Inclination lemmas with dominated convergence

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/87/002327-11 $ 3.70/0 Vo[ 32. March 1987 �9 Birkhfiuser Verlag Basel, 1987

Inclination lemmas with dominated convergence

By Hans-Otto Walther, Mathematisches Institut, Universitfit Miinchen, Theresienstr. 39, D-8000 Mfinchen 2, W. Germany

Introduction

Inclination lemmas help to study dynamical systems in the neighborhood of a saddle point. The content of the first version, the 2-1emma of Palis [3, 4], may be rephrased as follows. Suppose g: U ~ E is a Cl-diffeomorphism in a finite- dimensional space E with 9(0) = 0 which is hyperbolic at x = 0, i.e. no eigen- value of T = Dg(O) lies on the unit circle. Suppose also that g is normalized so that the local stable and unstable manifolds of x = 0 are contained in the stable and unstable linear spaces Q and P of T, respectively. Define the slope of a vector t e e with components p t ~ P , 0 4: q t ~ Q as s ( t ) = Iptl/]qtl, and consider a transversal H to P: Backward iterates H k = g-k(H) converge to Q as k tends to infinity, and slopes of tangent vectors t ~ TxHk, t + O, tend to zero uniformly with respect to x ~ Hg.

It is not hard to extend Palis' result to diffeomorphisms in arbitrary Banach spaces, see [3] and Proposition 10.4 in [2]. Recently Hale and Lin [1] obtained inclination lemmas also for maps which are not necessarily reversible.

With an application to functional differential equations in mind, we shall consider this most general situation, too. The aim of the present note is to show how one can improve uniform convergence for slopes of tangents by estimates of the speed of convergence.

For C2-maps we derive geometric convergence

sup {s(t): 0 :~ t ~ TxHg, x ~ Hk} <= const , f i -k

for all k ~ N, with some fi > 1, and in special cases, in particular for dim P -- 1, the much sharper pointwise estimate

( t ,x) s(t) < const . ]px[

for all t ~ TxHk\{O }, x ~ Hk, k ~ N - see Lemma 2.1. Estimate (t, x) is the key to an investigation of stability in a nonlocal bifur-

cation problem which we now briefly describe. Consider a state variable on a

328 H.-O. Walther ZAMP

circle, with one attractive rest point and a delayed reaction to deviations. A simple differential equation for such a system is ) ? ( t ) = f ( y ( t - e)), or equiva- lently

= o f(x(t - 1 ) )

where f: R ~ R is periodic, with zeros A < 0 < B, period B - A, and with 0 < f in (A, 0), f < 0 in (0, B). Under additional hypotheses on f, there is a heteroclinic solution from A to B at some critical parameter eo which bifurcates to solutions representing periodic rotation around the circle, for e > co. Existence was proved in [7], but questions of uniqueness and stability for the bifurcating "periodic solutions of the second kind" remained open.

Estimate (t, x) makes it possible to give a positive answer. Details will be found in [9].

For earlier results on nonlocal bifurcation, from homoclinic to periodic solutions of ordinary differential equations in R", we refer to Silnikov's papers [6, 5]. Lemma 3.3 in [5] on certain diffeomorphisms given by an O.D.E. appears to be related to the somewhat later 2-1emma of Palis. Estimate (3.5) in [5] may b~ compared to geometric convergence in Lemma 2.1 below.

Lemma 2.1 is stated in a way which yields uniform estimates also in case of parameterized maps, convenient for the application to bifurcation problems. On the other hand, it is restricted to a simplified situation where

a) slopes of tangents to the given set H are already small and b) the linearized map T is an expansion on P and a contraction on Q.

These hypotheses will not present essential difficulties in applications but have the advantage to allow a relatively short proof concentrated on the basic estimates which lead to dominated convergence.

At the end we formulate a generalization of the result on geometric convergence which holds without hypothesis a) and with b) replaced by the more natural assumption that Dg(O) is hyperbolic, see Lemma 3.1. For reasons of length we do not give a proof but refer to the preprint [8]. Lemma 3.1 will not be used in the study of nonlocal bifurcation mentioned above.

