On the Passage through Resonance

SIAM J. MATH. ANAL.Vol. 10, No. 6, November 1979

1979 Society for Industrial and Applied Mathematics

0036-1410/79/1006-0011 $01.00/0

ON THE PASSAGE THROUGH RESONANCE*

JAN SANDERSt

Abstract. The phenomenon of "passage through resonance" is studied from the asymptotic point ofview. Using averaging techniques we’ll describe this process and prove the validity of the approximationsobtained, on their natural time-scale. The theory is then applied to a model problem, proposed by Kevorkian(1974).

1. Introduction. The aim of this paper is to study the phenomenon of "passagethrough resonance" from the asymptotic point of view. Using averaging techniqueswe’ll describe this process and prove the validity of the approximations obtained, ontheir natural time-scale.

The organization of the paper is as follows:Section 2" The differential equations under consideration here are introduced,

together with a formal theory of averaging.Section 3: The proof of validity of this method (i.e. averaging) is given.Section 4: The behavior of the solution near the so-called resonance manifold is

considered here.Section 5: Since there remains a gap between the two regions for which estimates

have been obtained, a bridge between the results of the two preceding sections isestablished.

Section 6: The inner-outer vector field is introduced.Section 7: The final estimates for the composite expansion are given. This section

concludes the general theory.Section 8: The theory can be immediately applied to a model problem, proposed by

Kevorkian (1974). This problem is stated and as much information as possible withoutnumerical analysis is given here.

The reader is warned that the terms inner and outer do not refer to the time-variable, but to the space-variables. Kevorkian’s (1974) paper may provide the readerwith some of the intuition behind the theory developed here.

2. The differential equations and formal averaging. (If the reader does not havea nontrivial example in mind of the problem posed in this section, he is advised to read8 first, where he will find one.)

Consider the following local representation of a one-parameter family of C-vector fields, with "small" parameter e 6 (0, e0], on T M pr, where T is them-torus, M" and P are m- and r-dimensional manifolds respectively, pr is serving as akind of control space, in the sense that the vector field restricted to pr is a vector field onpr (it forms an independent subsystem).

2RO"-X(’lTl, bl)"]"e 1(0, m, L/) O

(2.1) rh E 2 " io(Ob m, u)3r E2 Yl(Oi,i m, u)+ e 3R2(0, m, u) trt G

=o i=o

., j=0, 1, is understood to be of the form Y Y](m, u), The dot meansdifferentiation with respect to a parameter, called time.

* Received by the editors September 26, 1977 and in revised form September 10, 1978.? Mathematisch Institut, Rijksuniversiteit Utrecht, Netherlands.

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PASSAGE THROUGH RESONANCE 1221

We refer to the Oi’s, 1,. , m, as the angular variables. The .{(., m, u)" SRn, 1,..., m, ] 0, 1, are restricted to be finite Fourier-series, with mean valuezero. This because of their nice product rules.

In the sequel we shall need to know how to handle the combination tones Oi + Oj,i, j 1, , m, and therefore we add their differential equations to (2.1). We will referto this extended equation as (2.1)*. This may change m into some m*>m, butotherwise doesn’t change the form of the equations. Take a compact domain D c [n/rand let (00, too, u0) Tm* x D be the initial values for (2.1)* Next we assume a splitting[kt[] of n, such that for m"-’(W,X)[]ko[l, one has X(m,u)=X(x,u), and,furthermore, the second order averaged equation (to be defined in 3) restricted toT’* l Rr is an independent subsystem.

We shall restrict ourselves to the case 1, since this keeps the discussion as simpleas possible. The assumption is satisfied in the model problem. This simplifies mattersbecause we don’t have to choose new coordinates when studying the flow in theneighborhood of the resonance manifold (cf. 4). Writing

(Yi.(Oi, lTl, u)) i=1,’’"(2.2) .2..(0i, m, bl)

\Z.(Oi, m, u) f O, 1,m

one has the equations"2R0 X(x, u) + 1(0, w, x, u)

e Yo (w, x, u) + e Y (Oi, w, x, u) + eaR2(O, w, x, u)i=1

2 3R(2.3) .i=eZ(x,u)+e Zo(Oi, w,x,u)+e Z(Oi, w,x,u)+e 3(0, w,x,u)i=1 i=0

i eW(u).

What one would like to do now, in order to simplify this system, is to average overthe angular variables and forget the remainder terms Ri, 1, 2, 3. This can be done asfollows: define a domain D)c ,+r (where 6 is some order function of e) and a map

" T’*D)T’*D, such that there is a vector field on T’*D) of the form(2.1)*, but with Y j= 0 for =.1, , m;/" 0, 1, which is -related to (2.3).

(Two vector fields v and t are b-related iff v(c/)(x))= d4)(x)g)(x); if & is invertible, tis merely the pull-back of v along b.)

Usually one finds by substituting some formal development in the definition of-relatedness and equating terms with the same order of magnitude in e. In general

this procedure does not lead to a unique result (neither in the map, nor in the vectorfield), which probably accounts for the vast number of different theories on this subject.

Having obtained this vector field, one forgets about the remainder terms. Theresulting vector field is usually called the averaged vector field.

Trying to define , everywhere on T’* + [,/r is likely to produce trouble, namelythe small-divisor problem, which is also called resonance. It follows frorh the formalcomputations that resonance occurs iff one of the Xi becomes zero. Therefore weintroduce some concepts in order to define a nice domain D, before presenting theformulae for and the averaged vector field. First we assume X to have the followingstructure:

(2.4) X(x, u)=x

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1222 JAN SANDERS

the "i-th resonance manifold" (a manifold only in the sense that it is the solution of someequations; it may not be locally Euclidean everywhere). Condition (2.4) assures us thatthe resonance manifolds are not slowly moving, which makes matters simpler.

