha he bi u ca ion is degene a e. In ac , we al eady know ha jus in he collision he only equilib ium poin ha exis s o ρ > 0 exploi s in a con inuum o pseudo-equilib ia, and no equilib ia emain o ρ < 0. The bi u ca ion se in (ρ, ν)-plane and he s a e space diag ams a e shown in Fig. 3 and Fig. 4 espec i ely. In egion 1 we ha e a s able equilib ium o cen e ype which loses i s s abili y in passing o egion 2 and gi es ise o a s able sliding limi cycle su ounding he uns able equilib ium. This bi u ca ion is simila he Focus-Cen e-Limi Cycle (F-C-LC) bi u ca ion s udied in (F ei e, Ponce and Ros, 1999). As men ioned ea lie , while a con inuum o pseudo-equilib ia is de ec ed on he line {ρ= 0}, no equilib ia pe sis o ρ < 0. 5.3. Case S udy 3: Bounda y-Hop Bi u ca ion. Le us conside a case e y simila in p inciple o he p e ious one, by subs i u ing only he linea ec o ield esponsible o he non-hype bolic poin and choosing ins ead he ec o ield associa ed o a canonical Hop singula i y, namely (18) (1)(x, y) = −y+x(ν−x2−y2) x+y(ν−x2−y2), 20 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 1.5 ρ ν F−C−LC 1 2 4 Con inuum o pE 3 Figu e 3. Bi u ca ion se in he plane (ρ, ν) o he linea cen e plus a cons an ec o ield case. which again only ha e he o igin as possible eal equilib ium poin . We know in ad ance ha o ρ > 0 and h ough a Hop bi u ca ion a ν= 0 he o igin passes om being a s able ocus o be o ν > 0 a uns able ocus su ounded by a s able limi cycle, which is a ci cle o adius √ν. The sliding se Σsis, by ecalling (15), he se o poin s wi h x=ρ, y − x(ν−x2−y2)≤0, namely Σs= Σs(ρ, ν) = (x, y)∈R2:x=ρ, ρy2+y+ρ(ρ2−ν)≤0. No e ha his se is, o ρ > 0,bounded when non-emp y, and unbounded and possibly disconnec ed o ρ < 0.I ρ= 0 he se is he nega i e y-axis. A comple e desc ip ion o he sliding se is gi en in Tab. 2, whe e when hey a e well de ined, we in oduce he alues yl=−1 2ρ− ν−ρ2+1 4ρ2, yu=−1 2ρ+ ν−ρ2+1 4ρ2. P oceeding u he , we see ha D∇h(x), (2)(x)− (1)(x)E= 1 −y+x(ν−x2−y2), NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 21 −2 −1 0 1 2 −2 −1 0 1 2 x y −2 −1 0 1 2 −2 −1 0 1 2 x y −2 −1 0 1 2 −2 −1 0 1 2 x y ν = − 1 ρ = 1 ν = 0 ρ = 1 ν = 0.2 ρ = 1 ν = 1 ρ = 0 −2 −1 0 1 2 −2 −1 0 1 2 x y ν = 1 ρ = −1 AE AE LC 12 34 Figu e 4. Some s a e space diag ams in he plane (x, y) o he linea cen e plus a cons an ec o ield case. Pa ame e egion Sliding pa Σso Σ ρ > 0, ν < ρ2−1 4ρ2∅ ρ > 0, ν ≥ρ2−1 4ρ2A bounded segmen , wi h yl≤y≤yu ρ= 0 The hal -axis y≤0 ρ < 0, ν > ρ2−1 4ρ2The wo unbounded pa s, y≤yland yu≤y ρ < 0, ν ≤ρ2−1 4ρ2The comple e line x=ρ, ha is Σs= Σ Table 2. De ini ion o he sliding egion Σs o a ious pa ame e alues and om (6) we ge (19) Fs(x) = 1 y−x(ν−x2−y2)−10 x+y(ν−x2−y2). Thus, he dynamics on he sliding se Σsis go e ned by he nonlinea di - e en ial equa ion ˙y=ρ+y(ν−ρ2−y2) y−ρ(ν−ρ2−y2)−1. 