Acknowledgement. The author wishes to thank Seminar fiir Angewandte Mathematik, ETH Zfirich, for a pleasant stay which intensely furthered the project [9] and as a part thereof, the present note.

1. Preliminaries

I. Let E be a Banach space. We begin with C2-maps.~: [.7 -~ E with

([7) 0e [7 , [ T c E o p e n , ~ ( 0 ) = 0 .

Vol. 38, 1987 Inclination lemmas with dominated convergence 329

Let T: = D0(0) be an expansion-contraction. That is,

E = P | Q with T-invariant subspaces P and Q, (1)

(~f17) ITqxl<c~lqx j , f l l p x l < l r p x l < = T I p x [ fo ra l l x ~ E

with contants c~ < 1, fl > 1, 7 > fi.P and q denote the projections of E onto P and Q given by (1).

In addition, we assume

0([7 c~ P) c P, O(0 c~ Q) c Q. (2)

In applications, this can be arranged by a change of coordinates which uses local stable and unstable manifolds.

For the remainder term R E C2(U, E),O = T + R, and for the partial deriva- tives Dp and Dq with respect to the decomposi t ion E = P | Q, we find

(R) n(o) = O, DR(O) = O, R(U n P) c P, R(U ~ Q) c Q,

DrqR(x ) = 0 for all x ~ 0 c~ P,

DqpR(x) = 0 for all x ~ U n Q.

II. For applications of Lemma 2.1 to bifurcation problems where p and q may depend on parameters it is convenient not to consider the map (s) 0 but restrictions g to sets U c E with

(U) U c / _ T o p e n , OEU, p x + s q x ~ ( J , s p x + q x e ~ 7 for all

x ~ U and s~ [0 ,1 ] .

Let c > 0. Set c%:= 0~ + c, t ic:= fl - c, 7c '= 7 + c. If

(c) ]DpR(x)] < c and ]Dqe(x)] < c for all x e [7,

then

(g, c) [qg(x)] < e~ [qx[, fi~ Jpx[ < [pg(x)[ < 7~ [px] for all x ~ g .

Proof. For x E U, (U) implies that the straight line from x to p x is contained in [7. We have Iqg(x)l = I rqx + qe(x)l, and I q R ( x ) l - - I q e ( x ) - q e ( p x ) l < c Ix - p x l = c [qx[, by (c) and (U) and the mean value theorem. (0q87) now shows the first inequality. The others follow in the same way.

Remark. Proper ty (c) yields ]DijR(x)l <= c on U, for i,j ~ {p, q}. III. Let m > 0, ~ > 0 be given with

(m) ID(DqpR) (x)l < m for all x e L~,

(~, m) Ipxl < ~/m for all x ~ U.

Then ]DqpR(x)l < m ]px[ < ~ for all x ~ U.


P r o o f The straight line from x ~ U to q x is contained in U so that the mean value theorem gives [DqpR(x ) [ = [ D q p R ( x ) -- D q p R ( q x ) l < rn Ix -- qx] -- m Ipxl .

IV. We shall consider tangents to preimages of some set H ~ U under g. These preimages, and the set H itself, are not necessarily manifolds. A vector t ~ E is called tangent to a set S in E at a point x ~ S if there is a differentiable curve r ( - 1, 1) ~ g with r = x, r 1, 1)) ~ S and t = De(0) (1). T~S denotes the set of all tangents to S at x. As usual, the hypotheses S ~ W, ~: W--, E differentiable, ~ (S) ~ S', t ~ T~ S altogether imply O ~ (x) t ~ T o ~)S ' .

Preimages of a set S ~ E with respect to a map~,: W - ~ E , W c E, are defined by So : = S, Sk + 1 : = r 1 (Sk) for all k e No.