Define e[/[i((e)) by

(2.6) /[i((8))- { (1TI, U) E Rn+r a(e) }[Xi(x, u)]o(1) for e $ 0

where is some order function of s with 8(s) o(1) and E/(2(E) O(1) as s $ 0 (3 liessomewhere between s 1/2 and 1).

Now let Da be the intersection of the local coordinate domain and fq/--1 tli(8(s)).Da may be disconnected.In the sequel it is assumed that only the zeros ofX are of importance and that the

zeros of the 12i, if they exist at all, cannot become important due to the initial conditionsand the time-scale of interest to us.

We introduce for (b, y, z, u) E T"* xD the map

(2.7) m*2 2 W(1)()i, y, Z, U)i=1

’SThe reason for the appearance of the w(x) is the following: In the (e)-neighborhood of i the second variable y is slowly varying with respect to the naturaltime-scale 1/x/, and therefore can be approximated by a constant.

The only remaining problem is to approximate the first and third component of thesolution in this neighborhood.

To do this we need sufficient accuracy of the approximate initial values provided bythe outer expansion. It follows from the estimates in the next section that one needs toknow the third variable very accurately in order to get a reasonable approximation tothe first (here one needs the splitting assumption). This cannot be done without theW(1) S, as the reader may want to verify.

Then, differentiating the relations

0 "- e 2 U(o (,b, 3’, z, u)

w y + e Y’. V(o)(qSi, y, z, u) etc.

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we obtain the following equations:

--X(z, U)+x3(z u)R(, y, z, u; e) I

eYe’o)(y, z, U)+xZ(z u)Rz($, y, z, u; e) y R’3

,;---eZ(?o)(Z,U)+e:Z)(z,U)+x(z,u)Rs(, y,z,u;e) ze

W(u) u

where the Ri, 1, , 3, are uniformly bounded in all variables on T’* D x (0, eo]and

1 m* 1(2.10)

X"(z, u)- ]X(z, u)l""Here we have incorporated the assumptions on the averaged equations which charac-terized the splitting of Rn. Introducing the following notation:

(2.11)

o (;)(, y, z, u)=W (j) ()b y, Z,

Oy Oyl OZ (9 1’

Ou O-u1 O-u Om Oyl Oy Oz

we see that (2.9) holds when the following choices have been made"

O(0)(tl, y, Z, U)Xi(z U)

o(Oi, Y, Z, U) di

U(o)(ti, y, z, u)--’

(2.12)

Xi(z, u)W(o)(Oi, Y, Z, U) X(z, u) di

/ Z?o ]

f,o,(y, z, u) Is VD(0)(ti’ y’ Z, U)-’O--Z(o ((i, Y, Z, U) d&i

2’1) (z, u) z (y, z, u) + Y 9’’(y, z, u)i=1

[. .., &,-.’,l+j,k--- o(0)

Om .,ojOi + Ok t#

0 90,iW(o) dO.:-m w() ouHere the integrals are taken in such a way that U(o) and have mean value zerotl(o)

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1224 JAN SANDERS

We will now proceed to define approximations for a solution of the initial valueproblem (2.3) starting in some domain D1 (that is, the distance to any resonancemanifold is Os(1)), going through a resonance manifold until it is again in (anothercomponent) of D1. This is the so-called "passage through resonance". The neighbor-hood of a resonance manifold is called the inner domain and its complement the outer

domain.In the outer domain the natural time scales are 1 and 1/e. in the inner domain it is

1/x/, or, if there is a saddlepoint, 1/x/log 1/e depending on how close to thissaddlepoint one permits the solution to come. All estimates in the x/-neighborhood ofthe resonance manifold will be on a time-scale V1/x/, but the Gronwall-type estimateallows one to take the logarithmic factor into account.

3. Estimates for the outer expansion. Now consider the "outer equation""

, X(z, u)

e Yo)(Y, z, u)

eZ(o)(z, u)+ e2)(z, u)

=eW(u).

The solution of this equation is called the outer expansion. In this section we are going toestimate the difference between the solutions of (2.3) and an outer expansion with

appropriate initial conditions, as long as they are in the outer domain. Therefore thetime-scale of validity has to be at least 1 !e. Suppose one knows the initial conditions for(2.3) within some order of accuracy. Then the first problem is to show how one can usethese approximate values to solve (2.9) or (3.1) without losing too much accuracy.

Let (0o, Wo, Xo, Uo) be the exact initial conditions and (0o, o, o, Uo) some approx-imation with:

(0o, rPo, o, Uo) Do()

(3.2) Ilwo- oll- o(=())6o(e)=O(1)

li(E)"- 0(0), i= 1,..., 3

EIlxo-oll-- 0(’3()) 6g(e--o(1)

as e ,1, O.

(In the end we will consider the special case where all initial conditions are knownexactly; this case follows from the present discussion.) Since u0 is always known exactly,we will restrict our attention to the other variables. This is easily done, since ,,restricted to pr, is the identity.

Define to be

(3.3) (0o, 1o,.o)- 0o, Wo, Xo--E 2 W(o)(Oio, 1o,o, Uo)i=1

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and let (bo,)70, ZTo) be the image of (0o, fro, o) under .One has, by abuse of notation"

(I)e((e (0--0, 10, 30)) -,(I)e((I)l(0o, Wo, Xo))

O(1) + O(00)O(6z)+O(-oo) as e 0onDo.

