Applications of the Euler characteristic in bifurcation theory

Publicacions Ma emá iques, Vol 35 (1991), 527-535 . APPLICATIONSOF THE EULER CHARACTERISTIC IN BIFURCATION THEORY Abs ac SLAWOMIR RYBICKI Le : R" x R k - . R" be a con inuous map such ha (O,, ) = 0 o all AE Rk . In his a icle we o mula e, in e ms o he Eule cha ac e is ic o algeb aic se s, su icien condi ions o he exis en e o bi u ca ion poin s o he equa ion (x, A) = 0 . Mo eo e we apply hese esul s in bi u ca ion heo y o o dina y di e en ial equa ions . I is wo h o poin ou ha in he las pa ag aph we show how o e i y, by compu e , he assump ions o he heo ems o his pape . 0 . In oduc ion In [A] Alexande has de ined an in a ian which non i iali y implies he exis en e o a bi u ca ion poin o a con inuous map : R' x R k --> R' such ha (0, A) = 0 o all A E R k . This in a ian is an elemen o he g oup 7 k_1(GL(n)) . Gene ally i is di icul o e i y i his in a ian is a non i ial elemen in 7 k-I(GL(n)) . K asnosielski in [K] has p o ed a 1-pa ame e bi u ca ion heo emwhich is a e y use ul ool in bi u ca ion heo y . This heo em gi es su icien condi ions o he exis en e o a bi u ca ion poin o in he case k = 1 . Many au ho s ha e p o ed gene aliza ions o he classical K asnosielski he- o em (see [MA], [R1], [R2]) . We a e in e es ed in o mula ing su icien condi ions o he exis ence .o bi u ca ion poin s o in case he dimension o he pa ame e space is g ea he han one and when he Alexande in a ian can no be applied . In [S1], [S2] and [W] he au ho s Na e p o ed e y in e es ing o mulas o a compu a ion o he Eule cha ac e is ic o algeb aic se s in e ms o he B ouwe opological deg ee o sui able maps . In he i s pa o his pape (using hese o mulas) we o mula e and p o e su icien condi ions o he exis en e o bi u ca ion poin s o . Namely, we de ine (De . 1 .4 .) a se o essen ial maps (Ess (n, k)) and p o e ha 0 -E R is no an isola ed bi u ca ion poin o any essen ial map (P op . 1 .1 .) . 52 8  S . RYBICKI We also de ine (De . 1 .3 .) a se o egula maps (Reg (n, k)) and show how o e i y i a egula map is an essen ial map (P op . 1 .3 .) . In P oposi ion 1 .4 . we o mula e su icien condi ion o he exis ence o a bi u ca ion poin o a homogeneous map ( o de ini ion o a homogeneous map, see De . 1 .2 .) . As he las case we conside a se o e en maps (E en (n, k), De . 1 .5 .) . In P oposi ion 1 .5 . we gi e su icien condi ions o he exis ence o a bi u ca ion poin o e en map . No ice ha he assump ions o P oposi ions 1 .4 ., 1 .5 ., 1 .6 . a e exp essed in e ms o he B ouwe opological deg ee o polynomial maps . F om his poin o iew i is impo an o compu e he B ouwe opological deg ee o polynomial maps . Nie enbe g has o mula ed in [N] án in eg al de ini ion o he B ouwe opo- logical deg ee . . We ha e w i en a compu e p og am whichcompu es he opo- logical