Suppose that the C2-mapg: L? ~ E and U ~ r satisfy (U), (1) and (~[3~) for T = DO(0), (2), (U), (c) and 1 < [3~ = [3 - c. Let H c U and x e H k , the k-th preimage with respect to 9, k e N. Then x j ~ H j for the k + 1 points given by x k : = x, x j_ 1 : = g (x j) for j = k, . . . , 1 ; and (9, c) yields

7s j IpXk-fl [px[ fi -J IPXk-jl for all j = 0, . . . , k.

2. Dominated convergence for slopes of tangents

In order to formulate a result which yields uniform bounds in parameterized problems with families of maps~, we introduce the set B c R 7 of vectors b = (c~, [3, 7, c, m , p ~ , p z ) with c~, [3, 7, c, m positive, Pl and P2 nonnegative and

(B) (ec + c = ) ~ + 2c < l < fi - c ( = flc), p l < p2 �9

For b ~ B given we set tic: = (tic + 1)/2 ~ (1, tic) and

fc f l o - - I t i c - 1 g : = m i n ' f l c + l ' f i c + l - - - [ [ 3 c - Zc]} > o.

Lemma 2.1. Let b e B. There is a constant c b > 0 such that for all C2-maps ~: ---, g and all sets U ~ ~7 with (U), (1) and (efl7) for T = D0(0), (2), (U), (c), (m),

(g, m) and for every set H c U with

(p) pl lpxl ((g)) q t +-O and s(t) <=

for all x e H, t �9 T~H\{O},

we have:

(p, k) pl y j k ~ [pxl ~ p213 j k,

(t, tic) q t =i = 0 and s(t) < Cbfi2 -k,


and in case

(*) 0 < p l and ( a + 3 c ) Tc/flc < 1 ,

( t , x ) q t 4 : 0 and s ( t ) ~ c blpX]

for all preimages Hk, k e N o, of H under 9, and all x ~ Hk, t e TxHk\{O }.

Remarks . Condit ions (1) and (afiT) say that E = P | Q, with T = D0(0) an expansion on P and a contract ion on Q. Condi t ion (2) means that 0 is normalized, with local stable and unstable manifolds contained in Q and P, respectively.

Typical applications will start with maps .~: ~ ~ E which are given and satisfy ~(0) = 0, (1), (~flT), ~ < 1 < fl < 7, and (2) with ~ instead of ~7. Then a small constant c > 0 is chosen so that (B) holds. It is now easy to find a constant m and neighborhoods LT, U of 0 so that the remaining conditions in Lemma 2.1 on 0 := ~ ] U and U, U are satisfied for all vectors (c~,/~, 7, c, m, ", �9 ) e B. In this sense conditions (c), (m), (g, m), (U) are not restrictive.

Inequali ty (p) and ((g)) express a weak kind of transversality of H and P. We do not assume that H is smooth, and we do not exclude cases with dim H < dim Q, H n Q :# 0, H ~ P = {0}. However, hypothesis ((g)) requires that slopes of tangent vectors to H are "sufficiently small".

Part IV of section I shows that (t, x) is a better estimate than (t, tic). Inequali ty (*) is easily verified in applications with dim P = 1" If a

C2-ma~p~: U-->E is given with ~(0) = 0, (1) and (c~fl7), e < 1 < fl < 7, and (2) with U instead of U then dim P = 1 allows to choose fl = 7, and we have v.~,/fl = ~ < 1 so that (*) holds for small c.

Finally, note that without hypothesis Pl > 0 estimate (t, x) will not be true: Consider b e B with (*), a direct sum E = P | Q of nontrivial Banach spaces, H : = ( - e, 0 t with t e E , 0 < e [pt[ ~ P2, qt =t = 0 4= p t , s(t) < g. Then t e Toll0, 0 < s ( t ) , pO = O.

P r o o f o f L e m m a 2.1. Let b e B and ~, U and H be given, satisfying the hypotheses of Lemma 2.1. We consider the restricted map 9: U ---> E and preimages H k of H c U under 9- Part IV of section I implies (p, k).

