The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems without Sliding

The Focus-Cen e -Limi Cycle Bi u ca ion in Discon inuous Plana Piecewise Linea Sys ems wi hou Sliding En ique Ponce, Ja ie Ros and El´ısabe Vela Abs ac Plana discon inuous piecewise linea sys ems wi h wo linea i y zones, one o hem being o ocus ype, a e conside ed. By using an adequa e canonical o m unde ce ain hypo heses, he bi u ca ion o a limi cycle, when he ocus chan- ges i s s abili y a e becoming a linea cen e , is comple ely cha ac e ized. Analy ic exp essions o he ampli ude, pe iod and cha ac e is ic mul iplie o he bi u ca ing limi cycle a e p o ided. The s udied bi u ca ion appea s in eal wo ld applica ions, as shown wi h he analysis o an elec onic Wien b idge oscilla o wi hou symme- y. 1 In oduc ion and main esul s Nowadays, he analysis o discon inuous piecewise-linea sys ems is an ac i e ield o esea ch since ce ain mode n de ices a e well modeled by his class o sys ems, see [1]. Fo he simples si ua ion howe e , as is he case o he agg ega ion o wo plana linea sys ems, he e a e bi u ca ions ha s ill equi e a ho ough analysis. Recen ly, in [2] i has been p oposed a canonical o m o he case o plana discon inuous sys ems wi h wo zones o linea i y, o be deno ed D2PWLS2 o sho . In he quo ed pape , he e a e shown some bi u ca ion esul s o he case when bo h linea dynamics a e o ocus ype wi hou isible angencies, ha is, he e a e no eal equilib ium poin s in he in e io o each hal -plane. He e, by eso ing o he canonical o m gi en in [2], we conside a di e en si ua ion when we ha e an equilib ium poin o ocus ype in he in e io o a hal plane wi hou speci ying he linea dynamics ype in he o he hal plane. Ou goal is o desc ibe quali a i ely and quan i a i ely he possible bi u ca ion o limi cycles h ough he change o En ique Ponce, e-mail: eponce[email p o ec ed] ·Ja ie Ros, e-mail: ja ie [email protected] ·El´ısabe Vela, e- mail: eli [email p o ec ed] Depa amen o Ma em´a ica Aplicada II, E.T.S. Ingenie ´ıa, 41092-Se illa (Spain) 1 2 En ique Ponce, Ja ie Ros and El´ısabe Vela s abili y o such an equilib ium poin . Thus, his wo k is a ele an gene aliza ion o discon inuous ec o ields o he bi u ca ion s udied in [3] o he con inuous case. To begin wi h, we assume wi hou loss o gene ali y ha he linea i y egions in he phase plane a e he le and igh hal -planes, S−={(x,y):x<0},S+={(x,y):x>0}, sepa a ed by he s aigh line Σ ={(x,y):x=0}. The sys ems o be s udied become ˙x =(F+ 1(x),F+ 2(x)T=A+x+b+,i x∈S+, F− 1(x),F− 2(x)T=A−x+b−,i x∈S−,(1) whe e x= (x,y)T∈R2,A+= (a+ ij)and A−= (a− ij)a e 2×2 cons an ma ices, b+= (b+ 1,b+ 2)T,b−= (b− 1,b− 2)Ta e cons an ec o s o R2, and he de ini ion o he ec o ield in Σ is no ele an o ou pu poses. We assume he gene ic condi ion a+ 12a− 12 >0, which means ha o bi s can c oss he discon inui y