deg ee o polynomial maps, in a e sion gi en by Nie enbe g . The impo an ques ion is how o e i y ha E Reg (n, k) . In o he wo ds we mus e i y i 0 E Rk is an isola ed poin in IP -1 (0) . The e a e compu e algo i hms which allow us o check i 0 E R k is an isola ed ze o o he polynomial map, T : R k --> R k . The e algo i hms a e based on he Eisenbud and Le ine esul s (see [E.L .]) . The e is a compu e p og am, w i en by And zej Lecki om Uni e si y o Gdansk, which is based on such kinds o algo i hms . Using his p og am we can e i y i E Reg (n, k) . Acknowledgemen . The au ho wishes o hanks o And zej Lecki o se e al help ul commen s . In he second pa ag aph we apply opological esul s o his a icle o he bi u ca ion heo y o o dina y di e en ial equa ions (Th . 2 .1 ., 2 .2 ., 2 .3 .) . In he las pa o his pape we show how o e i y by compu e he assump- ions o he heo ems o his pápe . 1 . Resul s Deno e by X and Y Banach spaces and by  X x R k -> Y a con inuous ope a o such ha (0, A) = 0 o all a E R k . De ini ion 1 .1 . A poin ao E R k is said o be a bi u ca ion poin o he equa ion (*) (x,, ) = 0, i (0, o) E  closu e {(x, . ) E X x R k : (x, .1) = 0and x :,A 0} . The se o bi u ca ion poin s ó he equa ion (*) will be deno ed by Bi ( ) . Conside a C 1 -map : R' x R k - > R' such ha (0, .~) = 0 o all A E R and de ine a map (D : R k -> R by ~P (A) = de (D ., (0, A)) . De ini ion 1 .2 . A map is said o be a homogeneous map, i -P(A) is a homogeneous polynomial o deg ee g ea he han 1 . APPLICATIONS OF THE EULER CHARACTERISTIC  529 The se o homogeneous maps will be deno ed by Hom (n, k) . De ini ion 1 .3 . A map : R" x R k -> R' is said o be a egula map, i 1) E Hom (n, k), 2) <D has an isola ed c i ical poin a he o igin . The se o egula maps will be deno ed by Reg (n, k) . De ine a map IP : R k --> R k by he o mula 2 + .p(a) 2 ,. . . , La 4) (A)] 2 + ~(a)2 and no ice ha i T(Ao) = 0, hen xP( - Xo) = 0 o all E R . Rema k 1 .1 . The map ~¿ has an isola ed c i ical poin a he o igin i I P-1(0) = {0} . Using his ema k we will show ha a map conside ed in Example 3 .2 . is a egula map . De ini ion 1 .4 . Amap : Rn x R k -> Rn is said o be an essen ial map, i 1) E Reg (n, k), 2) P-1(0) =,A { 0 } . The se o essen ial maps will be deno ed by Ess (n, k) . P oposi ion 1 .1 . I E Ess (n, k), hen Bi ( ) = ~D -1 (0) . In .pa icula , Bi ( ) =~ 0 . P oo : : The se <I) -1 (0) - {0} is a (k - 1)-dimensional mani old . Fix Ao E ob-1(0)-{0} and 1-dimensional mani old N(A0) such ha mani olds N(Ao) and (D-1(0)-{0} a e ans e sal a he poin Ao . I is easy o see ha he map <D changes sign, a he poin Ao, along N(Ao) . So, applying K asnosielski heo em (see [K]) we show ha Ao E Bi ( ) . The se Bi ( ) is closed, ha is why0eBi ( ) . a Deno e by d(D : R k - R k he g adien o he map I¿ . I E Reg (n, k) hen dD -1 (0) = {0} . Deno e by X(B) he Eule cha ac e - is ic o he mani old B and by deg H he B ouwe opological deg ee o d(D wi h espec o a disc cen e ed a he o igin . P oposi ion 1 .2 . I E Ess (n, k), hen o any E > 0 i) D-1(0) n SÉ -1 is a (k - 2)-dimensional mani old, ü) X ( , P -1(0) n SE -1 ) _ (1 + ( -1 ) k ) ' ( 1 - degd<D) . P oo :: No ice ha <P -1 (0)-{0} is a (k-1)-dimensional mani old ans e sal o SÉ -1 . Tha is why <I> - '(O) n SÉ -1 is a (k - 2)-dimensional mani old . Le us pu he ollowing no a ions A + _ {A E SÉ -1 : ID(A) >_ 0}, A_ _ {A E SÉ -1 : -P(A) < 0}, L= {R E SÉ -1 : -P(A) = 0} .  , 53 0  S . RYBICKI C .T .C . Wall in [W] has shown ha X(A+) = 1 + (-1) k-1 - degdP and x(A_) = 1 - deg d-D . I is well known ha X(L) = x(A + ) + X(A_) - X(Sk-1) . Using he abo e o mula and C .T .C . Wall esul s we ob ain he hesis . P oposi ion 1 .3 . I E Reg (n, k) and (1+ (-1) k ) - (1-deg dD) 7~ 0, hen E Ess (n, k) . P oo .- I E Reg (n, k), hen ~P -1 (0)nSÉ -1 is a (k-2)-dimensional mani old o is an emp y se . F om P oposi ion 1 .2 . and om he assump ions i ollows ha ~¿ -1 (0) n SÉ -1 is no emp y se . So ou p oo is comple ed . In his pa o he pa ag aph we o mula e su icien condi ions o he exis- ence o bi u ca ion poin s, in a case when E Hom (n, k) . Le us deno e B i = {A E Si -1 : (-1) i + 1 - -D(A) >_ 0} o i = 1, 2 and le m be he deg ee o he polynomial P(A) . I m is an odd numbe hen ob has o change a sign nea he o igin ha is why i is enough o conside only he case o e en m . De ine polynomials Fi : Rk -> R by he o mula o i = 1, 2 . By d i : R k , R k we deno e he g adien o Fi . By deg (Q, DÉ , 0) we will deno e he B ouwe opological deg ee o he map Q compu ed on he disc DÉ wi h espec o he o igin . P oposi ion 1 .4 .  The e is a posi i e numbe E > 0 such ha dF~ 1 (0) n DÉ _ {0} .  I deg (dFi,DÉ,0) ~ (-1) k o i = 1, 2, hen o any el > 0 Bi ( ) n5, k , -1 =,A 0, in pa icula , 0 E Bi ( ) . Mo eo e he opological dimension o he se Bi ( ) n 5, k , -1 is equal o k - 2 . P oo . . Fi s o all we will show ha X(Bi) 7~ X (Sk -1 ) o i = 1, 2 . Sza aniec has shown in [S2] ha he e is s  >  0 such ha dFZ 1 (0) n DÉ  =  {0} . I deg (dFi, Dk , 0) =~ ( - 1) k , hen 1 - deg(dF i , DÉ , 0) 5 E 1 + (-1)k -1 .  I is well known ac ha X(S k-1 ) = 1 + (-1) k-1 . F om Co olla y 1 in [S2] i ol- lows ha X(Bi) = 1-deg(dF i , DÉ, 0) . So, we ha e x(Bi)  X(Sk -1 ) . F om he abo e i ollows ha Bi  S k-1 o i = 1, 2 . Fix Ai E Sk-1 - Bi o i = 1, 2 . Le [0,1] -> S k-1 deno e a con inuous map such ha l ;'(0) _ A1 and j(1) = A2 . Conside a Pmposi ion <P o 1 : [0,1] --> R . I is ob ious ha (~P o )(~1) (4> o 1) (A2) < 0 . So, om K asnosielski heo em (see [K]) i ollows ha Bi ( ) n «[0,1]) =,A Q) . I is easy o e i y ha i Ao E Bi ( ), hen - Ao E Bi ( ) o all E R . So, ou p oo is comple ed . In he las pa o his sec ion we u n o a case when a map is such ha d>(A) is no necessa y an homogeneous polynomial . De ini ion 1 .5 . A map : R'E x R k -> R" , is said o be an e en map, i -D(A) is a polynomial such ha 4)(-A) = The se o e en maps will be deno ed by E en (n, k) . Fix E E en (n, k) and deno e by d he deg eee o <> . Choose any e > 0 . De ine a map A, : DÉ+I -, R as ollows Rema k 1 .2 .  