I. Let x e U, t e E. Set u : = D g ( x ) t. Part III of section 1 yields

]pul = Ip D g ( x ) t[ = [ Tp t + D p R ( x ) [pt + qt][ >__ fl ]pt[ -- c Iptl - ~ ]qtl

= tic Ipt[ - 6[qt[.

In addition,

]qul < "" < ~c [qt] + c I p t l .

II. We show

(Ak) q t + O and s(t)<~y~kofl--~-i for all k e N o , x e H k , t e T x H k \ { O }. i

332 H. -O. W a l t h e r Z A M P

11.1. Hypothesis fig)) gives (Ao). 11.2. Suppose (Ak) for some k e N o. Let x e Hk+l, t e TxHk+l\{O }. Set

u" = D g (x) t e Tg(~) H k. II.2.1. Proof of q t # - O : Assume q t = O . By I, [ p u [ > f i r

From (Ak), q u + O . Again by I, 0 < ] q u l < c ~ [ q t [ + c [ p t l = c [ p t [ . Hence 1 < fldc < s(u) < ~Zkof i[ ~ < ~/(1 -- fi~-a) = ~(fic + l)/(fl~ -- 1) which contra-

i

dicts ~ < (tic - 1)/(fi~ + 1). II.2.2. In case u = 0 , we find 0 = Ipul > ~ ] p t l - ~ l q t l , s ( t ) < o / ~

__< e Z ~ + ~ L -,. i

II.2.3. In case u # O, (Ak) gives qu + 0 and s(u) < O Z ~ f i [ ~ < e/(l -- f ifz) = ~(flc + 1)/(fi~ - 1). F rom I we obtain i

G - cs(u)) s(t) < ~ + ~ s ( u ) .

Hence

c - c e ~ 7 i - l l s( t) < e + s(u) .

The definition of ~ implies

~cs(t) < e + s(u).

Using (Ak), we get the desired estimate for (Ak+ ,). III. Note that II implies q t=t =0 and s(t) < l for all k ~ N o, X e H k ,

t c T~Hg\{0}. IV. I f x e U, 0 =t = t e E , Dg(x) t - - 0, qt .4 = 0 then s(t) < raft2-* [px].

Proof. An estimate as in I yields 0 = [pDg(x) t[ >= tic [pt[ - I D p g ( x ) qt[ > tic [ptJ - m [px[ [qt[, with III (section 1).

V. If k e N , x e Hk, 0 # t e TxHk, 0 :# D g(x) t = : u then qt ~= 0 ~= qu and

s(t) < __~c + c s(u) + m

Proof. We obtain u ~ Tg(~)Hk_l\{O}. By Ill , qu ~- O, qt ~- O, s(t) < 1. Es- timates as in I give [pul > tic [pt[ - m [pxl Iqtl, ]qu[ < G [qt[ + c Ipt[. Hence

S(U) >= tics(t) -- m lpx[ c~c + cs(t) ' (c~c + c l ) s(u) >= ~cs(t) - m Ipxl .

- k VI. We show s(t) <= bfic , with

2m 1 ~b'= 1 +

/~- p21 - (~ + 2c)' for all k ~ No; x ~ H k, t e TxHk\{O } .

VIA. Note qt :4= 0 and s(t) < g < 1 <= Cb for all x e H, t E T~H\{O}.


VI.2.1. Let k ~ N , x ~ Hk, 0 =1= t ~ TxH k. Consider the points and tangent vectors defined by Xk " = X, X~_ I := 9(x~), tk " = t, t i_ ~ "= D g(x~) t~ for i = k , . . . , 1. Then x~ ~ H~ and t i ~ T ~ H i for i = k, . . . , 0. By III, q t ,t = O.