Thus (b approximates -1 in a certain sense. The mean-value theorem gives thefollowing estimate for the difference between (bo, ;o, $o) and (bo, yo, zo);1(00, WO, Xo)"

I](D0-- 011 O(1) -I- O(-02)

][Z0-- 0[[ O(3) "- O() -- O(EIIf we started with exact conditions, (3.5) would read"

(3.6) IlY-)7ll (oo)Ilzo- oll- o

This completes the estimates for the initial conditions.We now turn to the solutions.Let (4, Y, z, u)(. be the solution of (2.9) with initial conditions (4o, yo, zo, uo) and

let (4, ;, ,f, u)(-) be the solution of (3.1) with initial conditions (4o, 17o, o, uo). Note that(rb, y, z, u)(t)= (0, w, x, u)(t) since (2.9) and (2.3) are -related and have unique

solutions. (This is a general line of argument: Let v, and 4 be as in 2; suppose x is an

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1226 JAN SANDERS

orbit of tS, i.e. 2(t)= O(x(t)). Define 4(x)(" by (&(x))(t)= 4)(x(t)) and put y 4(x).Then" )(t)= d4)(x(t))(x(t))= v(cb(x(t)))= v(4)(x)(t))= v(y(t)). Thus y is an orbit of v.Let be another orbit of v with initial conditions (0) $(x(0)). Then, by uniqueness,:=y.)

We now make the followingASSUMPTION. Lef m , m 2. The

Iot e dz C( 1 1 )(3.7)X"(e(r), u(r))- Xn-(e0, u0) +x"-l(e(t), u(t))

This assumption has to be made to insure that the solution enters (leaves) the resonancefast enough. For suppose Zo) is zero or very small; then (3.7) is not likely to hold,because the natural time-scale for will be much longer than 1/e in that case.

Of course one can develop a theory for such cases by changing the assumption andputting the appropriate time-scale in formula (3.7).

Note that the problem with Z0 0 and 2) # 0 is not only of theoretical interest,but it has an application in the spin-orbit resonances in 2-body problems with loweccentricity (for the equations, see Kyner (1969)).

In 8 we show that it is not always impossible to verify (3.7) in concrete situations;this is because and u are known explicitly, at least formally.

LEMMA. With the notations as above and with assumption (3.7) satisfied, thefollowing estimaw holds"

IlO (t) (t)ll o()+o()+ o()+ o(@)+ o()N(t)ll

(3.8) Ilx(t)- e(t)l[ o +o] + o(&)

x(t)-e(t)-e Z w(m((t), ;(t),e(t), u(t))i=1

as long as ONetNL and (, ,i, u)(t)D.Pro@ In this proof appear a number of G, 1, 2, , all e-independent. These

will be denoted by the universal constant C to simplify notation. Suppose is somefunction from N [0, 1] and let

(3.10)a(O) g(O)z (0)+ (1 g(O))zT(O) .(0)+ g(O)(z (0)- .(0)) .(0)+ 0(a3),

=zT(0)+o(1) ase$0.

Thus there is a nonempty interval [0, r*), with 0 < er*<- L, such that

1 2(3.11)X(a(t), u(t)) <X’e’t’,tt) u(t))

e [0, r*)

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because . (0) D and 1/X behaves more or less nicely there. Suppose there is some 0osuch that, if sCo Ooz + (1- 0o), then

(3.12)1 2

X(o(*), u (*)) x(e(*), u (*))"

Combining (2.9), (3.1) and the mean-value theorem, one obtains the following esti-mate"

z(t) z(O) + eZo (z(o.), u(o.)) + e22) (z(o.), u(o.))

-I" X4(Z (O’), U (0"))g3(b(o’), y(o.), z(o.), u(o.); e)} do’,

.(t) (0)+ {eZo) ((o’), u (o.)) + e 22)((o’), u (o.))} do’,

which implies

]]z(t)-(t)l}<-llz(O)-e(o)ll/ ll/Po) (z(), u(,))-Zo ((), u())ll

+e 3 1 }X4(z(ff), u())llR3((), y(), z(), u(); e)l[ d

2(3 13)

+ X4(z(), u())

fot fot Ce3

In proving (3.13) we have used implicitly the following estimates"

=o(1) ase$0x2(&,(c), u(r))0 X.((cr), u(c)) 0

(3.14) (tJ= O* in (3.11))

X4(z(o’), u(o’))= O X4(i(), u()) ((t) 1 in (3.11)).

The estimate for the derivative ofZ follows from the definition ofZ in (2.12).Here we use the fact that Z0 and2 do not depend on y. The existence of 0*

follows from the mean-value theorem. We apply Gronwall’s lemma to (3.13) to get

Using (3.7) gives

(3.16) [Iz(t)-(t)l[ c [[z(O)-(o)ll+X(z(t), +

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1228 JAN SANDERS

and thus

(3.17)

(3.18)

=o(1) ase$0.

Applying (3.17) to (3.12) we see that

2x((*), u (z*))

1x(0(r*), u (*))

-x(e(*l, u(*ll x(o(*l, u(r*l x(e(r*, u(*ll

1-x(e(*), u(*)) x(e(*), u(*))

x(e(*), u (*))1 +

x(e(*), u (r*))]

1(1+o(1)) ase$0,

x(e(r*),

which contradicts the existence of o and -*. Thus (3.18) implies that (3.15) is valid onO<_et<_L.