Nó ice ha A, (A) is ahomogeneous polynomial and ha sgn A, (A) = sgn D(P(A)) o A E SÉ , whe e P : SÉ -> DÉ is he p ojec ion gi en by he o mula P(A) = (al, . . . , k) . De ine polynomials Ei : Rk+I -> R as ollows o i = 1,2 . By d _' : Rk+I --> Rk+I we will deno e he g adien o Es, P oposi ion  1 .5 . Fo su jicien ly small El  >  0,  i deg (d E', D k 1 , 0) (-1)k+ 1 o i = 1, 2, hen Bi ( ) n D k  :,A 0 ; mo eo e he opological dimension o he se Bi ( ) n D k  is equal o k - 1 . A p oo o his p oposi ion is a consequence o P oposi ion 1 .4 . and Rema k 1 .2 . In his sec ion we will use he no a ions o he i s sec ion . Conside a C l -map g : R x R' x R k -+ Rn and assume ha g( , x, A) can be exp essed in he o m whe e APPLICATIONS OF THE EULER CHARACTERISTIC  531 . _1 AE(a) =AE(a1, . . .,ak+I) = II~IId+2  CP 11 Eñl ' . .'' II~1I EA k J L i (A) _ (-1)'AE(A) - (al +  . . + ñk+I)`~'+I 2 . Applica ions g( , x, A) = A(A)x + (p( , x, A), 1) A(A) is a n x n-ma ix such ha A(0) = 0, 2)  W( , 0, A) = 0 o all ( , 0, A) E R x Rn x R k , 3) D x cp( , 0, A) = 0 o all ( , 0, A) E R x Rn x R k . We a e in e es ed in desc ibing he se o bi u ca ion poin s o he ollowing bounda y alue p oblem (*)  x ( ) = g ( , x ( ), ~) x(0) = x(1) . 532  S . RYBICKI Le us deno e X = {x E C l ([0,1]) : x(0) = x(1)} and Y=C o ([0,1]) . De ine an ope a o F : X x Rk -> Y by he o mula F(x( ), A) = L(A)(x( )) - cp( , x( ), A), whe e L(A)(x( )) = x( ) - A(A)x( ) . No ice ha ze oes o he ope a o a e in one- o-one co espondence wi h solu ions o he p oblem (*) . Le : R' x R k ---> R'' be he map de ined by (x, A) = A(A)x . Now we a e in a posi ion o o mula e he main heo em o ou pape . Theo em 2 .1 . I E Ess (n, k) and E is a su ccien ly small posi i e numóe , hen 1) Bi (F) n DÉ = 4 -1 (0) n DÉ , 2) Bi (F) n SÉ -1 is a (k - 2)-dimensional mani old, 3) x(Bi (F) n SÉ -1 ) = x(D -1 (0) n SE -1 ) = (1 + (-1) k ) . (1- degdD) . P oo . No ice ha L(0) : X --> Y de ined by L(0)(x( )) = x( ) is a F edholm ope a o wi h F edholm index 0 and ha X = Xo ® Xl and Y = Yo ® Y,, whe e Xo= ke L(0) = R n = {subspace o cons an unc ions}, (see [M] o mo e de ails) . Yo = R' = {subspace o cons an unc ions}, 1 Yl = im L(0) = {x E C o [0,1] : x(s) ds = 0}, 0 We begin wi h he Lyapuno -Schmid educ ion . Le Po(x) = o x(s) ds and P, (x) = x - ó x(s) ds deno e he p ojec ions o Y on o Yo, Y,, espec i ely . Then he equa ion F(x, A) = 0 is equi alen o he sys em o equa ions Po(F(xo + xl, A)) = 0, P,(F(xo + x1,A)) = 0, whe e x = xo +x1, xo E Xo, -xl E Xl . No ice ha he map P l oF : Xo ®Xl ®R k --+Y, is con inuously di e en iable nea (0, 0, 0) E Xo ® Xl ® R k , P l oF(0, 0,, ) = 0, and he FYéche de i a i e o Pl o F wi h espec o xl a (0, 0, 0), D xl Pl o F(0, 0, 0) is an isomo phism o Xl on o Y, . The e o e by he implici unc ion heo em, he e is an upen neighbou hood U o (0, 0) E Xo ®R k and xl E C l (U, X I ) such ha he ze os o F nea (0, 0, 0) a e gi en by (xo, xl (xo, A), A) o (xo, A) E Xo ® Rk . I is easy o see ha xl(xo, A) = 0(Ilxo11) a xo = 0, uni o mly o A nea 0 . APPLICATIONS OF THE EULER CHARACTERISTIC  53 3 F om he abo e i ollows ha ze os o F a e in one- o-one co espondence wi h ze os o a ini e dimensional map Q : U -+ Yo de ined by he o mula Q(xo , A) = (Po - F) (xo + xl (xo, A) , A) . I is no di icul o e i y ha Q(xo, A) is a map o he o m Q(xo, A) = - A(A)x0 +T (xo, a), whe e T(0, A) = 0 and D,,T(0, A) = 0 . No ice ha Q E Ess (n, k) . The es is a consequence o P oposi ions 1 .1 . and 1 .2 . The nex heo ems gi e su icien condi ions o he exis en e o bi u ca ion poin s o he ope a o F o mo e gene al class maps han he class Ess (n, k) . Theo em 2 .2 . I E Hom (n, k) and deg (d i, DÉ, 0) ~ (-1)k o i = 1, 2, hen o any el G e Bi (F) 1 SÉ - , 1 0 . In pa icula , 0 E Bi (F) . Mo eo e he opological dimension o he se Bi (F) n SÉ ; l is equal o k - 2 . This heo em is a consequence o P oposi ion 1 .4 . A p oo o his heo em is simila o he p oo o Theo em 2 .1 . Theo em 2 .3 . Le E E en (n, k) .  Then o su icien ly small E i deg (d Es, DÉ+1, 0)  (-1)k+1 o i = 1, 2, hen Bi (F) n D Ek 0 .  Mo e- ¡ o e he opological dimension o he se Bi (F) n DÉ is equal o k - 1 ; in pa icula , 0 E Bi ( ) . This heo em is a consequence o P oposi ion 1 .5 . A p oo o his heo em is simila o he p oo o Theo em 2 .1 . 3 . Examples We shall emind well known esul s om singula i y heo y (see [E .L .]) . Le 0,, be he ing o ge ms o eal analy ic unc ions a 0 E R' . I g1, . . . . gn E B n , le us deno e by (gl, . . . . g  ,) he ideal in 9  , gene a ed by he elemen s 91, . . . . g n . Fo g = (91, . . . . g n ) : (Rn, 0) --, (Rn, 0) we pu Q(g) =e .M1, -. . , gn)- I g is ini e, in he sense ha Q(g) is ini e dimensional eal ec o space, hen 0 E Rn is isola ed in g-1 (0) . Conside he ollowing bounda y alue p oblem : ( ) = g ( , x( ), x(0) = x(1) 53 4  S . RYBICKI and assume ha he map g sa is ies all he assump ions o he p e ious pa a- g aph . So we can exp ess g in he o m g( , x, A) = A(A)x+W( , x, A) . Theo ems 2 .1 ., 2 .2 ., 2.3 . show ha i is enough o examine only he ma ix A(A) . Example 3 .1 . Assume ha n= k = 3 and ha he ma ix A(A) is o he o m A1 + A2,  A2 + A3,  A 1 + A3 A(A) =  A 1 + A 2 + A3,  A l - A2,  2 -A l - 3 - A3 A3,  A1,  0 Using a compu e p og am we show ha deg (d i, D3,0) = 1 o i = 1, 2 . F om his