VI.2.2. Assume t ~ + 0 for i = k , k - l , . . . , k - j > 0 w i t h 1 < j < k , j e N . Then qt~ =# 0 for these indices, see III. We prove

(S,) s ( t k - j+ , ) <= t i c ' [(C~ + C)' S( tk_j) + m [PXk_j+ x I z ~ - l ( ~ + C) ~] ~C

for , = 1 . . . . . j :

F o r / = 1, V gives s ( tk_ j+ l) < ~ + C = tic

for some l~ { 1 , . . . , j - - 1}. Par t V gives

m s(tk_j+,+~) < ~c + c s(tk_j+,) + [PXk-j+,+~l

m < ~ + c s(tk J+,) + tic-' ]PXk-3+ [ (IV, section 1)

f i c ' - l(~c + C)'+ t S(tk_j) + f i c t - l ln [pXk_j+ l l

x E ~ - 1 + 1 (c~c + c) ~ + m f i [ ' - 1 [p xk-3 + 11 (with (&)) K

= tc( '+l ) [ (0~c "~ C) t + l S(tk_3) + m Ipxk-j+xl Eb+~-~(~ + c)~]. K

VI.2.3. Suppose t~ # 0 for all i = k . . . . ,0. Then VI.2.2 with j = k and ($i) imply

s(t) <= f i2 k l e + m f i ~ - l p 2 1 - - ( ~ + c) <= cbflc-k

(with ~ + 2c < 1, [PXl] < fi~-x IpXo[ , x o ~ H , S(to) _< e < c < 1). VI.2.4. Suppose there exists j E { 0 , . . . , k - 1 } with t~ + 0 for i = k, . . . , k - j

and with t k_ ;_ 1 = 0. By IV, S(tk_3) < mfi~ -1 Ipxk-jl. In case 1 < j , ($3) implies

m - - S ( tk - j ) + ~ IPXk-J+ i I 1. Suppose (S,) holds

s(t) <= t2J[(~r + c)Js(t, j) + m fpx~ j+,l EJo ~(~ + c) ~] K

[o ' 1 <-_flF j t~ - l l p x k - j l + m l p x ~ - ~ + ~ l l _ ( ~ c + c ) "

By IV (section 1), the last term is majorized by

t~ -~ t j l /~c (k-j~ Ipxol + rn tc (k-j+l~ Ipxol I - (~c + c

[mt 1 )l<r <= t ick c l p 2 + rnf iF~P2 I - (a~ + c =

334

In case j = 0 we use IV and find

s(t) < m f l f 1 [px] < m f l f l f l~kp2

< ~b~; -k.

H.-O. Walther ZAMP

(with IV, section 1)

(see V)

(hypothesis for tk-~)

(with IV, section 1)

(with 1 < 7c/fl~).

If j > 2 (and k > 2) then Ipq(xi)l = Ipx~-~l ~ c2s(t~_x) = c2s(Dg(xi) ti) for i = k , . . . , k - j + 2, and VII implies

S(tg) < ( ~xc + 2c'] j - 1 = ~ / s(t~_j+O.

s(tk) ~ (c~c + c) H2 - ~ S(tk- O + mH? 1 Ipxkl

( ~ + c) fl2-~ c2 ~ IpXk-~ I + mfl~ -1 Ipx[

< (~c + c) ~ - ~ c 2 ~ 7 ~ IPXk[ + m~; -~ Ipxkl

< c3 [pxkl

VII. We prepare the proof of (t, x). Set c 1 : = c/m, c2 :-- flccl > cl . Claim: k e N , x e H k, 0 ,t = t ~ TxHk, 0 + D g ( x ) t = : u , qu + 0 and

Ipg(x)l < c2s(u)

imply q t ~ 0 and

s(t) < __~c + 2c s(u). L

Proof. We have 0 ,t = u ~ To(x)Hk_ 1 . By III, q t =t = O. Use V and

~ Ipxl < Ipg(x)l < CzS(U) = H~(c/m) s(u).