line in opposi e di ec ions, allowing he exis ence o pe iod o bi s ha use he wo hal planes. By applying P oposi ion 3.1 o [2], sys em (1) can be w i en in he canonical o m ˙x =T−−1 D−0x−0 a−i x∈S−, ˙x =T+−1 D+0x−−b a+i x∈S+, (2) whe e a−=a− 12b− 2−a− 22b− 1,b=a− 12 a+ 12 b+ 1−b− 1,a+=a− 12 a+ 12 (a+ 12b+ 2−a+ 22b+ 1), and T±= (A±),D±=de (A±)a e he linea in a ian s o each zone. The canonical o m (2) has se en pa ame e s; apa om he men ioned linea in a ian s,we ind he wo pa ame e sa+and a− ela ed o he posi ion o equilib ia, and a pa ame e bwhich is esponsible o he exis ence o a sliding se , whe e bo h ec o ields canno be conca ena ed in a na u al way. In ac , he e exis s a sliding se which is a segmen joining he o igin and he poin (0,b), see [2] o mo e de ails. These wo endpoin s a e angency poin s, he o igin o he le egion and he poin (0,b) o he igh one. Fu he mo e, he sliding se becomes a ac i e o b<0 and epulsi e o b>0, sh inking o he o igin when b=0. By compu ing he sign o ¨xa he angency poin s, we ob ain ¨x|(x,y)=(0,0)=a−, ¨x|(x,y)=(0,b)=a+,so ha he le ( igh ) angency is called isible o a−<0 (a+>0), being in isible o a−>0 (a+<0), see [2]. Thus he a±pa ame e s a e ela ed o he isibili y o he angencies, and when hey anish we ha e boun- da y equilib ium poin s, see [4] and also [5]. The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 3 The possible equilib ia ( eal o i ual) a e loca ed a (a−/D−,a−T−/D−)and (a+/D+,b+a+T+/D+)whe e i is assumed D+D−6=0. Wi hou loss o gene ali y, we assume ha he e exis s an equilib ium o ocus ype in he le zone, ha is, T2 −−4D−<0 (wha implies D−>0), and a−<0. Ou in e es is o s udy wha happens when he ace T−passes h ough he c i- ical alue ze o, ha is when he ocus passes om s able o uns able o ice e sa. No e ha o T−=0 in he le hal plane we ha e a cen e con igu a ion which e - mina es in a isible angency a he o igin. To a oid o he non-local phenomena i is hen na u al o impose ha in he igh zone we ha e also a angency a he o igin o in isible cha ac e , wha amoun s o equi e b=0. In ac , i we allow o mo e his pa ame e bin a neighbo hood o ze o, hen we should ha e he possibili y o new bi u ca ions, namely he collision o angencies, which has been epo ed in [4]. Thus, ou s udy can be seen as a i s s ep in he analysis o he codimension- wo bi u ca ion ha appea s when he pa ame e bis allowed o be mo ed. Such codimension- wo bi u ca ion will be he aim o a u u e wo k, since i u ns ou o be a non-gene ic case, no included in he ex ensi e analysis done in [6]. Thus, he e a e no p ope sliding se no jumps in he igh dynamics wi h espec o he c i ical cen e . Unde hese assump ions ou i s esul is he ollowing. P oposi ion 1. Unde he hypo heses T2 −−4D−<0, a−<0(le ocus dynamics wi h isible angency a he o igin) and assuming ha in he igh zone we ha e an in isible angency a he o igin, ha is, b =0and a+<0, sys em (2) is opologically equi alen o he sys em ˙x=Tx−y, ˙y=Dx+a,i x >0,˙x=2 γ x−y, ˙y= (1+ γ 2)(x+1),i x <0,(3) whe e T =T+, D=D+, γ = α / ω , wi h 2 α =T−, ω >0issuch ha 4D−−T2 −=4 ω 2, and a =D−a+/( ω a−)>0. The p oo o P oposi ion 1 appea s in Sec . 