on nonlinear delay differential equations€¦ · on nonlinear delay differential equations a....

transactions of theamerican mathematical societyVolume 344, Number 1, July 1994

ON NONLINEAR DELAY DIFFERENTIAL EQUATIONS

A. ISERLES

Abstract. We examine qualitative behaviour of delay differential equations of

the form

y'(t) = h(y(t),y(qt)), y(0)=y0,

where A is a given function and q > 0. We commence by investigating ex-

istence of periodic solutions in the case of h(u, v) = f{u) + p(v), where /

is an analytic function and p a polynomial. In that case we prove that, un-

less q is a rational number of a fairly simple form, no nonconstant periodic

solutions exist. In particular, in the special case when / is a linear function,

we rule out periodicity except for the case when q = 1/degp . If, in addition,

p is a quadratic or a quartic, we show that this result is the best possible and

that a nonconstant periodic solution exists for q = ^ or | , respectively. Pro-

vided that g is a bivariate polynomial, we investigate solutions of the delay

differential equation by expanding them into Dirichlet series. Coefficients and

arguments of these series are derived by means of a recurrence relation and their

index set is isomorphic to a subset of planar graphs. Convergence results for

these Dirichlet series rely heavily upon the derivation of generating functions

of such graphs, counted with respect to certain recursively-defined functionals.

We prove existence and convergence of Dirichlet series under different general

conditions, thereby deducing much useful information about global behaviour

of the solution.

1. Introduction

The conceptual point of departure for this paper is a proportional-delay vari-ant of the familiar Riccati equation, namely

(1.1) y'(t) = by(t) + ay(qt)(l-y(qt)), y(0) = y0,

where a, b, y0 e C and q > 0, q ^ 1. Note that q e (0, 1) yields a retardedequation, whereas q > 1 results in an advanced equation. Our main interest

is in the retarded case, but, as some of the results can be readily extended to

advanced equations and others are valid only in the advanced case, we presentthem without restricting q > 0.

Existence and uniqueness of the solution of ( 1.1 ) follows easily in the case

q e (0, 1 ) by replicating the proof of the familiar Picard-Lindelöf theorem.

On the other hand, the existence of the solution in the advanced case is wide

open and a matter of conjecture. It is known, however, that even in the linear

case y'(t) - by(t) + ay(qt) there might be an infinite number of solutions inthe advanced case q > 1 [5].

Received by the editors January 27, 1993.

1991 Mathematics Subject Classification. Primary 34K05; Secondary 34K20.

© 1994 American Mathematical Society

0002-9947/94 $1.00+ $.25 per page

441

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

442 A. ISERLES

Our interest lies in qualitative features of the solution of (1.1):

• Uniform boundedness: max,>o |y(r)| < oo ;

• Asymptotic stability: lim,_>00y(i) = 0;

• Periodicity: There exists T > 0 such that y(t + T) = y(t) for all t > 0 ;• Almost periodicity: For every e > 0 there exists Te, lim infe_o+ F£ > 0,

such that \y(t + Te) - y(t)\ <e for all t > 0.Frequently we are able to present our results in a considerably more general

framework. Thus, in §2 we explore the existence of periodic solutions to the

equation

(1.2) y'(t) = f(y(t))+P(y(qt)), y(0)=y0,

where / is an analytic function and p is a polynomial. As long as q < 1,

existence and uniqueness of the solution can be again justified by extending the

Picard-Lindelöf theorem. We prove in §2 that, unless q is a rational number of

the form * , where I e {1, 2, ... , degp}, the equation (1.2) has no periodic

solutions. In the special case (1.1) we have p(x) - ax(l - x), hence q must

be either an integer or half an integer. Moreover, if / is, like in (1.1), a linear

function then we prove that necessarily k = 1, / = degp. Frequently wecan prove this to be the best possible result. For example, letting a, b e R,

~\ < a < I ' we Prove that there exists a periodic solution to ( 1.1 ) when q = \.

A particularly useful means to analyse the behaviour of delay differential

equations with the lag function 6(t) = qt is an expansion into Dirichlet series[3, 4]. Section 3 is devoted to brief introduction into global analysis of Dirichlet

series. This analysis is put into effect in §4, to explore solutions of

(L3) y'(t) = h(y(t),y(qt)), y(0) = y0,

where « is a bivariate polynomial of total degree m and, without loss of gen-

erality, «(0, 0) = 0. The Dirichlet expansion of (1.3) reads

(1.4) y(0 = 5>r*Wri,T6T

where b e C\{0} and T is a countable set of indices. The coefficients dj

and arguments XT can be defined by nonlinear recursion. Needless to say,

(1.4) is a formal expression, and our focus is on its convergence for t > 0.

The main step in our analysis is in establishing that a very good choice of the

index set T is a subset of connected planar trees. With this interpretation,

the nonlinear recurrence is reduced to operation on graphs. Moreover, the

question of convergence is subsumed in the problem of analyticity of generating

functions, that count elements of T with certain features. In other words, we

use graph theory to explicate the behaviour of a functional differential equation

and algebraic combinatorics to analyse the aforementioned graphs...

The theory of §4 requires that the set {¿7-: T e T\{F0}}, where T0 is theunique tree with one vertex, be bounded away from 1. In §5 we explore a variety

of different equations (1.3) in regard to that feature.

The theory of this paper can be extended to more elaborate equations and in§6 we focus on the rational equation

'M-TÄ- '(°>^where a, b, yo e C.


ON NONLINEAR DELAY DIFFERENTIAL EQUATIONS 443

The approach of Dirichlet series has many advantages—knowledge of a

Dirichlet expansion is sufficient to explicate global behaviour of the solution,

provide realistic bounds on its growth and even help in its numerical modelling

—but also a critical shortcoming. To initialize the recursive procedure

{dj}reTm, we need to assign a value to a single coefficient, dj0, say. This

leads to an expansion (1.4) with the initial condition y(0) = Y,TefmdT • ^n

other words, the procedure produces a solution of (1.3) with some initial con-

dition but the theory falls short of proving that the 'right' value of dj0 can beselected for every y0 in a specific subset of the complex plane. The most we

can claim is the existence of y* > 0 such that |yn| < y* results in a convergent

Dirichlet expansion. However, since y* is defined by restricting the magnitudeof dr0 , it is provided only in an implicit fashion.

This paper lies no claims to resolve completely the question of the global

behaviour of even the simplest delay Riccati equation (1.1). For example, we

can prove that the zero solution is stable, say, for specific regimes of a, b e C,

but fall short of determining the basin of attraction with respect to yn e C.Interestingly enough, numerical modelling by standard discretizations (in the

retarded case q e (0, 1)) affirms that stability is lost when |yn| becomes too

large. This implies that our analysis is on the right track and delivers the right

kind of qualitative information, but much remains to be done to resolve the

problem completely.In line with §6, our theory can be extended to a wider range of similar prob-

lems. It requires a degree of care, since generality tends to interfere with con-

vergence estimates. Nevertheless, it may lead to powerful new results and will

undoubtedly be a subject for future research.

2. Periodic solutions

The theme of this section is the existence—or otherwise—of periodic solu-tions to the equation

(2.1) y'(t)=f(y(t))+p(y(qt)), y(0) = y0eC,

where / is a given analytic function, p is an mth degree complex polynomial

and q > 0. Let us assume that (2.1) possesses a nonconstant periodic solution

and that T > 0 is the minimal period. Thus, y(t + T) = y(t), t > 0, and this

implies y'(t + T)-f(y(t + T)) = y'(t) - f(y(t)). We now invoke (2.1) to argue

that p(y(q(t + T))) = p(y(qt)), t > 0. Changing the variable qth^t,we have

(2.2) p(y(t + qT))=p(y(t)), t>0.

Lemma 1. There exist positive integers r e {1, 2, ... , m} and w such that

Proof. Let xk(t) := y(t + kqT), k e Z+, and p(z) = Z7=oPiz' > Pm ¿ 0.Our intermediate aim is to prove that there exists je{l,2,...,m} such that

Xs — Xo .