VIII. Suppose b ~ B satisfies (*) andp~ > 0. Let k ~ N o, x ~ H k, t ~ T~ Hk\{0 }. VIII.1. I f k = 0 then qt ,t = 0 and s(t) < g < g p ? l Ipx[. VIII.2. Assume k e N. We consider sequences of points x i ~ H i and tangent

Vectors ti ~ Tx, Hi, i = k, . . . , O, as in VI.2.1. Because of III, qtk = q t =t = O. VIII.2.1. Suppose there exists j E { 1 , . . . , k } such that for all i~

{k, . . . , k - j + 1},

ti :~ 0 and ]pxil ~ C2S(ti)

while tk_ J ~= 0 and C2S(tk_~) < [pXk-~[. Then S(tk) <= Ca [PXk] where

"= + 7c. fie C2

Proof. (By III, qti +- 0, and s(ti) is defined, for i = k, . . . , k - j . ) I f j = 1 then


By V,

S(tk_j+ 0 < (~ + c) f i21S(tk- j) + raft21 [PXk-j+ll -1 < ((G + c) fi~-lc 2 + mf i [ 2) Ipxk-jI

(with (9, c), hypothesis for i = k - j )

< (.. .) 72 ]pXk] (with IV, section 1).

Together,

S(tk) < (. . .) 7c [(~Z~ + 2C) '~cflc-1] J-1 [pxk[ ,

and (*) implies the assertion. VIII.2.2. The case t k_ 1 = 0: Par t III and t k 4= 0 give q t k ~ O. Par t IV yields

s(t) < mfiy 1 [px[. VIII.2.3. In case tk_ 1 + 0 and s(t) < c ; 1 Ipxl there is nothing to prove. VIII.2.4. If ti 4= 0 (and qti 4= 0) and [pxi] < c2s(ti) for i = k, . . . , 0 then VII

implies s(t) = S(tk) < ((G + 2C)/f ly S(to). With

S(to) ~ C ~ C p l I IpXO[ ~ Cp117kc ]pXkl (IV, section 1),

we obtain

s(t) <= ~p2-1 [(~ § 2c) Vc/fi~] k [px[ ~ ~p? l [px[ (with (*)).

VIII.2.5. Suppose tk_ 1 ~ 0 and [PXk] ~ C2S(tk) , and that there is a smallest integer j ~ {1, . . . , k} such that the s tatement "tk_j 4= 0 and qtk_ j ,# 0 and lPXk-j] < c2s(tk_j)" is false.

VIII.2.5.1. Suppose in addit ion tk_ j = 0. We have

S(tk) ~= ((O:c if- 2C)/fic)J-l s( tk- j+ i) (see VII)

and

s( tk_j+l) <= m f l c I [pxk_j+ll (see IV).

With IV (section 1),

s(t) = s(tk) < mfl~-l[(~c + 2c) 7 J f l y -1 Ipxkl <= mfls 1 ]pxl (see (*)).

VIII.2.5.2. Suppose tk_ j 4: O. Then qtk_ j =t = 0, by III. It follows that C2S(tk_j) < lpxk_j]. This allows to use VIII.2.1:

sG) <-_ c3 Ipxkl.

VIII.2.6. Altogether,

s(t) <__ G Ipxl

with d b = cpl t + mflc I q_ c~1 + c3"


3. A result for hyperbol ic m a p s

A C 1 - map9 : V--. E on an open set V in a Banach space E with g(0) = 0 is called hyperbolic (at x = 0) if o- n S 1 = 0 for the spectrum o- of T = Dg(O). Then E = P @ Q with closed T-invariant subspaces P and Q so that the induced maps Tp: P ~ P, To:Q-- , Q have spectra o-e:= {z ~ o-: lzl > l } and o-Q:= {z e o-: Iz[ < 1}, respectively. Let p and q denote the project ions onto P and Q defined by the decomposi t ion of E.