3. Wi h his esul , we manage o desc ibe he le dynamics wi h only one pa ame e , needing o he h ee pa ame e s o deal wi h he igh egion. The ollowing ema k should be aken in o accoun . Rema k 1. Sys em (3) ep esen s in gene al a discon inuous ec o ield since o x=0 we will ha e gene ically a6=1+ γ 2. The o iginal sys em (2) is con inuousonly in he case a+=a−and b=0, bu e en in such non-gene ic case, he new sys em p o ided by P oposi ion 1 will be discon inuous. Thus, he analysis o sys em (3) is ele an o some con inuous cases ha could be also s udied by al e na i eme hods wi hin a con inuous ec o ield con ex . This ac will be illus a ed la e in Sec . 2. Rega ding equilib ium poin s, in he zone x<0, he e exis s a ocus a (x,y) = (−1,−2 γ ), o be s able o γ <0 and uns able o γ >0. When γ =0, we ha e a linea cen e . In he zone x>0, since a>0, he e will be no equilib ium poin s i D=0; o he wise, hepossible equilib iumpoin will be loca ed a (−a/D,−aT/D). We will ake γ in (3) as he bi u ca ion pa ame e , ha ing i s c i ical alue a γ =0, whe e he cen e con igu a ion akes place, see Figu e 1. Ou i s main esul is he ollowing. 4 En ique Ponce, Ja ie Ros and El´ısabe Vela 01 Y 0011 Y bx0 bx1 Fig. 1 (Le ) The c i ical si ua ion o γ =0. (Righ ) The bi u ca ing limi cycle o γ T<0 and | γ | small. Theo em 1. Conside sys em (3) wi h a >0and unde he assump ion T 6=0. The linea cen e con igu a ion es ic ed o he zone x ≤0, ha exis s o γ =0gi es place o a unique pe iodic oscilla ion o γ T<0and | γ |su icien ly small. Mo e p ecisely, o T <0 he limi cycle bi u ca es o γ >0and i is s able, while o T >0 he limi cycle bi u ca es o γ <0and i is uns able. I we deno e wi h bx0= (0,by0)T he lowe c ossing poin o he bi u ca ing limi cycle, hen he peak- o- peak ampli ude App in x, he pe iod P o he pe iodic oscilla ion, he cha ac e is ic mul iplie ρ o he pe iodic o bi and he coo dina e by0a e analy ic unc ions a 0 in he a iable γ 1/3. Namely, o γ T<0and | γ |su icien ly small we ha e App =2+(3 π )2/3 2 1+a (T2a)1/3 γ 2/3+O( γ 4/3), P=2 π +2(3 π )1/3a−1 (a2T)1/3 γ 1/3−2 π 15 (15a2−12D+T2−a(3D+T2)) a2T γ +O( γ 4/3), ρ =1−2(3 π )1/3T a2/3 γ 1/3+2(3 π )2/3T a4/3 γ 2/3+ +2 π 15 15a2+12D−31T2 a2 γ +O( γ 4/3), and by0=3 π a T1/3 γ 1/3+ π 2T 3a1/3 γ 2/3+ π 15a2+3D+T2 15aT γ +O( γ 4/3). I should be no ed ha he sign o he pa ame e Dis no ele an o he bi u ca- ion and hus ou esul co e s he cases he ocus-an isaddle case (D>0), he case o D=0 (pa abolic case) and he saddle case (D<0). O cou se, he uniqueness o he bi u ca ing limi cycle is e e ed o