We rewrite (2.2) as

m-\

(2.3) 5>+i(*í+I-*¿+1)3 0.1=0


444 A. ISERLES

Since / is analytic, so is y and we may deduce from (2.3) that either xx = xo,

t > 0, orm-l

1=0 i'o+ii >0i'o+i'i=/

We continue by induction. Thus, suppose that, for some se {1,2,...,m-

1} , it is true that xk £ x0 , k = 1,2, ... , s , and that

m—s

(2.4) 5>+* £ x$x?---x¡<=0.1=0 i0,i¡,...,is>0

ío+/i+•••+/,=/

Note that we have already seen that, for s = I, xx £ xo implies (2.4).Since xk(t) = y(t + kqT), k e Z+ , we are allowed to step the indices of

xo, ... , xs up in (2.4) by one unit, hence

J>+J £ **...*«**,=(>./=0 io,ii,...,/i>0

¿0+Í1+—+lj=/

Subtracting (2.4) from the latter identity yields

m—s—l

(2.5) £>'+*+! £ (x;°+V-40+V!'---*i5 = o-/=0 io,ii,...,is>0

¡0 + !'l+--' + ij=/

Clearly, xs+x = xo is a solution of (2.5). Suppose, however, that xs+x ^ xq.

Since y is analytic, so is xk, k e Z+ . Therefore,

m-j-l ¡o+l/^ _ vk+ht\

y Pi+s+l y w'm °, rxHt)---xHt)=oj\ , : ¿"i >0 %lW -X0(t) X S"1=0 Jo >'l.'s^U

¿0+l'l+"-t-íi=/

holds for t e £7~ ç [0, oo), where 3~ is a continuum. Thus,

m-s-1 v/'o+l _ Y-'o+l

X>«+i £ T ° *i'-*,^o.^^ ^^ „ X°+X — XO1=0 i0,U,...,is>0 s+> u

¡o+h+—+'s-l

Trivial manipulation affirms that this is identical to (2.4) with í replaced by

5 + 1. This advances the induction one step.

The identity (2.4) cannot be true for s = m, since pm^0 and y ^ 0. Thus,

we deduce that xr = Xo for some re{l,2,...,m}. In other words, there

exists 1 < r < m such that

y(t + rqT)=y(t), r>0,

and y is periodic of both period T and period rqT. However, it is trivial toverify that if a function y is periodic of distinct periods Fi and T2, say, thenit possesses period \T2-TX\. Consequently, and unless rq is an integer, y isperiodic of period (qr - [qr])T e (0, T), and this contradicts our assumptionthat T is the minimal period. Hence, q is a rational number of the form

stipulated in the statement of the lemma. D



Lemma 2. Let w and r be relatively prime. In the special case f(z) = fo+fz,i.e.

(2.6) y'(t) = fo + fy(t)+p(y(qt)), y(0)=y0,

necessarily w = 1 in Lemma 1. In other words, periodicity implies that q = \

for some r e {1,2, ... , m} .

Proof. We assume again that T > 0 is the minimal period. Since y is analytic,

so is its derivative. Hence the Fourier expansions of y and of y' are bothpointwise convergent [6] and

v^ j 2nint _ _y(t) = 2^d„exp—Y-, t>0.

nez

Substitution into (2.1) yields

2ltinty/(t) := y'(t) - f(y(t)) = £ ¥n exp

Tn€Z

, v^ (2nint r\ j 2nint(2.7) =-fo + ̂ \-^r-f)dnïvç>^r

nez

2ni(kx H-\-k¡)wt= A) + £p/ E dktdkl---dklexv-

rT1=1 k,,k2,...,kieZ

Comparing coefficients in (2.7), we deduce that for every « e Z either y/n = 0

or there exist I e {1, 2, ... , m} and kx, k2,... , k¡ e Z such that

/

rn = w £ rc;.

7=1

u; and r being relatively prime, the latter implies that w divides « . Hence

v^ 2ninwt¥(t) = 2_, Vnw exp —-—

«ez

and it is periodic of period ^ . Our contention is that so is y. Because of(2.7), it is clear that, unless « is divisible by w ,

-fx)dn = 0.(2nin -\\rr-fx)

Thus, for this range of «, necessarily dn = 0, except that there might exist a

unique « , say, such that f =2niñ/T. We deduce that

v^ i 2ninwt , 2nihty(t) = 2^ dwn exp —-— + dn exp —=-.

nez

Thus,

T\ , , „ 2niht(2.8) y(? + -)-y(0 = /3exp

where

w 7 r

2ir tn/? :=exp—— - 1.

wT


446 A. ISERLES

Iterating (2.8) we obtain

^ , s n 2niñty(t +T)= y(t) + wß exp ~j^-

Since F is a period, it follows that ß = 0. Consequently, y is also periodicof period ^ . Unless w = 1, this contradicts the assumption that T > 0 is the

minimal period. D

Lemmas 1 and 2 can be combined to spell out the main result of this section.

Theorem 3. The equation

y'(t) = f(y(t)) + P(y(qt)), y(0) =y0eC,

where q > 0, f is an analytic function and p is an mth degree polynomial,

possesses a periodic nonconstant solution only if q is a rational number of the

form q = j, where re {1,2.m}. Moreover, if f is a linear function

then w = 1. D

The result of Theorem 3 is frequently the best possible. Thus, suppose that

p is quadratic in (2.6). In that case (and excluding the no-delay case q = 1)

Lemma 2 leaves out just one choice of q, namely q = \- The equation reads

y'(t) = fo + fy(t)+Piy({-)+P2y2({), p2¿0.

Letting z(t) := y(t) + j(l+px), t > 0, we obtain

(2.9) z'(t) = a0 + axz(t) + ßz({)(l - z(¿)),

where

ao = fo-$l+P$(fi+P\) + \(l+Pi)2P2, oii= fi, ß = -p2.

We seek a solution of the form

z(t) = j + a cosXt + b sin Xt.

Substitution in (2.9) and comparison of coefficients results, after some manip-

ulation, in

ao+x2ax + \ß = \ß(a2 + b2),

bX = axa-\ß(a2 -b2), -aX = axb- ßab.

It is straightforward to verify that the solution is

Needless to say, we require for periodicity a real value of X, and this is the case

if and only if

2tHi+4t)(HH-In the special case ao = 0, periodic solution of the stipulated form exists for

//, ,,-vn 1 at 1

(2.10) -4<j<r



For example,

z'(t) = \z(t) + z({)(l - z({)) =» z(t) = \ + icos^ + f sin^ ;

z'(t) = -\z(t) + z(L)(l - *(<)) => z(t) = I - i cos ^ + ^ sin ^.

Of course, we make no claim that these are the only periodic solutions of

(2.9), that (2.10) represents the broadest possible range of parameters giving

rise to periodic z or that the only initial value that is consistent with periodicityis z(0) = 5 + 2ax/ß . Numerical results, however, support the conjecture thatall these statements might be true.

Similar construction can be sometimes extended to higher degree equations.

For example, long and tedious calculation verifies that the equation

y'(t)=po+pxy({) + P2y2(L4)+Piy\L4)+P4y\L4), p^o,

where

lp\ l P2P3 * Pi 1P2P3 1 p\Po 8/J4 32 p2 ~ 5l2p¡' Px 2 p4 Sp2'

and provided that 3p2 > 8P2P4, possesses the periodic solution y(t) = a +bsinXt with

1 f -5 ■* 1/2 \ ( \ ~\ ̂ l2

a=-û' b=±p-4Up2i-p2P4\ ' x=±%p¡wl~pm\ •

Computer experiments indicate that a periodic solution of (2.9), when it ex-ists, is of a different nature to the familiar periodic solutions of ordinary differ-

ential equations. Firstly, it is unstable in the initial value and any z(0) = zo ^

\-\-2ax/ß , ao+axzo+ßz0(l-zo) ^0, appears to lead to an oscillatory solution

with asymptotically growing amplitude, which cannot be uniformly bounded.