The only difficulty in this proof was that in order to estimate IIz(t)-2(t)ll oneneeded an estimate for 1/(Xn(z(t), u(t)))in terms of 1/(Xn(2(t), u(t))), which was easyenough, once one had an estimate for IIz(t)-2(t)ll, etc.

The remaining estimates follow easily:

(3.19)

Using (3.5) in (3.19) gives:-- o,

lie(t) (t)]] O()+O(o)+O()+ (-o3)+

In the same way one obtains

The derived estimate (3.8) now follows from (2.7) (using (3.11) and applying thetriangle inequality).

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Suppose the initial conditions are known exactly and 60 1; then (3.8) becomes

t )IlO(t)- (t)ll 0 -(3.22) [[w(t)- ;(t)l] 0()[Ix(t)-e(t)ll o()

x(t) (t) e E W(o)(qb(t), ;(t), .(t), u(t))i=1

This is the case of a solution starting in D1, coming to the region of resonance

4. Estimates for the inner expansion. The results in the preceding section hold inDa. That leaves us with the question" what happens if the solution does not stay in Da,but enters the inner domain?

To answer this question we introduce the inner vector field. To estimate thedifference between the exact solution and the inner expansion (i.e., the solution of theinner vector field) is not easy. We do it in two steps" first in a x/-neighborhood of aresonance manifold, then in the next section in 6-neighborhood minus the /-neighborhood.

The first step is straightforward:Suppose we have already constructed an approximation that is entering the

/-neighborhood of Vi. Then we can average over all angles except Oi. Since we knowthat we are working in a /-neighborhood of Ni, we scale the coordinate transversal toW and develop the vector field by Taylor-expansion. Throwing away the higher orderterms, we call the remaining equation the "inner vector field" and its solutions innerexpansions.

m*So let D denote the intersection of the local coordinate, domain and f3 ij o/[/[i(6)and define

xt% T"* D Tm*D

where Uo), Vo) and Wo are defined as in (2.11) and (2.12).This transformation is intermediate between first and second order averaging and

has to be used since first order averaging doesn’t give the desired results. It is notobvious why first order averaging does not suffice. The technical reasons for its failurewill appear only in the next section.

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1230 JAN SANDERS

The -related vector field is"

2Rd)=X(z,u)+ (4), y,z,u" e)

:ZR=eY(y,z,u)+eYio(i, y,z,u)+e 2(qb, y,z,u" e)(4.2)

:2 eZ(z, u)+ eZio (cbi, y, z, u)+ e2R3(b, y, z, u; e)

a=eW(u)

where the gi are, as usual, uniformly bounded and

(4.3) X.(z, u)lx 0.

Let z be a zero ofX and scale

(4.4) z z/ x/sr.We have sr O(1), since we are in a x/-neighborhood of A;.. We assume that (OXi/Oz)(z, u) : 0 (cf. (3.7)).

Following the outline in the Introduction we write down the inner equations"

X(zo, uo)+ 4-i(zo, uo)

p=O(4.5) : 4-i2(, Yo, Zo, Uo)

a=sW(u)

where =Z +ZJo. The initial conditions Uo and yo used in (4.5) are initial withrespect to the moment of entering the x/-neighborhood. In the sequel both z and sr areused, but one should keep in mind that (4.4) always holds, although not explicitlywritten down on every occasion.

Let (b, y, z, u)(. be the solution of (4.2) with initial conditions (4o, yo, Zo, Uo)and let (b, 17, st, u) be the solution of (4.5) with initial conditions (b, 37o, sro, Uo).

(Since the vector field is autonomous, one can always translate in time and call anyto zero; thus the zero-time in this section need not be the zero-time in another section.)

Using again Gronwall’s lemma, we see that the following estimate holds"

(11, (t) i(t)ll + Ily (t) 37(t)11 + IIz (t) 2 (t)ll)(4.6)

--< 116;o- 4oll / Ily0- 37011 / llzo- coil / c e

Moreover

(4.7) [[y (t)- 37(011 -< C(llyo- Poll + et).

These estimates can be obtained by a slight modification of the method of analysisgiven in Sanders (1978).

Let (0, w, x, u) be the solution of (2.3) with initial conditions (0o, Wo, Xo, u0) and let(, y, z, u) be the solution of (4.2) with the same initial conditions; then it is not dicultto see that the difference between these solutions is of O(s).

Together with (4.6) this gives us the desired estimate for the difference between(0i, w, x, u) and (., p, 2, u), on a time scale 0 NtNL. It is not clear how to extend thistime-scale using Gronwall-type estimates, other than with logarithmic factors in e.

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PASSAGE THROUGH RESONANCE 1.231

5. Extension of the time-scale for the inner expansion. Were it not for the fact thate/62= o(1) as e $ 0, the preceding analysis would have given the complete picture ofwhat is happening. The problem remains to extend the validity of the inner (or theouter, or some other) expansion to a 6 (e )- neighborhood of the resonance manifold.The difficulty of extending the validity of the inner expansion is in the fact that itsnatural time-scale is of O(1/,,/-e) which is not enough to go through the larger6- neighborhood.

Following an idea due to Eckhaus (1976), we prove in this section the validity of theinner expansion on the larger time scale 6/e.

It follows from (4.2) by a straightforward argument that one has the followingestimates on its solutions:

’(t) O()(5.1) u(t)=uo+O(6) on [0, L]

y(t) y0+ O(6).