a neighbo hood o he mos ex e nal pe- iodic o bi o he linea cen e exis ing o γ =0, and so i is a local uniqueness. The case T=0 is explici ly excluded om ou esul because hen we should ha e The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 5 a global cen e o he c i ical alue γ =0; he possible bi u ca ion o limi cycles o such non-gene ic case equi es di e en echniques, as conside ed in [7]. Undoing he changes o a iables in oduced in he p oo o P oposi ion 1, i is easy now o ge a simila esul o sys em (2) wi h b=0 and adequa e hypo heses. Theo em 2. Conside sys em (2) unde he hypo heses T2 −−4D−<0, a−<0, a+<0, T+6=0and assume ha in he zone x >0we ha e an in isible angency a he o igin, ha is b =0. The linea cen e con igu a ion es ic ed o he zone x ≤0, ha exis s o T−=0gi es place o a unique pe iodic oscilla ion o T−T+<0and |T−|su icien ly small. Mo e p ecisely, o T+<0 he limi cycle bi u ca es o T−>0and i is s able, while o T+>0 he limi cycle bi u ca es o T−<0and i is uns able. I we deno e by bx0= (0,by0)T he lowe c ossing poin o he bi u ca ing limi cycle, he peak- o-peak ampli ude App in x, he pe iod P, he cha ac e is ic mul iplie ρ and by0 a e analy ic unc ions a 0in he a iable T1/3 −. Namely, o T−T+<0and |T−| su icien ly small we ha e App =−2a− D−−1 23 π 22/3a++a− D−a− a+1/3T− T+2/3 +O(T−)4/3,(4) P=2 π √D−−(12 π )1/3a−−a+ a1/3 −a2/3 +√D−T− T+1/3 + + π 15 15a2 +D−−3D+(4a2 −+a−a+)+ T2 +a−(a−−a+) a2 +D3/2 − T− T+ +O(T−)4/3,(5) ρ =1+(12 π )1/3 √D−T+a− a+2/3−T− T+1/3 +(18 π 2)1/3T2 + D−a− a+4/3T− T+2/3 + + π 15 45a2 ++a2 −(12D+−31T2 +) a2 +D3/2 − T−+O(T−)4/3, and by0=3 π 21/3a1/3 +a2/3 − √D−−T− T+1/3 − π 2 121/3a4/3 −T+ a1/3 +D−T− T+2/3 − − π 30 15a2 +D−+a2 −(3D++T2 +) a+D3/2 − T− T+ +O(T−)4/3. (6) This heo em is an ex ension o he esul s ob ained in [3], whe e only he con i- nuous case was analyzed. Consequen ly,i we pu in he abo eexp essions a+=a− he con inuous case is eco e ed. The exp essions (4)-(7) ha e been compu ed wi h he help o symbolic compu a ion sys ems (bo h Ma hema ica [8] and Maple [9]) and only he i s coe icien s o he se ies a e explici ly shown. Mo e e ms can be compu ed wi h he same echniques, i equi ed. 6 En ique Ponce, Ja ie Ros and El´ısabe Vela The es o he pape is o ganized as ollows. An applica ion o Theo em 2 o he s udy o he dynamics o an elec onic ci cui is gi en in Sec . 2. The p oo s o main esul s along wi h o he echnical lemmas appea inally in Sec . 3. 