Secondly, it is typical of periodic solutions of ordinary differential equations,parametrized by their parameters, to occur at a Hopf bifurcation, when the

behaviour changes from an unstable to stable oscillation (or, in a phase space,

from an outward-oriented spiral to an inward-oriented spiral). As indicated bynumerical modelling, this is not the case for (2.9), when the parameter ax,

say, is varied within a regime consistent with (2.10). If this is done for fixedz(0) then periodicity occurs at a 'transition' from an unstable spiral to anotherunstable spiral. On the other hand, for every such qi there exists, by our anal-

ysis, an initial value that gives rise to periodicity—in this strict sense, periodicbehaviour is robust.

Periodicity and tendence to a limit (the latter being the main theme in the

sequel) are not the only phenomena that can be encountered in the context of

delay equations. It is known [4] that the solution of the linear equation

y'(t) = iay(t) + by(qt), y(0)#0,

where a e R\{0}, b e C, \b\ < \a\ and q e (0, 1), is an almost periodicfunction [6]. Another phenomenon occurs when the equation (2.9) is solved at

the left end of the range (2.10), namely when ax = -\ß. Computer experi-

mentation indicates that, at least for a0 = 0, ß > 0 and z(0) e (0, |), the

solution exhibits chaotic character. Figure 1 displays the solution of

(2.11) z'(t) = -\z(t) + z(lí)(l-z({)), z(0) = ¿.


448 A. ISERLES

Figure 1. The solution of z'(t) = -\z(t) + z(t/2)

•(1 - z(t/2)), z(0) = \, for 0 < t < 105lines denote z = 0 and z = 1

Horizontal

The exact nature of the solution of (2.11 )—and of solutions of similar equations

—is at present a subject of ongoing research.

3. Dirichlet expansions

A Dirichlet expansion is (in general, countably infinite) linear combination

of exponentials [3]. Let

(3.1) f(t):=Y^dTebXT>, t>0,Tef

where b e C\{0} , XT > 0 and dT e C for all F in a countable set T. ' We

assume that

(3.2) d* := £ \dT\ < oo.reT

The purpose of the present section is to provide a brief review of asymptotic

features of Dirichlet series that are of interest in the remainder of this paper.

We do not dwell on the more usual applications of Dirichlet series, e.g. in

number theory [3].Let X* := liminfretAr > 0. It follows at once from (3.1) and (3.2) that

(3.3) \f(t)\<d*eKebr', t>0.

We deduce that (3.1) is well defined and bounded for all t > 0. Moreover,

X* > 0 and Re¿> < 0 imply that lim,-^/^) = 0 and / is asymptoticallystable. However, X* > 0 is not necessary for asymptotic stability. The following

result generalizes a method of proof implicit in [3] to cater for the case X* = 0.

1 Of course, we may assume without loss of generality that the index set T coincides with

Z+ . This, however, is unhelpful to the analysis in the sequel. We demonstrate in §§3 and 4 that a

more 'exotic' form of T is advantageous, since it is more in tune with the recursive technique of

implicitly defining the arguments Xt > T e T .



Proposition 4. Let Re b < 0. Then lim,-^ f(t) = 0 and f is asymptoticallystable.

Proof. We wish to show that for every small e > 0 there exists tE > 0 such

that

(3.4) |/(0I <e, t>tE

Since the sequence {dr}teT is h , there exists a partition of T = Ti U T2 such

that Ti n T2 = 0, T2 is finite and

E 1^1 > d* - \.ret2

The latter inequality, X* > 0 and Re b < 0 imply that

(3.5) E dTeblT'

reT,

< E 1^1 <rex.

r>0.

Since T2 is finite, necessarily A(£> := min7-eT2 Xj > 0. Consequently,

(3.6) E dTebXTt

Teii

< ElrfrkReM<,,/<rfVReW("', t>0.TeTi

Thus, choosing

h:=loge I (2d*)

£- A<£)Reè '(3.5) and (3.6) imply the inequality (3.4). Hence asymptotic stability. D

The behaviour of Dirichlet series is known also on their stability boundary.

Proposition 5. IfReb = 0 then y is almost periodic for t > 0.

Proof. According to a theorem of H. Bohr, almost periodic functions are

Loo[0, 00) limits of trigonometric polynomials [6]. In other words, y is almost

periodic if and only if there exist functions {yn}^Lx such that

yW(t) = Y, dlTn]eb*T>, «eZ+,

Tei„

where T„ is of finite cardinality, and lim^ocy1"1^) = y(t) uniformly for all

t >0.Since YIt \dr\ = d* < 00, we can choose finite sets {T,,}^ such that

1y\dT\>d*--, « = i,2,....

TeTn

We lety[»l(r):= Y, dTebkTt, « = 1,2,....

reTfi

Let S[nl := T\Tt"l. Then Reb = 0 implies that

1\y(t)-yn(t)\< E \dr\<-

resi"i


450 A. ISERLES

uniformly for t > 0. This implies that y is almost periodic. D

It has been demonstrated in [4] that the linear vector equation

y'(t) = Ay(t) + By(qt), y(0)=y0,

possesses an almost periodic Dirichlet series, provided that A is invertible,maxRea(^) = 0, p(A~xB) < 1, and y0 is not orthogonal to the space spanned

by the eigenvectors of pure imaginary eigenvalues of A. Much depends on

whether q e (0, 1) is rational, since in the latter case there exist rotational

almost symmetries in the 'phase space' (Rey, Im y). Many geometric featuresof such solutions are reported in [4], others are subject of an ongoing research.

4. General equations y'(t) = h(y(t), y(qt)), h a polynomial

It is easy to formulate necessary conditions for asymptotic stability for delay

differential equations, by following techniques intimately related to the standard

variational equation approach. Thus, consider the equation

(4.1) y'(t) = h(y(t),y(qt)), y(0) = y„ e C.

As long as qe(0, 1) and « is locally Lipschitz in each of its variables, existence

and uniqueness of the solution can be deduced by standard methods. Moreover,

the set of all fixed points of (4.1) coincides with /:={}»£ C: h(y, y) = 0} .Suppose that « is smoothly differentiable in both variables and that y e ^ is

attractive. In other words, there exists ô > 0 such that |y0 - y\ < Ô implies

that lim,_00y(/) = y. Let e(t) := y(t) -y and suppose that / is sufficiently

large, so that \e(t)\ < n for all t > qto . Substitution into (4.1) yields

(4.2) e'(t) = hx(y,y)e(t) + h2(y,y)e(qt) + cf(n2), t > t0.

Here «i and «2 are the derivatives of « with respect to the first and the second

variable, respectively.Truncation of the cf(r\2) term in (4.2) results in the pantograph equation

(4.3) v'(t) = bv(t) + av(qt),

where b = h\(y, y) and a = h2(y, y). Asymptotic behaviour of (4.3) has been

determined in [4]:

1. If Reb < 0 and \a\ < \b\ then lim^oo^í) = 0;2. If either Reb > 0 or \a\ > \b\ then, as long as v(0) ^ 0, limsup^^ \v(t)\

= 00 ;

3. If Reb = 0 and |a| < \b\ then v is almost periodic;4. If Reb < 0 and |a| = |Z?| then, under a logarithmic change of variables,

v tends to an almost periodic curve. (The latter case has not been derived in

[4]. It will feature in a future paper.)

Hence, if Re«i(y, y) > 0 or \h2(y, y)\ > \hx(y, y)| then it is impossible for e

to tend to zero in (4.2).

Theorem 6. A necessary condition for y e ¡%? being an asymptotically stable

fixed point of (4.1) is that Rehx(y, y) < 0 and \h2(y, y)\ < \hx(y, y)\. D

It is important to emphasize that there is nothing in our analysis to support

the view that the condition of Theorem 6 is sufficient for stability. The re-

mainder of this paper is devoted in the main to the exploration of sufficient

conditions for asymptotic stability.