Using (5.1) we expand (4.2) to get"

Differentiating 4 gives

,; 4x"(z), Uo) +(4-6 + e)R,(cb, y, , u; e)Oz

e(z}, Uo)2(4, yo, z, uo)+ eR,(6, y, , u;Oz

(Here we have used the "one-and-a-half-order averaging").Integrating yields

bil"2 1L-- f/’ OXi fot-z.(0)2= e (Ji) dJi + e8 Rai(r) dr(o) OZ

(5.4) eH(j, i(0), yo, z, Uo)

+e d(l,((l, y(l, ((, u(l; el .Note: H is not a function on the circle but on its covering space, since Z need not

be zero.Consider the equation"

(5.5) 1/24; 1/2(0)2= eH(,bi, 4).(0), yo, z{, Uo).

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1232 JAN SANDERS

Let 4* be the solution of (5.5) with 4*(0)= 4.(0) and let t* be the solution of

e3 b*(t*(z))Rl,i(qb*(t*(z)), y(z), r(z), u(r); e) dr(5.6) .2= 1 + 1/24i(0)2 + en(qb*(t*), b.(0), yo, Z/o, Uo)

with initial condition such that 4*(t*(0))= bj(0).Define

(5.7) .(t) qS*(t*(t));

z db*(5.8) cbi i*dt*

(5.10)

1*(t*b(0)2 + eH(b (t)), qSi(0), Yo, z, Uo)

+et qb*(t*(z))Rl,i(cb*(t*(z)), ,) dr

=2 ’(O)2+eH((t)’cb(O)’Y’Z’u)+e q(’)R13(t(z)’ Y (z)’ "’) dz"

4 therefore obeys (5.4) and has as initial condition 4(0)= b*(t*(0))- b(0).Furthermore

IqS* (t* (t))- cb*(t)l <- ftt*(t)

Assume

I,*(z)l dr Ca(e)lt*(t)- tl.

s=o()(5.11)

*(t*cbi(O) + eH(4 ), bi(0), yo, z, Uo)2which is more or less equivalent to e/l. O(1). Thus 1/r2= O(1), implying that one

has to stay outside the /-neighborhood of the resonance manifold.From (5.6), using (5.11), we obtain the following estimate:

(,4 [ ,](5.12) It*(t)- t] <- C -+ bio with 6 0,-Lewhich implies"

8 )(5.13) [i(t)-*(t)l<-C -+ 1/o[[bo- qo[

It is easy to see that on the one hand if we have a solution of (4.5) with initialconditions (4o, 7o, r-o, Uo), then its bi?component can b.e identified with b*; while on theother hand b’ has initial conditions bi(O) bi(O) and ’/(0) (0), which implies thatis equal to the b.-component of the solution of (4.2) with initial conditions(bo, yo, Zo, Uo) by uniqueness of the solutions of this equation.

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I’ASSAGF THROUGH RZSOrqANCE 1233

We now have two estimates ((5.14) and (5.15)) for the difference between b. andb., one for the ingoing and one for the outgoing solution. (We call a solution ingoing if itenters the resonance region from outside, i.e. from D.) The first estimate, assuming14; ol is

(5.14) Ib;(t)- b-,(t)l C(/ 14’;o-ol)while the second, assuming Iiol x/, is"

(5.15)

Using (4.5) and (5.2) we obtain

(5.16)

Here {1, 8/x/} means 1 for the ingoing and 8/x/ for the outgoing estimate.Note that 8a/x/ is small compared with 86/(eax/), since e/8a= o(1). Our final

estimate is the easiest:

(5.17) [[y(t)- 37(t)ll C(, + Ilyo- o11).

The connection of the approximations found here with the original equation hasbeen discussed at the end of the preceding section.

6. The inner-outer vector field. The inner-outer vector field may have Iittle to dowith the original equations, since it is the inner expansion of the outer vector field andthe outer expansion of the inner vector field.

This means that one is "assuming" two contradictory facts at once" that one is neara resonance manifold and far away from all of them.

But the flow of the inner-outer vector field (the inner-outer expansion) doesapproximate the inner expansion in the outer region and the outer expansion near theresonance manifold. Thus it is possible to make composite expansions like"

(6.1) Xc xz + Xo Xzo

where xz, Xo and Xzo are the inner, outer and inner-outer expansion. In the outerdomain the estimates run as follows:

(6.2) IIx-xcllllx-xoll+llxi-x,oll--o(1) as e $ 0

and in the inner domain"

(6.3) IIx-xcllllx-x,ll+llxo-xioll o(1) as e $ 0

where x is the exact solution and all approximations have the right initial conditions (orasymptotic behavior).

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1234 JAN SANDERS

The inner-outer equation is"

X(z o, uo)+ uo)OZ

(6.4) ) =0

=eW(u).

Integrating (6.4) one shows that ,o(t) is a parabola which takes an extremal valueat the resonance manifold under consideration. (All the inner-outer talk is about onepassage through one resonance; this is not a global inner-outer field, as is indicated bythe presence of z/0 in (6.4).)

To get the right inner-outer expansions we proceed as follows" (X ino denotes the

ingoing outer solution, etc.).Let x(0) x(0) (x(0) is the given initial condition of the exact solution). Suppose

in(T)" this determines Xoat some r, xo(r)ec Let xino(r) X o (0). Then let x,(0)inX,o(0). All estimates are given in the next section, proving that this is a possible way to

proceed. On [0, r) the composite expansion is defined byinin (t)+x(t) Xto(t).(6.5) Xc(t)=x,

On the next interval It, -?), where ? is to be defined below, let

(6.6) xtc(t)= xz(t).

Suppose at some time " the inner solution has a distance of Os(1) to the resonancemanifold.