2 Applica ion o he dynamics o a Wien b idge oscilla o C2 R1C1 Vo R2 + + - - + - RSRF OA Fig. 2 Scheme o he Wien b idge oscilla o . The powe supply needed o bias he ope a ional ampli ie is no shown. In [10] a ealis ic model o an ope a ional ampli ie is p oposed o illus a e he jump ansi ion o oscilla ion in he Wien b idge oscilla o . The p oblem leads o a idealized symme ic model ha can be se led down in a con inuous ec o ield con ex , see [3]. He e, we will assume a mo e gene al si ua ion in he sense ha he nonlinea cha ac e is ic o he elec onic de ice is allowed o ha e some lack o symme y. This assump ion implies ha in he bi h o oscilla ions only wo linea zones a e in ol ed, and we can disca d one o he mos ex e nal egions. Thus we assume ha he ci cui model is gi en by ¨ +2 Γ ˙ + =0, >1, ¨ −2 Γε ˙ + =0, ≤1, whe e is he dimensionless inpu ol age in he ope a ional ampli ie , Γ is a posi- i e cons an pa ame e which depends on he passi e elemen s o he ci cui , and ε is aken as he bi u ca ion pa ame e , see [10] o mo e de ails. The abo e equa ions, a e he ansla ion =x+1, can be w i en in he o m ˙x=y,˙y=−x−2 Γ y−1,i x>0, −x+2 εΓ y−1,i x≤0.(7) Sys em (8) is in he o mula ion (1.1)gi en in [2] and we can apply he P oposi ion 3.1 o ha pape in o de o pu sys em (8) in Li´ena d o m. The homeomo phism The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 7 ex=1 0 −2 Γ −1xi x>0,ex=1 0 2 εΓ −1xi x≤0, ans o ms sys em (8) in ˙x=−2 Γ x−y,i x>0, 2 εΓ x−y,i x<0,˙y=x+1,(8) whe e he ildes ha e been d opped o he sake o simplici y. Now, we can apply Theo em 2 o sys em (9) by aking T+=−2 Γ ,T−=2 εΓ , D+=D−=1, a+=a−=−1 and b=0, ob aining ha he bi u ca ion akes place o ε =0, he limi cycle exis s o su icien ly small ε >0 and i is s able. The peak- o-peak ampli ude, he pe iod and he cha ac e is ic mul iplie o he limi cycle a e App =2+3 π 22/3 ε 2/3− π 4/3(99+52 Γ 2) 40·181/3 ε 4/3+O( ε 5/3), P=2 π +4(18 π 5)1/3 5 ε 5/3+O( ε 2), ρ =1−2(12 π )1/3 Γε 1/3+4(18 π 2)1/3 Γ 2 ε 2/3+2 π 15 Γ (27−124 Γ 2) ε + +4 54 π 91/3 Γ (34 πΓ 3−27 πΓ −5) ε 4/3+O( ε 5/3). We no e ha he o de ε 5/3 e m in he las exp essiono he pe iod Pis no explici ly shown in Theo em 1. I has been compu ed wi h he same p ocedu e o he p oo o Theo em 1, inc easing he numbe o e ms in (18). We emphasize ha he abo e se ies a e e y use ul o desc ibe accu a ely all he ea u es o he nonlinea oscilla ion o pa ame e s nea he c i ical alue co espon- ding o he bi u ca ion.O cou se, we could ha e ob ained he same esul s applying P oposi ion 1 and Theo em 1. 3 P oo s o main esul s We gi e i s he p oo o P oposi ion 1. P oo (P oo o P oposi ion 1). Clea ly, he assump ion on igh angency a he o igin, ha is ˙x|(x,y)=(0,0)=0, educes o b=0, and his angency will be in isible o a+<0. Unde he hypo heses, i we de ine ω >0 such ha ω 2=D−−T2 −/4 and α =T−/2, he eigen alues o he linea pa a S−in (2) a e α ±i ω . We make i s he change X= ω x,Y=y, τ = ω o he a iables in he hal plane S−, wi - hou al e ing a iables and ime in S+. No e ha we do no change he coo dina e y, so ha pe iodic o bi s using bo h hal planes a e p ese ed. Then, we ge 8 En ique Ponce, Ja ie Ros and El´ısabe Vela dX d τ =1 ω dX d =dx d =X ω T−−Y, dY d τ =1 ω dY d =1 ω dy d =1 ω D−X ω −a−=D− ω 2X−a− ω . In oducing