We next assume that « is a bivariate polynomial of total degree m,

m k

h(u,v) = YYJak,iulvk-1.k=0 1=0

In the spirit of our discussion so far, we stipulate that «(0,0) = 0 and thatthe fixed point of interest lies at the origin, otherwise we subject y to a linear

transformation. Finally, we require b := axx ^ 0. In other words, the focus

of our attention is on the equation

m k

(4.4) y'(t) = by(t) + aXyoy(qt) + YJ2ak,iy'(t)yk~l(Qt), y(0)=y0,b¿0.k=2 1=0

Subsitution of the Dirichlet solution (3.1), that is y(t) = Yrei djebXTt, into

(4.4) yields

(4.5)b E dT(XT - l)ebXrt = «i ,o E dTebqXTt

ret rex

+ EEflM E dTxdTl-- d^explbÍY^ + Q E ^ )'f "k=2l=0 Ti.Tkel I \/=l i=/+l / J

As usual, an empty sum is assumed to be naught.Let A := {Xj: Tel}. We need to choose A so that each nonvanishing

exponential on the right of (4.5) can be matched with such an exponential onthe left. This gives rise to the following composition rules, which provide a

recursive definition of A.

1. 1 eA;2.1f ke{l,2,...,m}, le{0,l,...,k}, (k,l)í(l,l), akyl¿0 and

Xti , XTl, ... , XTk e A, then

/ k

(4.6) E^ + «EAr(GA.¡=i i=/+i

Proposition 7. Each X e A is a polynomial in q with integer nonnegative coef-

ficients.

Proof. Follows from the composition rules by straightforward induction. D

It is highly useful to identify members of the index set T with multiply-rooted

planar trees. We remind the reader that a tree is a connected graph G = (V, E)

where each two distinct vertices vx, v2 e V are joined by precisely one path

of edges from E and it is planar if distinct trees with the same topology (e.g.

mirror images) are counted separately [1,2]. For the purpose of this paper we

define a multiply-rooted tree as the ordered pair (G, R), where the multiroot

R C V has the property that each vertex v e V\R is linked to exactly one

element of R by exactly one path, that does not pass through any other vertex

from R.2 We call such a path the principal path of v and extend this definition

to R by stipulating that the principal path of v e R is the zero-length path

2 It is easy to prove that the restriction of G to R is itself a connected tree, consisting of a

single path. This plays no role in our analysis and is left to the reader.


452 A. ISERLES

consisting of v itself. Principal paths define partial ordering on (G, R) (atrivial extension of a monotone ordering of a rooted tree).

The order of a graph is the number of its vertices. We denote the order of

a multiply-rooted planar tree T by ord T and note that there exists a unique

such tree of order 1, T0, say.

We say that v e V is a top vertex if it does not lie on any principal path of a

vertex from V\{v} . In other words, a top vertex has no children in the partial

ordering of (G, R). Moreover, lenu denotes the length of the principal pathof a top vertex v e V.

Let W be the set of all multiply-rooted planar trees. We define two operations

on elements of W :

Addition. Let T¡ = (Gi,R¡) e W, Gi = (Vi,Ri), i = 1, 2,..., s, s > 2. Westipulate that F, n F, = 0 for i / j. Then

T=TX-T2-TS = (G,R), G = (V,E),

is defined as follows: G = (jsi=lG¡, R = [fi=l R¡, whereas E = (jsi=l E¡ U F»and the set F* is made out of 5 - 1 edges linking the rightmost element of R¡

to the leftmost element of Ri+X, i = 1, 2, ... , s. (Note that 'leftmost' and

'rightmost' are well defined for planar trees.) For example,

tv,Y.tvfWe observe that addition in W is not commutative and that ord T = ord Fi +

-1- ord Ts.

Rooting. Given V e W, 7" = (O, R'), G' = (V, E'), we define

V

T= i=(G,R), G = (V,E),

by letting V = V U v , say, where v £ V , R = {v} and E = EaL)Eß , whereEa C E' is made up by removing any edges that are wholly in R', whereas

Eß ç (V, v) contains edges linking each element of R' with v . For example,

r = I_y =*► t = v^

We note that ord T = ord T + 1.It is trivial to verify that W is closed under addition and rooting.

We map W into the set of polynomials in q with nonnegative integer coef-

ficients by letting

(4.7) XT:= E 4len"-

v is a top vertex of T



Note that the map T h-> Xt is an injection: in general, many members of W

are mapped to the same polynomial. We will return to this point in the sequel.

The composition rules may now be reformulated in tree terminology. Thus,

T is constructed out of elements from W such that

1. T0 e T •

2. If A: €{1,2.m), I e {0, 1,..., k}, (k, I) # (1, 1), akyl^0 andTx, T2, ... , Tk e T then

Ti+i T[+2 Tk

(4.8)

r = 2i—t2-

It follows at once from (4.7) that

/ k

(4.9) XT = Y^ + qYÀTr1=1 i=l+\

Hence, by (4.6), this construction corresponds to the second composition rule.

We stress three important points. Firstly, not every element of W is bound to

appear in T. Secondly, many Xt 's (even a countable infinity) may be identical,

although possibly with different coefficients dr. Indeed, the same element of

W might well feature several times in T. Thirdly, the exact contents of T

depend on m and on the set

{(k,l):akyl¿0,k=l,2,...,m,l = 0, 1, ... , k, (k, I) # (1, 1)}.

These points, however cause no difficulty.

Let coky¡:=akyi/b, k = 1,2,..., m, 1 = 0, 1,... ,k, (k, 1)^(1, I).

Proposition 8. There exist a function a : T -> 1, dependent just on q, and

functions

ß,yiyj:7^Z+, i=l,2,...,m,j = 0, 1, ...,i,(i, j) ± (1, 1),

independent of q and the coefficients aiy¡, such that

i m i

(4.10) dT=^-d^i[ n <r> ^t-y ' ¡=i ;=o

(¡.MM)

Moreover, ß(T) is the number of top vertices of T eT.

Proof. Substitution of (4.10) into (4.5) affirms that the proposition is true, as

well as providing explicit recurrence relations for the evaluation of the under-

lying functions. Again, we follow the composition rules.

1. a(T0) = ß(T0) = 1, ykyl(T0) = 0 by definition.

2. Suppose that T eT, ord T > 2, is constructed from TX,T2, ... ,TkeTas in (4.8). Then it follows from (4.5) that

k

(XT- \)dT = (ok iY[dTsj=i


454 A. ISERLES

and substituting (4.10) yields

1 -Am na(T)~T° H H ""WK ) i=l j=0

w.y(r)

<0k,i jZLßWn TT w£>-;<r').m ;

r~,dTo "' II nlls=la(Ts) i=l ;=0

(¿,i¥(i>i)

Therefore

(4.11) a(r) = (Ar-i)n«(^);5=1

it

(4.12) ß(T) = Yß(Ts)ls=l

(4.13) yf);(r)=|

Y,;=xYlj(Ts):(i,j)ï(k,l),

x7k,i(Ts) + l:(i,j) = (k,l),

1 = 1,..., m,j = 0, ... , i, (i,j)¿(l, 1).

Hence (4.10) is true.

To complete the proof we need to demonstrate that ß(T) is the number of

top vertices of T. This follows at once by induction from the composition

rules and (4.12). D

Corollary. ß(T) is the sum of coefficients in Xt ■ In other words, letting Xt =

Xr(q), it is true that /?r = Xt(1) .

Proof. A straightforward inductive argument applied to (4.9). D

We letm i

(=1 7=0

(i,MU)

Proposition 9. The function a(T) can be represented in the form

r(T)

a(T)=Y[(XSr{T)-l),r=l

where SX(T), S2(T),..., Sy{T)(T) e T\{T0} .