Let out out () .. Let Xo() Xzo() Then theXzo (’)= xz(?). Now ? is defined by Xzooutgoing composite expansion is, on [?, T), given by

out out(t(6.7) x cUt (t) xz(t) + x o (t) xo

Collecting all our results we can prove that the composite expansion obtained by

Xc(t) O<-_t<r

(6.8) Xc(t) {xc(t) r <= < "/out

Xc (t) <=t<T

is in fact an approximation of the exact solution.

7. Estimates for the composite expansion. We will now give the estimates for (6.4).inUsing the notation adopted there, the solution (,)7, 5) in 3 will read (0ion, Wo, xn)

here.First using (3.22) we obtain

IIo(t) O (t)[I in

(7.1) IIw(t)- wo(t)ll 0 0 <= <= . < r

Ix(t)-Xo(t)l=O

where 6i, is an order function as defined in (2.6). Using (3.22) again, estimates the

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difference between the inner and the ingoing inner-outer solution:

in (t)ll O()(7.2) [[w, (t)- W,o (t)[I- O 0 _-< , < z.

Ix/(/) Xzoin (/)l O(_.n)The estimate for the composite expansion now follows from (6.2).We know that the outer and the ingoing inner-outer solution are equal at z.

Using (4.6) and (4.7) we get the following backward estimates:

(7.3) ( Ozo(t)l/llwo(t)- wzoIozo(t)- in in (t)[[1 in+ --e lXo(t) x zo t) for 4lt- 1--< L.

Using (5.15), (5.16) and (5.17) we have

-Oj.,o(t)[=O

E(7.4) Ilwo(t)- W inlo (t)ll-- O(ain) [t- z[ < L.

6in (_i/[xo(t) xzo (/)l- O

Combining (7.1), (7.2) and (7.4), we have, at the moment - of entering thegin-neighborhood of W., the following estimate

(7.5) IIw() w,()ll -[" O(in)

Using (5.14), (5.16) and (5.17) gives

10(t)- 0;,,(t)l- O +

(7.6) IIw(t)- wI(t)[[-- O(in) -[" O--.n)[x(t)-x,(t)]=O +

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1236 JAN SANDERS

until the solution reaches the /-neighborhood of Y.; here the estimates become, using(4.6) and (4.7):

(47)in -[\8i6n8/lO(t)- o,z(t)l O +O2-/

(7.7) IIw(,)- wz(t)ll O(ain) -I- O(L)6

+0\E /

We can do the same thing for the difference between the outer solution and theinner-outer solution, but that has already been estimated in (7.4).

At this point we choose 8in; in fact we can take different 6in’S for each of thecomponents, since we work with a composite expansion.

Thus we get

IO,(t)-O,.c(t)l O(e 1/14)

(7.8) IIw(t)- wc(t)ll o()

with (e)= o(1) as e 0,

Ix(t)- Xc(t)l O(417).Again using (5.15), (5.16) and (5.17) yields:

o(,:ut ,,14)IOi(t) Oi ,(t) x/ + 0(8

(7.9) IIw(t)- Wz(t)ll=O(out)+O (6 2

-Xl(t,, + o ’outThe difference between the inner solution and the outgoing inner-outer solution is

in the outer domain estimated by using (3.18)

,o,,(t, out,,Io(t)l O aout(7.10) lWl(t) outwo (t)ll O

outIXl(t)- xo (t)] O

In the inner domain the difference between the outgoing outer solution and theoutgoing inner-outer solution is given by (4.6) and (5.15), (5.16) and (5.17):

, out o,,o(t)l 0,.o(t) ou,

(7.11> iiwUt(t outw io (t)ll o(aout)

out out (t)l O(utxo (t}-x,o ,]"

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Therefore if the solution leaves the out-neighborhood of Yj, one has at thismoment ? (combining (7.9), (7.10) and (7.11)):

(7.12)

[0j(’) I],,OzOut (’)[ O(E) -[- O() -["O(E 1/14)

out o(E) (4)IIw()-wo ()ll=O(ou,)+ +o()

6out ()l O() +O() 4/7) (:ut 1/14)Ix(t-xo +O(e +0 e

(7.13)

Using (3.16), (3.17) and (3.13), we have:

() (E1/14)]Oi(t)_out(t) (4out .+. 0E

t,,O 0\--, +O ’ut +"

[[w(t) wOoUt(t)l} O(out)+ O(out) + 0 (e)

O(6ut-’[- \U]-[-" O(E 4/7) -[- O E

out ( E ) (to6ut 4/7) (o2ut 1/14)Ix(t)-xo (t)l=O +O,--,+O(e +O\-eCombining (7.10) and (7.13) and choosing tout optimally:

(7.14)

IIw(t)- wc(t)ll o (, for any s.t. /(e)= a(1) as e ,[, 0

Ix(t)-xc(t)l- 0(Ell/21).This procedure does not give a nice estimate for the angular variable(s), although

there seems to be no fundamental reason against this.Here we conclude the more theoretical part of this paper and turn to the

application promised in the Introduction.Remark. More experienced readers may wonder what happened to the more

classical methods like matched asymptotic expansions; those readers are stronglyencouraged to provide (as an "exercise") approximations and estimates of validityalong those classical lines.

The reason for the approach given here is that it does not depend upon explicitrealizations of solutions (to be expanded in the matching process). Furthermore, theestimates for the difference between the inner, respectively outer, solution and theinner-outer solutions were already obtained in earlier sections. One of the difficulties inthe matched asymptotic expansion approach is that one needs an expansion of theinner solution, which is not explicitly known. A glance at Kevorkian’s 1974 paper mayconvince the reader that it is indeed possible to overcome this difficulty, if only he iswilling to pay a price in mathematical elegance.