22 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE 5.3.1. Exis ence and s abili y o pseudo-equilib ia. Using GCE condi ion (16), which now is equal o x+y(ν−x2−y2) = 0, we a i e a he condi ion (20) q(y) := y3+ (ρ2−ν)y−ρ= 0, which mus be ul illed by any pseudo-equilib ium candida e, o which we mus also check i s belonging o Σs.No e ha , due o he special o m o (2), his equa ion co esponds exac ly wi h he ho izon al nullcline o (1). Equa ion (20) can ha e up o h ee eal solu ions and he ollowing auxilia y esul will be use ul. Lemma 1. Fo e e y nega i e oo o (20), he poin (ρ, y)belongs o Σs, while all posi i e solu ions a e such ha (ρ, y)is in Σ Σs. P oo . The case ρ= 0 is di ec . Assume ρ6= 0, and le ybe a nega i e oo o (20). By mul iplying his equa ion by ρ, we can w i e 0 = ρy3+ρ(ρ2−ν)y−ρ2, which is equi alen , a e adding y2and a anging e ms, o ρ2+y2=ρy3+ρ(ρ2−ν)y+y2=yρy2+y+ρ(ρ2−ν). Now, since he le hand side is posi i e and y < 0 by assump ion, we conclude ha ρy2+y+ρ(ρ2−ν)<0, and so (ρ, y)∈Σs.The a gumen o posi i e oo is comple ely analogue and he lemma ollows.  The numbe o pseudo-equilib ia could change by mo ing pa ame e s. In pa icula , om he abo e lemma, we should s udy he possible change in he numbe o nega i e solu ions o (20). By compu ing i s de i a i e, we ob ain q′(y) = 3y2+ρ2−ν. I o ins ance ρ2> ν, hen he unc ion q(y),is inc easing, ha ing only one eal oo , since i s de i a i e is always posi i e. I u he mo e ρ > 0, such oo is posi i e as q(0) = −ρ < 0,and hen he e a e no pseudo-equilib ia, acco ding o he lemma. Eq. (20) will s a o ha e nega i e oo s i we choose he pa ame e s o ge a maximum in a poin y∗<0 whe e bo h q(y∗) = 0 and q′(y∗) = 0, o hen he g aph o qin he plane (y, q) has a angency wi h he nega i e ho izon al axis. The exis ence o such a poin will imply a saddle-node bi u ca ion o pseudo-equilib ia (SNpE), o hen a change o pa ame e s should lead o he appea ance o wo nega i e oo s o (20). Thus, we will elimina e yin he sys em o med by he wo equa ions q(y) = 0, q′(y) = 0, ying o ob ain a pa ame e ela ion ha co esponds NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 23 wi h he SNpE bi u ca ion. F om q′(y) = 0, we ob ain he ela ion ν= 3y2+ρ2, and subs i u ing in (20), we ge −2y3−ρ= 0. By assuming ρ > 0,we ha e y∗=−ρ 2 1 3 and eplacing his alue in ν= 3y2+ρ2, we inally a i e a he condi ion ν=ρ2+ 3 ρ 22/3 . This cu e in he pa ame e plane (ρ, ν) de e mines whe e he SNpE akes place. The phenomenon is associa ed o he ac ha he bounda y Σ in e sec s he ho izon al nullcline in wo poin s, a e becoming angen o i a he poin (x, y) = (ρ, y∗). I we check he sign o ˙yon Σs, we obse e ha i coincides wi h he sign o q(y). The e o e, i is immedia e o conclude ha a e he SNpE bi u ca ion he uppe pseudo-equilib ium beha es as a saddle while he lowe one as a s able node. No e ha , when his angency be ween he nullcline and he bounda y is de ec ed, he limi cycle, su ounding he uns able equilib ium, canno con inue o exis . In pa icula , o he cycle o exis pas his angency, i should de elop a sliding segmen lying on he e ical line x=ρ. Such a sliding segmen should be con ined o he egion gi en by −3ρ 22/3≤y≤3ρ 22/3 . Such egion, hough, ce ainly con ains he poin y∗, whe e wo pseudo- equilib ia a e c ea ed a he angency. So, clea ly he cycle canno coexis wi h such new in a ian se s and he e o e ceases o exis . The wo pseudo-equilib ia will exis o ν > ρ2+ 3 ρ 22/3 p o ided ha ρ > 0; i we le ρ end o ze o hen om (20) we deduce ha he s able pseudo-node ends o (0,√ν) while he pseudo-saddle ends o he o igin whe e we will ha e a bounda y equilib ium collision. 