Proof. It follows from (4.11) that a(T) can be factorized as a product of the

formf{T)

a(T)=H(XSr(T)-l),r=l

where p : T —> Z+ obeys

k

(4.14) fi(T0) = l, p(T) = Y,MTs) + l,5=1



whereby, like before, T originates from TX,T2, ... ,Tk by the second compo-

sition rule. However, according to (4.12), (4.13) and the definition of y(T), the

latter function obeys (4.14). Since y and p obey the same recurrence relation

and share an identical initial value, they necessarily coincide. D

Table 1 (see next page) displays all distinct members of T of order < 4,

subject to the assumption that no coefficients akl vanish. Several important

features of our construction are illustrated by this table. Firstly, elements of

T are multiply-rooted planar trees together with their functions a and y ! It

is perfectly possible (and, indeed, likely) for the composition rules to produce

identical members of W with different values of a and y . Thus, letting

ïi:=I, :r2:=_, r3:=L.,

we obtain (subject to all the 'right' aky¡ 's being nonzero)

L~ = 7i—T0—T0 (7Cn = 2, a(T) = q2-l)

= Tx—T2 (7(71 = 3, a(r) = ç2-l)

= Ta—T0 (700 = 3, a(T) = q(q2 - 1)).

As long as the values of y and a are clear (or do not matter), we will continue to

refer to members of T as 'trees' and employ the same notation as for elements

of W. However, we must bear in mind that composition attaches two 'labels'

to each tree.

Secondly, the same tree, with identical values of y and a, can be frequently

composed in a number of ways. Thus

T =, 1 , = To—T'a = Ti—To where T4 := «_[ .

We choose T4 such that y(T4) = 2. In either case we obtain y(T) = 3,

a(T) = q(q2 - 1). In principle, we could have dealt with this by allocating

multiplicities, but this is hardly necessary, since members of T are counted (in

Proposition 10) indirectly, by employing composition rules.

Thirdly, an identical value of Ar may occur repeatedly in T. For example

(which requires a trivial extension of Table 1), letting F5 := • • • and

bearing in mind that y(F5) is either 1 or 2, we obtain inter alia the following

F e T, all with XT = 3q :

To T0 T0

a. a3,o^0: H' = ^^ ordT = 4,7T = l;

T , T5b. aly0, a3l3 ¿ 0: ^Y =[ , ordT = 4, 7T = 2;


456 A. ISERLES

c ai,o,a2,2 ¿ 0: ^V = 1 , ordT = 4, 7T = 3;

• • •d. oi,o, a3,i ¿ 0: I_V = Ti-V. ovdT = 5, 7r = 2;

e. ali0,a3i2 # 0: 111 = 2\-1\—1 > ordr = 6> 7r = 3;

f- ali0, a3,3 j¿ 0: 111 = 2\—Tx—Tx > ordT = 6, -yT = 4.

More trees such that XT = 3q can be composed, and this is left to the reader.

Table 1. All distinct FêW, ordF < 4, «j > 4,with their Xt , a(T), ß(T) and y(T) functions.

ordT= 1 :

Y V. ¿S iS U-.X=2q2 A=l+2? A=l+2c A=l+2ij A=2+5

a=(2q-l)(2q1-\) a=2q(2q-\) a=2q(2q-l) a=2q a=q2-l

0=2 0=3 0=3 0=3 0=3

7=2 7=2 7=2 7=1 7=3

T , . T T T Tè—•—• •—•—• •—«—• •—•—• •—4—•\=2+q X=2+q X=2+q X=2+q X=2+q

a=q(q2-l) a=q2-l a=q(q2-l) a=q2-l a=q(q+l)

0=3 0=3 0=3 0=3 0=3

7=3 7=2 7=3 7=2 7=2

_J _J _I W ^YX=2+q X=2+q X=2+q X=3q X=3q

a=q(q2-l) a=q2-\ a=q(q+l) a=2(3?-l) a=2(3?-l)

0=3 0=3 0=3 0=3 0=3

7=3 7=2 7=1 7=3 7=2

^

X=3q X=4 X=4 X=4

a=3g-l a=3 a=6 a=3

0=3 0=4 0=4 0=4

7=1 7=3 7=2 7=1

•

A=l

a=l

0=1

7=0



Table 1 (Continued)

ordT = 2A=,

a=q-

0=1

7=1

A=2

o=l

0=2

7=1

ordT = 3:

ordT = 4;

L J A YX=q2

a=(q-

0=1

7=2

X=l+q

l)(q2-l) a=q(q-l)

0=1

7=2

A=l+S

0=9(9-1)

0=2

7=2

A=l+9

a=9

í»=2

7=1

A=3

a=2

0=3

7=2

A=3

o=2

0=3

7=1

L J J

X=q+q2 *=q+q2 X=q+q2

a=»(ï-l) a=(,-l)(î2+î-l) 0=9(9-1)

<9J+9-l) 0=2 (q2+q-l)

0=2 7=2 0=2

7=3 7=3

X=q+q2

X=2q

a=2q-l

0=2

7=1

uX=q3 X=l+q2 X=l+q2 X=l+q2

c,=(,-l)(,J-l) a=,3(,-l)(,2-l) o=,J(,-l)(,*-l) c=q2(q-l)

x(«3-l) 0=2 0=2 0=2

0=1 7=3 7=3 7=2

C V v> v' v>

A=2g

a=(,-l)(2,-l)

0=2

7=2

A=9+92

o=(ï-l)(ï2+«-l) <*=í(9a+í-l)

0=2 0=2

7=2 7=2

It should be obvious by now that an important distinction needs to be drawn

between ß(T) on the one hand and y(T) and a(T) on the other. Whereas

ß(T) is a structural property of T e W and can be deduced directly from

the tree (and, indeed, from Xt) , y(T) and a(T) depend on the derivation of

F e T by a specific sequence of composition rules—indeed, an intuitive meaning

of y(T) is as the number of composition that have been required to produce

T eT.Let us suppose that there exists n = n(q) > 0 such that

(4.15) |Ar-l|>«,

Therefore, by Proposition 9,

1

\<*(T)\< r7(T)

TeT\{T0}.

TeT.

We set

a:=max{\coiyj\: i= 1,2, ... ,m,j = 0,1, ... , i, (i, j) ¿ (1, 1)}


458 A. ISERLES

and use (4.10) to deduce the inequality

(4.16) \dT\<\dT0\ß{T](a/ny^.

Inequality (4.16) is important to our goal of bounding

d* := E M-rex

To that end we partition T into the sets

MnyS:={TeT:ß(T) = n,y(T) = s},

where « = 1,2,... and 5 = 0,1,.... Let M„yS be the number of elements

in BnJ. Note that many M„yS's equal zero, but this creates no problems

whatsoever.

According to (4.16), we have

/-rr\7iT)

TcT \ I /

(4.17)

where

oo oo

= EEl^ol" ^ =\dTo\M(\dTo\,a/n),„_i <._n \ I /n=\ 5=0

M(x,y):=Y,Y,Mn>sxn~lys-n=l 5=0

In order to take advantage of (4.17) in the context of the analysis of §3,

we require (x,y), where x = \dr0\, y '■= o/q, to lie inside the domain of

convergence of the Taylor series of M. Thus, our next step is to take a closer

look at the bivariate function M. This will be done in several stages. In the next

three propositions we assume that all the underlying sums absolutely converge.

Proposition 10. For every «=1,2,...

(4.18) M„yo = SnyX,m k

(4.19) M„;i<E^ E E UM'J' s=l,2,....k=\ ii,...,/*>! j\.A>0 r=l

i,+-+ik=n j,+-+jk=s-\

Proof. It is a straightforward consequence of (4.13) that there is just one T eT

such that y(T) = 0, namely T0. Since ß(T0) = 1, this proves (4.18).

To prove (4.19) we observe from (4.5), (4.12) and (4.13) that any F €B/i.fj s > 1, 'originates' from some Tx ,T2, ... ,Tk , where Tr e B,,,Jr,

r = 1,2, ... , k, and £Í=i »V = n, £r=i jr = s - I. Moreover, by al-lowing k and / to range in {1,2, ... , m} and {0, 1, ... , k} respectively,

(k, I) ^ (1, 1), we can recover in this manner all TeB„iS. This proves the

required inequality. D

Let us define recursively for « = 1,2,...