8. The model problem. In order to illustrate the theory developed in the preced-ing sections, we look at a problem of reentry roll resonance, studied by Kevorkian(1974). In this paper, Kevorkian gives a formal treatment of the resonance problem

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1238 JAN SANDERS

with multiple-time-scale expansions. Lacking proof techniques, this was a most difficultthing to do, since he had to match terms with small divisors.

The reader is advised to study this paper for two reasons:1. It gives a lot of background information and references on the problem of

reentry roll resonance.2. It is a nice example of a complete treatment of a problem based on good

intuition.The only thing Kevorkian doesn’t show is that the phenomenon of sustained

resonance (on a time-scale l!e) is exhibited by the model problem.Having introduced the basic equations, Kevorkian restricts his attention to the

following model problem"

" + (p + o2)y 0 y(0) 0

Y(0) =xo

/0 eto2y sin p(0) P0

=1/2eo o(0) oo

4p (0) 0

We extend this system with

(8.2) (p2 + w2),/2 (0) O.

These equations are not in the standard form (2.1); therefore we make thefollowing transformation"

(8.3)sin (: +)

2)1/2(p2 + to cos ( +)

a(O) goPo(1 + 02)1/2

(The (y, 3))-notation is a bit sloppy, but one gets tired introducing new symbols allthe time.)

The induced equations are"

(p + o)/

2)-1 2d -ea (p2 + to to cos2 (: + ){1/2 + ap sin ( +) sin 4’}

2)-1c]=etoZ(pZ+to sin (so +) cos (: + ){1/2+ ap sin ( +) sin O}

p etoZa sin ( +) sin p

There are two fast variables ff and : and four slow ones" a, , p and to, where to isthe only one that doesn’t depend on the others.

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Equation (8.4) is still not in standard form; to put (8.4) in standard form we writethe right hand side in the form of finite Fourier-series and then introduce thecombination angles:- (p + w)

d -eaoo2(p2 + wz)-a{J(1 + 2 cos 2( + 4)) +-aP[cos ((3- 4)+ 3b)

-cos ((3 + ,)+,/,)+ cos ((’-)+,/,)-cos ((’ + )+ ,/,)]}

c ewZ(pZ+wZ)-l{1/4sin 2( + 4) +1/2ap[sin ((+) + b)

-sin ((-4,)+ 4)-sin ((3+ O)+ 34)+sin ((3- 0) + 3b)]}

O 1/2ewZa(cos ((- O) + b)-cos ((" + p) + 4))

Now define

O1 W1 a

02=3-- W2=-103 3+0 x =pw

04 --0 b/ ---0)

05=’+0.From (8.5) it follows that

O=u(l+x)/

0= u(3( + x)/-q-x)

63 u(3(1 + xZ) a/z +

O,= u((1 +xZ)/z-4-x)05 u((l + x2) a/2 +

(8.7) -ewe(1 + x)-{J(1 +cos (20 + 2w)) + kuwx[cos (0 + 3w)-cos (03 + 3w) + cos (04 + w2)- cos (0s+ w)]}

if2 e(1 + x2)-a{J sin (201 + 2w2)+ uwx[sin (0 + 3w2)

-sin (03 + 3w2)-sin (04 + w2) + sin (05+ w2)]}

i e{UWl[COS (04 + w)-cos (05+ w)]-x}

=e.Equation (8.7) is in standard form. The initial conditions for (8.7) are"

0(0) 0

w(o) xopo(1 + )1/(8.8) w(0) 0

x(0)= 1/

u(0) wo.

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1240 JAN SANDERS

Using the notation of 2, one hasai(U)--U i=1,...,5,

/( +x/3(1 + x 2) a/2 /x

X#(x) 3(1 + xe) 1/2 + 4-x( + x)/-4-Sx(1 + x)1/ + 45x

yo(W, X) (-1/4wa(l + x2)-a)o Z(x) -zx,1 2).(-wa cos (201 +2w2)’Y(01, w,x)=4(l+--- sin(2Ol+2We) 1’

uwx(-w cos (o + 3w)Y (02, W, X, U8(1 + x sin (0. + 3 W2) ]’

UWlX(WI\ COS (03 q- 3W2)y30 (03, w, x, u)8(1 + x -sin (03 + 3w2) 1’

Y (04, W, X, U) UWlX2)(--W1\ COS (04-- W2)8(1 + x -sin (04- we) 1’

Y (05, w, x, u) uwx)(+w\ cos (0+ w))8(1+x sin (05- w2) 1’

Zo =o,

1Z( uw cos (04 -1- w2),

1Z50 -uw1 cos (05+ w2).

It is clear that on the time-scale of interest, namelyl/e, "i(u)---Os(1); the onlyX#-component which can become zero is X4, where one has z 1. That this is theonly resonance is a special feature of the model problem.

The extended system has five more angular variables

06 5 --///

(8.10)

09 2sc +

and does not have any new resonances.We will now construct the outer expansion"The only thing one has to compute to get the outer expansion is 1). This turns out

to be zero, thus satisfying all assumptions on its structure.The outer equations for the model problem are"

(8.11)

in4 (1 + z2)l/2-4-z (4 (0)--" 0

J) 2)--1 in 02)1/2ey (1 + z y (0) ,oPo(1 +in))2-- 0 Y 2 (0) 0

Z"----eZ zin(0) 1/a

ll "--1/2EU uin(o) 0)0

We do not write down the equations for all the other angular variables.