he pa ame e γ = α / ω , we see ha T−=2 γω and D−= ( γ 2+1) ω 2. Making in he whole plane a homo he yo ac o k obe de ined below and emo ing he ac o kin he equa ions, we ha e o x<0, dx d τ =kdX d τ =k2 γ x k−y k=2 γ x−y, dy d τ =kdY d τ =kh( γ 2+1)x k−a− ω i= ( γ 2+1)x−ka− ω , while o x>0 we ha e ˙x=Tx −y, and ˙y=Dx−ka+. By imposing now ha k=−( γ 2+1) ω /a−>0, we ge he exp essions gi en in he s a emen . Be o e gi ing he p oo o Theo em 1, we need i s some echnical esul s. Le us conside sys em (1). We assume ha he o bi o ec o ield F+s a ing a a poin bx0= (0,by0)wi h F+ 1(bx0)>0, lies in S+and e en ually comes back o he discon inui y line a i ing ans e sally a he poin bx1= (0,by1), whe e F+ 1(bx1)<0. Lemma 1. I we deno e as b δ +( ) = (x( ,bx0),y( ,bx0)) he solu ion o ˙ x=F+(x) sa is ying b δ +(0) =bx0, hen a alue τ +>0exis ssuch ha x( ,bx0)>0 o 0< < τ + and x( τ +,bx0) = 0, y( τ +,bx0) = by1. Then we can de ine a igh Poinca ´ e map PRin a neighbo hood o he poin bx0such ha PR(by0) = by1and he i s de i a i e o map PRa by0is gi en by P′ R(by0) = F+ 1(bx0) F+ 1(bx1)expZ τ + 0di F+. P oo . Le us conside he di e en ial sys em ˙ x=F+(x)de ined in R2, and le us deno e as δ +( ) = (x( ,x0),y( ,x0)), he solu ion sa is ying δ +(0) = x0whe e x0= (x0,y0). Le us in oduce he equa ion Z( ,x0,y1) = 0, whe e Z( ,x0,y1) = x( ,x0) y( ,x0)−y1, and om ou hypo heses we ha e Z( τ +,bx0,by1) = 0and F+ 1(0,by1)6=0. Then he Jacobian ma ix ∂ (Z1,Z2) ∂ ( ,y1)( τ +,bx0,by1) =    ∂ x ∂ ∂ x ∂ y1 ∂ y ∂ ∂ y ∂ y1   ( τ +,bx0,by1) =F+ 1(0,by1)0 F+ 2(0,by1)−1 is nonsingula . The Focus-Cen e -Limi Cycle Bi u ca ion in D2PWLS2wi hou Sliding 9 Then, we conclude om he implici unc ion heo em he exis ence o wo un- c ions ϕ (x0), ψ (x0)de ined in a neighbo hood o x0, such ha x( ϕ (x0),x0) = 0,y( ϕ (x0),x0)− ψ (x0) = 0,(9) wi h ϕ (bx0) = τ +, ψ (bx0) = by1. Thus, we can de ine he igh Poinca ´e map PRin a neighbo hoodo he poin x0 as y1=PR(y0) = ψ (0,y0), which sa is ies by1=PR(by0) = ψ (bx0). To compu e he i s de i a i e o Poinca ´e map, we ake de i a i es wi h espec o (x0,y0)in equa ions (10) o ge ∂ Z ∂ ( ϕ (x0),x0, ψ (x0))·Dx ϕ (x0)+ B(x0)−0 Dx ψ (x0)=0,(10) whe e ∂ Z ∂ ( ϕ (x0),x0, ψ (x0)) =    ∂ x ∂ ( ϕ (x0),x0) ∂ y ∂ ( ϕ (x0),x0)  =F+(x1) and B(x0) = DxZ( ϕ (x0),x0, ψ (x0)) =    ∂ x ∂ x0( ϕ (x0),x0) ∂ x ∂ y0( ϕ (x0),x0) ∂ y ∂ x0( ϕ (x0),x0) ∂ y ∂ y0( ϕ (x0),x0)   sa is ies he equali y B(x0)F+(x0) = F+(x1),(11) due o he elemen a y p ope ies o a ia ional equa ions. By igh -mul iplying (11) by he ec o (0,1)T, we ge ϕ y(x0)F+(x1)+B(x0)0 1−0 ψ y(x0)=0, and so B(x0)0 −1=F+ 1(x1) ϕ y(x0) F+ 2(x1) ϕ y(x0)− ψ y(x0).(12) Taking in o accoun (12) and (13) we can w i e F+ 1(x1)0 F+ 2(x1)−11 ϕ y(x0) 0 ψ y(x0)=B(x0)F+ 1(x0)0 F+ 2(x0)−1 and aking de e minan s we a i e a he ela ion F+ 1(x1) ψ y(x0) = F+ 1(x0)expZ ϕ (x0) 0di (F+)d ,(13)