5.3.2. BEB o he o igin. Le us inally analyze he bounda y equilib ium bi u ca ion o he o igin a ρ= 0.As be o e, M anishes and N=Dρh(0,0)= 24 MARIO DI BERNARDO, DANIEL J. PAGANO AND ENRIQUE PONCE −1, so ha he ans e sali y condi ion is sa is ied. To check he non- degene acy condi ion o Theo em 1, we see now CTA−1B=1 0 ν−1 1ν−1−1 0= =1 1 + ν21 0 ν1 −1ν −1 0=−ν 1 + ν2, so ha depending on he sign o νwe ha e a di e en kind o bounda y bi u ca ion. Thus, o ν < 0 we ha e pe sis ence o ansi ion be ween one eal equilib ium and one pseudo-equilib ium, whils o ν > 0 he e is a non-smoo h old be ween hem. 5.3.3. G azing bi u ca ion. We also mus ake in o accoun ha h ough a Hop bi u ca ion a ν= 0 o ρ > 0, he o igin becomes su ounded o ν > 0 by a s able limi cycle, namely a ci cle o adius √ν. We could speak so o a g azing bi u ca ion when he discon inui y mani old Σ ouches he limi cycle. Clea ly, i happens o ρ > 0 a he cu e ν=ρ2o he plane (ρ, ν). Apa om he appea ance o a sliding segmen in he cycle, he e a e no o he quali a i e changes. Much mo e d ama ic is he change ha occu s by c ossing he cu e associa ed o he SNpE bi u ca ion; he s able limi cycle is i s ly ans o med in a loop con aining he saddle-node poin , and a e c ossing he SNpE cu e is o be b oken in di e en o bi s, so ha om all ini ial condi ions, excep ing he o igin and he pseudo-saddle, he o bi s end o app oach (in ini e ime) he s able pseudo-node. A new almos global a ac o is hen c ea ed. Pu ing all his in o ma ion oge he , we show in Fig. 5 he bi u ca ion se in he plane (ρ, ν). The numbe and ypes o in a ian se s exis ing in each o he egions in he wo-pa ame e space is gi en in Tab. 3. S a ing om he bo om igh -hand egion o he diag am in Fig. 5, we no ice he occu ence o a Hop bi u ca ion as we c oss in o egion 1. He e an uns able admissible equilib ium coexis wi h a s able limi cycle. Unde pa ame e a ia ions, as we en e egion 2, he cycle unde goes a g azing bi u ca ion which gi es ise a s able sliding limi cycle. Then, because o a saddle-node o pseudo-equilib ia, in egion 3 we obse e he coexis ence o one uns able admissible equilib ia and a pai o pseudo-equilib ia (a pseudo-saddle and a pseudo-node). The s able pseudo-saddle hen collides on he bounda y wi h he uns able admissible equilib ia so ha h ough a nonsmoo h- old associa ed o he BEB hey bo h disappea . Hence in egion 4, we ha e only one s able pseudo-node. Such a node unde goes ano he BEB along he cu e labelled as ETin Fig. 5 which gi e ise o a s able admissible equilib ium in egion 0. NONHYPERBOLIC BOUNDARY EQUILIBRIUM BIFURCATIONS 25 −1 −0.5 0 0.5 1 1.5 2 −2 −1 0 1 2 3 4 5 6 7 ρ ν 0 H G SNpE + HC 1 2 3 4 ET EA Figu e 5. Bi u ca ion se in he plane (ρ, ν) o he Hop + cons an ec o ield case. Region In a ian se s 0 1 s able AE 1 1 uns able AE + 1 sLC 2 1 uns able AE + 1 sLC wi h a sliding segmen 3 1 uns able AE + 1 pseudo-saddle + 1 s able pseudo-node 4 1 s able pseudo-node Table 3. In a ian se s in each o he egions depic ed in Fig. 5 5.4. Discussion. The h ee case s udies in es iga ed abo e show some gen- e al ea u es ha we conjec u e can be obse ed also in highe -dimensional Filippo sys ems, unde going simila ypes o bounda y-equilib ium bi u - ca ions. In pa icula , in all cases he un olding o he wo-pa ame e BEB poin shows nea by bi u ca ion cu es o wo ypes: (i) cu es associa ed o he speci ic ype o smoo h bi u ca ion exhibi ed by he sys em away om he