Mn,0 ■=Ön,\,

m k

(4-20) K,5-.= E* E E IlMU> s = i,2,....k=\ ii,...,ijt>l j\.jk>0 r=\

h+—+/*=« J\+—+h=s-\



We may deduce at once from (4.20) and Proposition 10 that

oo oo

(4.21) M(x,y)<M*(x,y):=YJYjM*nySxn-xf, x,y>0.n=\ 1=0

Our next result is a technical lemma that, in all likelihood, is already well

known. However, as its proof is absolutely straightforward, we include it for

completeness.

Proposition 11. Given integers k > 1 and r > 0 and sequences {aiyS: I =

1,2, ... , k, s = r, r+l, ...} and assuming that all underlying sums absolutelyconverge, it is true that

oo k k / oo \

(4.22) e e n^=nte«4s=r J\,-Jk>r < = 1 /=1 \s=r I

ji+—+jk=*

Proof. The identity (4.22) is trivial for k = 1, whereas for k > 2 we have,

after changing the order of summation,

oo k

E E IK,s=r h.h>r 1=1

ji+--+jk=s

oo s-(k-l)rs-(k-2)r-j, '-'-Um ^ (k-\ \

= E E E ••• E IIa'.i/«M-r'i*=' j\=r h=r jk-i=r \/=l /

oo oo oo oo /^~^ \

= EE-- E E [Uaijl)ak,s-T"-ijij\=rJ2=r h-\=rJ=r+V*_1 j¡ ^l=l ' ' '

l=\ \s=r /

Proposition 12. The functions

oo

n=\

obey the recurrence relation

(4.23) JTo*(x) = 1,m k

(4.24) jr;(x) = yjkxk-x E II•*/*(*)' *=i,2,....k=\ h,-,jk>0 1=1

j\+---+jk=s-i

Proof. Whereas (4.23) follows at once from (4.18), we prove (4.24) by multi-

plying (4.20) by xn~x and summing up for « = 1, 2, ... . This gives


460 A. ISERLES

oo m k

j?;(x) = Y,x"-1Y,k E E U^j,n=\ k=\ h,...,ik>\ h,...Jk>0 1=1

ii+-+ik=n ji+—+jk=s-l

m oo k

= E*k~lk E E E n^,*"-1-k=\ Ji,-Jk>0 n=\ h,...,ik>\ /=1

Ji+--+Jk=s-\ ii+—+ik=n

We now use Proposition 11 (with r = 1 and a¡j = M* jX''~l) to argue that

the recurrence (4.24) holds. D

Proposition 13. Given that x, y > 0 are sufficiently small, the function M*(x, y)

is the unique zero of the polynomial

m

dXyy(t) = l-t + yy£kxk-xtk

k=\

-l-t + yt-(Trjfji . x,yeR ,

that obeys M*(0,0) = 1.

Proof. Since M*(x,y) = \ZT=o^s*ix)ys > we multiply (4.24) by ys, sum up

for 5 = 1, 2, ... and add (4.23). This results in

oo m k

M*(x,y) = i+Y.Y.kxk~iys E II-W5=1 k=\ h./*>o /=1

h+-+jk=s-l

m oo k

= i+yE^/c~1E E rK;w-k=\ 5=1 ;'i,...,A>0/=l

j\+--+jk=s

We next employ Proposition 11 with r = 0 and a¡j = JK*(x)yl to argue that

m

M*(x,y) = l+yy^kxk'x(M*(x,y))k.

k=\

Therefore 6x>y(M*(x ,y)) = 0.As x, y —» 0, m - I zeros of 6Xyy travel to infinity and one to 1. Since

M*(0, 0) = 1, it is the latter that is identified with M*. This identification isvalid sufficiently near to (x, y) = (0, 0), since 6'0 0(1) ^ 0. D

Note that, because of (4.21), existence of a convergent Taylor expansion of

M* at (x, y) e K+ x E+ implies that this is also the case with M. The latter

in turn proves, by virtue of (4.18), that the Dirichlet series converges.Of course, we are interested in (x, y) e K+ x R+ away from (0,0). It

follows from standard theory of analytic functions that the bivariate Taylor

series for M converges (to the correct zero of Qx,y) for all (x,y) e *&m,

where the set .fm is the intersection of R+ x R+ with the natural projection of

the largest connected portion of the sheet of the covering Riemann surface of

6X y that crosses (0,0) at t = 1 and excludes branch points and branch cuts.



In other words, ^m is the largest subset of the nonnegative quadrant such that

for every (x, y) the solution tx>y of 6Xyy(tXyy) = 0 possesses a convergent

Taylor expansion and there exists a unique analytic continuation from in,o = 1

tO tXyy.

A long but straightforward method to produce &m is as follows: the bound-

ary of the set is either one of the lines x = 0 or y = 0, or it must be a

branch point.3 To determine the latter, apply the Euclidean algorithm to the

polynomials

m

(4.25) 6Xyy(t) = l-t + yYlkxk-xtk,

k=\

m-l

(4.26) d'Xyy(t) = -l+yy£(k + l)2xktk.

k=0

This results in a high-degree bivariate polynomial, Gm(x, y), say. All points of

(x, y) e C2 such that Gm(x, y) = 0 are branch points of 6Xyy . This defines a

family of curves. The set &m is the largest connected portion of K+ x R+ that

contains (0, 0) and excludes all solutions of Gm(x, y) = 0.

In the simplest nontrivial case m = 2 we have

C72(x,y) = (l-y)2-8xy,

therefore

&2 = {(x,y):0<xy<{(l-y)2,0<y<l}.

Likewise, a slightly more substantial calculation affirms that

G3(x, y) = 8y3 - 76xy2 + 243x2 - 28y2 + 108xy + 32y - 12

and

(x, y): 0 <x < ̂ ll9y -27 + J^^\ ,0<y<l

Larger values of m require symbolic manipulation. Thus, for example,

G4(x,y) = 3031040x3y2 - 1706995;c2y3 + 418100xy4 - 37000y5

+ 10911744*3y - 4440222x2y2 + 410700xy3 + 40700y4

+ 6137856x2y - 3127536xy2 + 309320y3 + 2925072xy

- 880452y2 + 855144y - 287712.

Since the Taylor coefficients of M* are all nonnegative, the domain of con-

vergence decreases with m . Letting m —► oo in the definition of dXyy we thus

obtain a lower bound on the set ^m . In other words,

^oo Q ■ ■ ■ Q &m+\ Q &m Q • • • •

To evaluate ^ we note that

lim 6Xyy(t) = l-t+ yt \xt\<l,m->oo ' (1 - xty

3 The phrase 'branch point' is a misnomer in the present context, since these points are not

isolated and they lie along 'branch curves'.


462 A. ISERLES

Figure 2. The sets "§„, for m = 2, 3, 4, oo

therefore

Goo(x, y) = 4x3 - x2y - 12x2 + 20;cy - 4y2 + 12jc + 8y - 4.

Simple calculation affirms that

&oo = \(x,y):0<x<l,0<y<^(s + 20x-x2- ^x(x + &)A j .

Figure 2 displays the sets &m for m = 2, 3, 4 and m = oo, the scale

being 0<x<|,0<y<l.All these sets are consistent with the assertion

(which, in its generality, is a tentative conjecture) that the boundary of ^m

in {(x, y): x, y > 0} is a strongly concave, strictly monotonically decreasing

curve, rapidly tending from above to the corresponding portion of d^ .

For all m > 2 and small x > 0 substitution into (4.25) and (4.26) yields

x (i-y)7

8y+ cf(x2), 0<y<l,

as the boundary of &m . Thus, asymptotically for small x, S?m is similar to &2

and, more importantly, for every y e (0,1) there exists ûmyy > 0 such that

(x,y)e ^m for all x e [0, ûm,y).Likewise, asymptotics, consistent with our conjecture, are available for x »

1. We assume that y ~ xx~m+x for some ^ € R. Substitution into (4.25)

and (4.26) yields 1 -1 + mxtm = o(l) and -1 + m2xtm~x = o(l) respectively.