i ollows ha deg (d i, DÉ, 0) A ( - 1) 3 , so he assump ion o The- o em 2 .2 . is ul illed . Applying Theo em 2 .2 we claim ha he e exis s e > 0 such ha o any el < e he in e sec ion o he se o bi u ca ion poin s o he p oblem (**) wi h SÉ, is no emp y, in pa icula 0 E Bi (F) . Mo eo e he opological dimension o his in e sec ion is equal o 1 . Example 3 .2 . Assume ha n = k = 2 and ha he ma ix A(A) is o he o m _  A 1 - A2,  Ai - A2 A(A) - [ -A1 + 2 - A 2 ,  A 1 1 A 2 - A2 ] . De ine a map : R 2 x R 2 -> R 2 by (x, A) = A(A)x . We will show ha E Ess (2,2) . Using a compu e p og am we show ha dim Q(YP) = 16, so om he esul s om singula i y heo y i ollows ha T -1 (0) = {0}, ha is why E Reg (2,2) . Compu ing he B ouwe opological deg ee (by compu e ) we ob ain deg d<D _ -2 and consequen ly X(D -1 (0) l SÉ) = (1 + (-1)k) - (1 - degd<D) = (1 + (-1)2) . (1 - (-2)) = 6 M-om his i ollows ha 4) -1 (0) =~ {0}, so E Ess (2,2) . So om Theo em 2 .1 i ollows ha Bi (F) (1 D 2 consis o exac ly six in e - als, which emana e om he o igin . Re e ences [A] J .C . ALEXANDER, Bi u ca ion o ze os o pa ame ized unc ions, J . Fune . Analy . 29 (1978), 37-53 . [E .L .] D . EISENBUD, H .I . LEVINE, An algeb aic o mula o he deg ee o a C°° map ge m, Ann . Ma h . 106 (1977), 19-44 . [I .R .1] M . IZYDOREK, S . RYBICKI, On he numbe o bi u ca ion b anches o C 2 -maps, accep ed o publica ion in JMAA (1991) . [I .R .2] M . IZYDOREK, S . RYBICIQ, On he numbe o b anches o bi u ca ion poin s, o appea . APPLICATIONS OF THE EULER CHARACTERISTIC  53 5 [K] M .A . KRASNOSIELSKI, "Topological me hods in he heo y o nonlinea in eg al equa ions," Pe gamon, 1964 . [M] J . MAWHIN, "Topological deg ee me hods in nonlinea bounda y alue p oblems," Con e ence boa d o he ma hema ical sciences, Regional con- e ence se ies in ma hema ics 40, Ame ican Ma hema ical Socie y, P o i- dence, Rhode Island, 1979 . [MA] A . MARINO, "La bi u cazione nel caso a iazionale," Con . Sem . Ma . Uni . Ba ¡ 132, 14 pp . MR50, 1068, 1973 . [N] L . NIERENBERC, "Topics in nonlinea analysis, N . Y . Y .," Lec u e No es, 1974 . [Rl] P . RABINOWITZ, Some aspec s o nonlinea eigen alue p oblems, Rocky Moun . J . o Ma h . 3 (1973), 161-202 . [R2] R . RABINOWITZ, A bi u ca ion heo em o po en ial ope a o s, J . Func . Analy . 26 (1977), 48-67 . [S1] Z . SZAFRANIEC, On he Eule cha ac e is ic o analy ic and algeb aic se s, Topology 25, 4 (1986), 411-414 . [S2] Z . SZAFRANIEC, The Eule cha ac e is ic o algeb aic comple e in e sec- ions, J . Reine Angew . Ma h . 397 (1989), 194-201 . [W] C .T .C . WALL, Topological in a iance o he Mi1no numbe mod 2, Topology 22 (1989), 345-350 . Depa men o Ma hema ics Technical Uni e si y o Gdansk ul . Majakowskiego 11/12 80-952 Gdansk POLAND P ime a e sió ebuda el 2 d'Oc ub e de 1990, da e a e sió ebuda el 3 de Juny de 1991