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(8.12)

We compute the ingoing outer solution explicitly:

2et ( l+x/l+a2et }in----Po 1+0 _-;-’ ]

04o(t)=(1-x/-)pot 2 2 x/l+a e +logE

(t) Xopo(l+)/ 1+Wlo 1 + 2eetwo(t) =0

in (t)1 -(1/2)etxo --e u(t) oe(1/t

where 7. is given by the equation

(8.13)

and thus

in (7.) 0XO

1(8.14) er log---5.

One has to check assumption (3.7). It suffices to show that:

e 1 1(8.15)

x4n (xi3(7.), u(7.))d7. C

X_ (x)(t), u(t))+ X4n-1 (xino(O), U0)

The following estimate holds on [1, oo):

1(1 + x)-> 4x -(1 + x2) 1/2 e (4- 1)(x 1).(s.16)

Using (8.12) and (8.15), it is not difficult to show that (8.15) holds.For the outgoing solution one can use the estimate:

1(8.17) 1 +x _->(1 +x2)1/2-’,/-x _->-(1-x) x el0, 1].

In the original coordinates for the model problem one has2

)1/4yoin(t) XoPo(1 +c 2) 1/2 l+al+ a2et.(8.18)

sinpo t-- x/l+a

2 et1+/1 +a-x/l+a e +logl+x/l+a

We will now turn our attention to the inner expansion.The inner equations are"

(4 ---U(T)

(8.19)

(8.20)

1/2/(- 1 + u(r)wno(r)cos (4)

fl 1/2eU,

u(r)=po

in 2)1/2( 1 d" O2 1/4

Wlo(7") XoPo(1 + a1 + a !

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1242 JAN SANDERS

Thus one has

(8.21)e in (r) cos 4 1) 0.4 -t-’U(T)(U(T)W 10

inThere are critical orbits (in the original coordinates) iff u(7-)w lO(7-) > 1. In that caseone is elliptic, one hyperbolic. Since the solution of (8.19) is not an elementary functionof time, one has to be content with numerical results.

Having done the numerical work, one may proceed with constructing the outgoingsolution, as described in 6.

The difficulty here is that one has to know the asymptotic behavior of the innersolution and thus the numerical integration has to be done on a time scale that is 1//times the natural time-scale of the inner equation. The same difficulty occurs when onestarts integrating since the initial conditions are given for outside the resonancemanifold. We are now going to compute these initial conditions"

The inner-outer equations are given by

(8.22)

fi =1/2eu.

The initial conditions have to be determined from the value of the outer expansion at 7-.

in 2p0 1 2p0 2 /+log ( 1+/ )}04IO(7") (1 /)log /1 + a 2e a e 1 +/1 +a

(8.23) 2) 1/4in 2) l+a,w 1,o (r) XoPo(1 + a

2in (7") 0W210

(ino( ) 0

u(r)=po.

Integrating backwards, we get

(8.24)

in,o(t) gxe(t r)

w xto(t) XoPo(1 + a2

in (t)= 0W210

=> sty(O) er

in0410(t) (1 /)por

1:: xI(O) 1 + log

2) 1/4

=):’ wit(O) XoPo(1 + a2) .! +2

= W2I(O) 0

2po{/l+a2_x/+log ( l+x/ )} Po2’e l+x/l+a 2-e

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which implies

(8.25) 04,(0)=(1-/)p07 2p0e 41+cel+x/l+a2 -2--z

This is as far as one can carry through the analytic treatment if one is interested inthe time-behavior of the solution. If one is interested in the orbits one may go further,using the fact that the inner vector field has an integral with a rather easy asymptoticbehavior. But this analysis should not be trusted too much, since the validity of theapproximation is only proven on the natural time-scale of the inner expansion. Then itmay happen that the "approximating" orbit goes out of resonance, while the exactsolution stays in resonance.

Comparing our results with the numerical work as represented in Figs. 1, 2 and 3 inKevorkian (1974) we can make the following remarks.

The theory applies to the phenomena in Fig. 1 and Fig. 3, but not to Fig. 2.In Fig. 2 we see that p on a time-scale l/e, with oscillations of period 1/.Since our estimates are valid on 1/, the theory does not apply to this picture.The period of the oscillations of p around suggests that the solution is oscillating

around the elliptic orbit (which may be slowly moving on 1/e) and does not stay near thehyperbolic one, as Kevorkian claims (cf. however Lewin and Kevorkian (1978)).

Anyway, it is not dicult to see that the solution cannot stay near the hyperbolic on1/e in the model problem, using the equations for the slow variables in the innerexpansion.

In conclusion we state that the problem of finding approximations valid on atime-scale 1/e and describing Fig. 2 in Kevorkian (1974) is still open.

REFERENCES

W. ECKHAUS (1976), Private communication.J. KEVORKIAN (1974), On a model ]:or reentry roll res.onance, SIAM J. Appl. Math., 26, pp. 638-669.W. T. KYNER (1969), Passage through resonance; Periodic Orbits, Stability and Resonances, G. E. 0.

Giacaglia, ed., Reidel Publishing Company, Dordrecht, Holland, 1970.L. LEWIN AND J. KEVORKIAN (1978), On the problem ofsustained resonance, SIAM J. Appl. Math., 35, pp.

738-754.J. A. SANDERS (1977), AJ higher order resonances really interesting? Celestial Mech., 16 (1978), pp.

421--440.

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