It is now easy to execute the Euclidean algorithm explicitly and derive x -



(m - l)m~x/mm+x. Thus, for x » 1 the 'upper' boundary of ^m is

As m -» oo, this portion of 3fm shrinks, ultimately disappearing in the bounded

As long as (x, y) e S'm, we can be assured that the Taylor series for Mconverges, hence d* can be bounded by (4.17). This, together with the theory

of §3, provides useful information on the asymptotic behaviour of (4.4).

Theorem 14. Suppose that there exists n > 0 such that \Xt - 11 > «, T ^ T0,

and that

(4.27) (\dT0\,a/n)eWm.

Then a solution of the equation (4.4) can be expanded into convergent Dirichlet

series for all í e [0, i*) for some t* > 0. Moreover, if Reb < 0 then t* = oo

anda. If Re b < 0 then the solution is asymptotically stable;

b. If Re b = 0, b 7¿ 0, then the solution is almost periodic.

Proof. An immediate consequence of the theory of §3. D

Note a major drawback in the last theorem, namely that, unlike a Taylor

expansion, a Dirichlet expansion is, in general, difficult to match with an initial

value. Thus, to generate a Dirichlet expansion we need to choose dja EC,

and this, in tandem with the recurrence (4.5), leads to A and to Dirichlet

coefficients 2 := {d^Tei ■ Finally, provided that 3 is known, we can recover

the initial value y(0) = J^TejdT- Hence, not much can be deduced from

Theorem 14 in the event when y(0) is given a priori.4 However, the recurrence(4.5) being homogeneous, dT0 = 0 implies y(0) = 0 and, by continuity, small

|ú?r0| corresponds to small |y(0)|. Suppose that a < «, hence y e (0, 1). Ithas been already proved that for every m > 2 and every y e (0, 1 ) there exists

um¡y > 0 such that ([0, ûm,y), y) c ^m. In other words, provided that |y(0)|

is sufficiently small, (4.27) holds.

Theorem 15. Suppose that Reb < 0, that there exists n > 0 such that \Xt~1\ >

n, T ^ T0 and that

max{|fl¿>_,-|: i = 1, 2,..., m, j = 0, I, ... , i, (i, j) ¿ (1, 1)} < n\b\.

Then there exists y* > 0 such that for all |y(0)| < y* the solution of (4.4) is

asymptotically stable (if Re b < 0) or almost periodic (if Re b = 0, b ^ 0). Inparticular, if Reb < 0 then 0 is an attractive fixed point of (4.4). D

4 The linear case m = 1 is an exception, since then the map dja -» y(0) is linear and invertible

[4].


464 A. ISERLES

The existence of « > 0 is critical to the last two theorems. It will be de-

bated further in the next section where we demonstrate that, unfortunately, the

assumption requires a restriction of the range of q > 0.

5. ON THE EXISTENCE OF Y] > 0

We commence this section by considering the equationm

(5.1) y'(t) = by(t) + Yjakyoyk(qt), y(0) = y0.k=\

The set T can be now derived in an explicit manner, since each T e T, ord F >

2, must be obtained by rooting a sum of I e {1,2, ... , m] members of T,

i.e.

Tx T2 ... Tk

Consequently,

k

(5.3) Ar = c?E^.5=1

The equation (5.1) is singled out by the fact that, unless / = 0 in (4.8), the

underlying dr must vanish. It is straightforward to deduce that T encompasses

all connected, m-ary rooted planar trees. Table 2 displays all members of T of

order < 5 in the case m = 3

Lemma 16. Suppose that akj = 0 for all k > 1 and I > 2, hence that (4.4)reduces to (5.1). Then

a.IfO<q<± then 0 < XT < mq < 1 for all T e T\{F0} ;

b. If ^ < q < 1 then 1 is an accumulation point of the set A ;

c. // 1 < q then XT > q > 1 for all T e T\{0} .

Proof. We already know that T is in the present case the set of all «z-ary rooted

planar trees.Let firstly q e (0, ¿]. We prove that XT < mq by induction on ord F > 2,

noting that Xt = q < mq for the unique F e T of order 2. Let the statement

be true for all T e T, 2 < ord T < n — 1 and suppose that ord T = « and

that F is constituted by rooting the sum of TX,T2, ... ,Tk as in (5.2). Since

ordTs < n — 1, s = I, ... , k , il follows by induction and (5.3) that

Xt < Q I E mq+ E * \ - mq-(ordr,>2 ord7", = l J

In the case of q e [^, 1] we commence by proving that for every s e Z+

and r e {1,2,...,«!*} there exists T e T such that Xt = rqs. This will



ordT= 1:

ordT = 2:

ordT = 3:

Table 2. All T eT with m = 3 and ord F < 5

f(T) = 0

IV(T) = 0

ordT = 4:

ordT = 5

V(T) = 0

V<p(T) = 1

Y </ v»^(r) = 0 V(T) = 1 y>(T)=l ?(T)=1 v(T) = 2

Y V V y^(T) = 0 ¥>(T)=1 Y>(T) = 1 tfT)=l v(r)=l

v J Y V ^y>(T)=l <p(T)=l <p(T) = 2 <p(T) = 2 <p(T) = 2

V(T) = 2 y>(T) = 2 dT) = 2

be done by induction on 5, noting that the statement is true for 5 = 0 since

Xt0 = 1. Assume the validity of the statement for 5 - 1, hence for every

i = 1, 2, ... , ms~x there exists F, 6 T such that Ar, = iqs~x ■ We form a new

tree F by appending a root to F,,, T¡2,..., T¡'m in a manner identical to (5.2).

Here m' e {1, 2, ... , m} and 1 < ix < i2 < ■■■ < im* < ms~x. Therefore

F e T and, by (5.3),

m'XT = qsYJ ij-

;=i

It is straightforward to verify that

0 {0,l,...,ms-x} = {0,l,...,ms}.


466 A. ISERLES

Consequently

1E h''• 1 < h < h < ■ ■ ■ < im' < ms x ,m' <m\ ={1,2,..., ms}

and F such that XT = rqs exists for all r = 1, 2, ... , ms.

Since mq > 1 and q < 1, there exists for every s > 1 a number rs e{1, 2,... , ms - 1} such that 1 e (rsqs, (rs + l)qs]. Thus, for every such 5

there exists a tree T* e T such that \Xt; - l\ < \qs and 1 is an accumulationpoint of the set {XT; :s=l,2,...}CT.

Finally, let q > 1. To prove that XT > q when ord F > 2 it is enough to

note that, by (5.3), the polynomial Xt is a multiple of q with integer nonneg-

ative coefficients, not all of which may vanish. This completes the proof of thelemma. D

Corollary. Expanding (5.1) into Dirichlet series, we may choose « = 1 - mq

f°r Q<m and *I = Q-1 for q>l. ü

The functions ß(T) and y(T) can be evaluated explicitly for (5.1). Let the

fertility frt v of a vertex v e T, where F e T, be defined as the number of itschildren. We set

<p(T):= E (fitf-l)+, FgT.v is a vertex of T

Proposition 17. For every T e T ß(T) = <p(T) + 1 and y(T) = ord F - tp(T) -1. D

The proof follows directly and in a straightforward manner from composition

rules in §4 and is left for the reader.

Next we consider the equation (4.4) under the assumption that there exist

1 < k < m, 2 < I < m and 1 < p < I such that ak,o, a¡,P ^ 0. This is analmost complementary situation to the case (5.1).

According to (4.8), we have

Fi T2 ■■■ Tk

(5.4)

TltT2,...,TkÇT =» "V" eT;

(5.5) Tp+i Tp+2

TuT2,...,TieJ => Ti-F2—•••—Tp-^ eT.

Lemma 18. Subject to the aforementioned conditions and stipulating that q < ¿,

necessarily either 1 is an accumulation point of A or there exists T e T\{T0}such that Xt = 1.

Proof. For every e > 0 we use the rules (5.4) and (5.5) recursively to construct

A£ c A that contains an element in an e-neighbourhood of 1.