L e m m a 3.1. Let a C2-mapg: V- - .E be given, hyperbolic at x = 0 with ap :# 0 :# o-o and g ( V n P) c P, g ( V n Q) c Q. Then there exist open neighborhoods W c U c V of x = 0 with the following property. F o r every C~-submanifold H c W with H n P = {x0}, with TxoH @ P = E and such that

(H) for every e > 0 there exists 6 > 0 with Ipx - x0[ <

for all x ~ H with [qx] < 6,

there are constants ~ > 0, h ~ (0, 1) and an integer l E N such that for every preimage H k of H under the restriction g[ U with k > l,

Ipxl < 5. h k,

qt =t= O and s(t) < 6. h k

for all x ~ Hk, t ~ TxHk\{O }. For a proof, see [8]. Hypothesis (H) is satisfied whenever H is the graph of

a C l -ma p of an open subset of Q into P. Note that the unstable space may be

infinite-dimensional, and that H n Q + 0 or x o = 0 are allowed.

References

[1] J. K. Hale and X. B. Lin, Symbolic dynamics and nonlinear semiflows. Preprint LCDS # 84-8. Providence (R.I.) 1984.

[2] J. K. Hale, L. T. Magalhaes and W. M. Oliva, An introduction to infinite dimensional dynamical systems - geometric theory. Springer. New York et al. 1984.

[3] J. Palls, On Morse-Smale dynamical systems. Topology 8, 385-404 (1969). [4] J. Palis and W. de Melo, Geometric theory of dynamical systems. Springer. New York et al.

1982. [5] L. P. Silnikov, On a Poincark-Birkhoffproblem. Math. USSR - Sbornik 3, 353-371 (1967)

(Transl. of Mat. Sbornik 74 (116), 1967). [6] L.P. Silnikov, On the generation of periodic motion from trajectories doubly asympotie to an

equilibrium state of saddle type. Math. USSR - Sbornik 6, 427-438 (1968) (Transl. of Mat. Sbornik 77 (119), 1968).

[7] H. O. Walther, Bifurcation from a heteroclinic solution in differential delay equations. Trans. AMS 290, 213-233 (1985).

[8] H.O. Walther, Inclination lemmas with dominated convergence. Research report 85-03, Sem. Angew. Math., ETH Ziirich 1985.

[9] H.O. Walther, Bifurcation from a saddle connection in functional differential equations: An approach with inclination Lemmas. Preprint 1986.


Summary

Inclination lemmas ()Memmas) serve as tools for the investigation of dynamical systems in the neighborhood of saddle points. They assert convergence of inclinations (slopes of tangents) when the system shifts a given transversal to the stable or unstable manifold towards equilibrium.

We derive estimates of the speed of this convergence, for preimages 9-k(H), k ~ N, of a transversal H to the unstable manifold. 9 is assumed to be a C2-map in a Banach space, not necessarily reversible, with a saddle point x = 0 and already normalized so that local stable and unstable manifolds are contained in linear spaces.

The estimates are needed particularly in a study of nonlocal bifurcation - from heteroclinic to periodic solutions of the second kind for parameterized functional differential equations

2 ( 0 = a h ( x ( t - 1))

which describe phase-locked loops

Zusammenfassung

Neigungslemmata (fl-Lemmata) dienen zur Untersuchung dynamischer Systeme in der N/ihe von Sattelpunkten. Sic garantieren Konvergenz von Neigungen (Tangentensteigungen), wenn das System gegebene Transversalen zur stabilen oder instabilen Mannigfaltigkeit zum Gleichgewicht hin transportiert.

Wir leiten Abschfitzungen der Geschwindigkeit dieser Konvergenz her, fiir Urbilder 9 k(H), k ~ N, einer Transversalen H zur instabilen Mannigfaltigkeit. 9 ist dabei eine C2-Abbildung in einem Banachraum, nicht notwendig umkehrbar, mit Sattelpunkt x = 0 und schon normalisiert, so dab lokale stabile und instabile Mannigfaltigkeit in linearen R/lumen liegen. Die Abschfitzungen werden insbesondere zu einer Untersuchung nichtlokaler Verzweigung - v o n heteroklinen zu periodischen L6sungen zweiter Art - f/Jr parametrisierte Funktionaldifferentialgleiehungen

2 ( 0 = a h ( x ( t - 1))

ben6tigt, die PLL-Schaltungen beschreiben.

(Received: November 2, 1985)

Documents

Inclination lemmas with dominated convergence