Let r and s in Z+ . We set F(°> := F0 and use (5.5) to generate

To To •-• T0

TU+i):=TU) — To-T0--V^ , 7 = 0,l,...,r-l.

Since XTm = 1 and

Arc/«) =ATu) + (p- 1)+ (/-/»)?, ;' = 0, 1,..., r- 1,

it follows that

xTu) = ((f -1) + (i-p)q)j +1, ; = o, 1,..., r.

Next we use (5.4) 5 times, letting

ru) r(i) ... yO)

T(i+i).= SVX , ¿ = r,r+l,...,r-M-l.

Since

Arü+i) = fc^Aro), j = r, r+1, ... , r + s - I,

it follows that

(5.6) Ar„+„ = (fcfl)4(((p - 1) + (/ - P)q)r + 1).

Let TryS := F<r+i> ,r,se1+ and set A, := {XTry, : r e1+} , s e1+ . Thus, if

kq = 1 then Xt0 , = 1, ToyS ̂ T0 for s > 1 and the lemma is true. Otherwise,

the assumption kq < 1 implies that Xt0 s < 1, whereas, by (5.6), Xt, , > 1

for sufficiently large r. Moreover, the elements of As, ordered by r, form a

monotonically increasing equidistant sequence,

Ar,+M - XTr,s = (kq)s((p - 1) + (/ -p)<7) = cs > 0,

say. Since kq < 1, sufficiently large 5 makes Cs as small as required. Thus,

for every e > 0 there exists se such that for all s > sE there is a member of

AjCA in an e-neighbourhood of 1. D

The import of Lemmas 16 and 18 is that Dirichlet expansions frequently fail

when q e (0, 1 ). Thus, in Lemma 18 it is only natural to choose k > 1 as the

least index such that aky0^0. For example, if aXyo ̂ 0 then Dirichlet series

fail for all q e (0, 1) .5 The disappointment is somewhat obviated, since it

is easy to prove that Dirichlet expansions exist in the advanced case q > 1.This has been already demonstrated for a special case in Lemma 16, but a more

general result is easy to derive.

5 Of course, the last statement requires m > 2. If m = 1 and the equation is linear then

Dirichlet series exists for every je(0, 1) [4].


468 A. ISERLES

Lemma 19.

a. q > 1 implies that XT > min{2, q} for all T e T\{T0} .b. If q > 2 then XT > ord F for all T eT.c. If akk ^ 0 for some k e {2,2, ... , m} then for every « > 1 such that

(« - 1) is divisible by (k - 1) there exists T eT such that Xt = ord F = « .

Proof. We prove the first statement by induction on ord F > 2. Table 1 affirms

that it is true for ord F = 2. Let us suppose that F 6 T has been derived from

Tx,T2,...,Tk via (4.8). Hence

/ k

(5.7) Ar = EAr,+tf E A^¿=i ;=/+i

Note that ord F, < ord F for i = 1, 2, ... , k. Moreover, Xt > 1 for allT eT, since q > 1 and Xt ^ 0 is a polynomial with nonnegative integer

coefficients in q . Therefore, I & {1,2,..., k— 1} and (5.7) imply that

Xt > q min Aj-, > q.l+\<i<k

On the other hand, in the case / = k, there are two possibilities. Firstly, if

F, = T0, i = 1,2, ... , k, then (5.7) gives XT = kXTo = k > 2, whereasif Tj ,¿ T0 for some j € {1,2,..., k] then we use (5.7) and the induction

assumption to argue that

Xt > Xtj > min{2, q}.

This completes the proof of our first assertion.

To prove that Xt > ord F for q > 2 we again use (5.7) and induction on

ord F. The statement is true for ord F = 1, since Xt0 = 1 = ord F0. Otherwise

F must have been derived from some Tx, ... ,Tk eT as in (4.8). Hence, by(5.7), q > 2 implies that

k k

Ar>E^+ E^-i=i i=/+i

If / < k - 1 then, by induction,

k k

E Ar,> E OTdT¡> 1,,=/+i ,=/+i

therefore (again by induction)

k

Ar>Eordr' + Li=i

Since / < k - 1,A:

ord F = E ord Ti + !i=i

(since exactly one extra vertex is added by composition) and this proves the

inductive step. In the remaining case, / = k , we have by induction



k k

Ar = EAr,>Eordr' = :r'1=1 1=1

because no vertices are added in the composition. The proof of the second partof the lemma follows.

Finally, to prove the last assertion, we let F(0) := F0 and define by induction

k—l times

TU+»:=TW-'T0-To-----To\ y = o, 1.

Since akk ^ 0, we have TU+O e T. It follows easily by induction that XTu) =

ord Fw = j(k - 1) + 1, j e 1+ . This completes the proof of the lemma. D

The implication of the last part of the lemma is that in most cases the bound

Xt > ord F for q > 2 is the best possible. However, the main significance ofLemma 19 is in establishing that we may take n = min{ 1, q - 1} > 0 whenever

q > 1. Needless to say, the larger the value of «, the more powerful are theresults of §4.

The set T and the functions ß and y can be worked out explicitly in special

cases and we have already seen this for the equation (5.1 ). Herewith we considertwo further examples. Firstly, the equation

(5.8) y'(t) = by(t) + a2y2y2(t) + aXy0y(qt), y(0) = y0.

Asserting ax y o, a2 y 2 # 0, we have

ri,r2€T => Tx—T2, ¡€T

and it is an easy matter to deduce that T = W, the set of all multiply rooted

planar trees.

To gain intuition on ß and y we examine in Table 3 all members of T oforder < 4, together with the corresponding values of y . For brevity, the value

of ß is not listed, but in all these instances it is consistent with the identityß(T) = ovdT-y(T).

Proposition 20. In the case of the equation (5.8), the value of y(T) for every

T eT equals the number of vertices of T that possess children, whereas ß(T) =ord F-y(F).

Proof. Let g>(T) denote the number of vertices of F that possess children and

let x(T) := ß(T) + y(T). Hence1. <p(To) = 0, x(T) = l;2. T=TX-T2^ <p(T) = tp(Tx) + tp(T2), X(T) = X(TX) + X(T2) ;3. T,

T = 1 => <p(T) = <p(Tx) + l, x(T) = x(Ti) + l,

and we deduce that, whilst both functions share the same recurrence relation

with y(T) and ordF, <p is consistent with the initial value of y, whereas;í(F0) = ordF0. G


470 A. ISERLES

Table 3. All F e T, ord F < 4, for equation (5.8)

ordT=l: •

l(T) = 0

ordT = 2:

ordT = 3:

ordT = 4:

I7(T) = 0 7(T)=1

• • •

l(T) = 0

• • • t

t(T) = 0

V.7(T) = 1

Vl(T) = 1

t(T) = 1

7(T)=1

L J

7(T) = 1

7(T)=1

Y7(T) = 2

u

7(T) = 1

a.7(T) = 1

7(T) = 2

7(T) = 2

7(T) = 1

V17(T) = 2

7(T) = 2 7(T) = 2 7(T) = 2 l(T) = 3

Our next—and last—example is the equation

(5.9) y'(t) = by(t) + a2yXy(t)y(qt), y(0) = y0.

We assume that a2yX ¿0.

Proposition 21. If T eT for (5.9) then necessarily ord F is odd.

Proof. Clearly, ord F0 = 1, an odd number. We continue by induction. The

only extant composition rule is

(5.10)Ti,T2€T Ti—* €T.

Therefore ord F = ord Fi + ord F2 + 1, and, provided that ord Fi and ord F2are odd, so is ord F. D

Table 4 displays all members of T of order < 7.

Proposition 22. Let TeT, ord F = 2« + 1. Then ß(T) = « + 1, y(T) = n .

Proof. Let us suppose that ord F, = 2/c, + 1, kx + k2 = « - 1 (recall that, byProposition 21, all TeT have odd order). Then, by the composition rule

(5.10) and induction on « ,

ß(T) = ß(Tx) + ß(T2) = (kx + l) + (k2 + l) = n+l ;

y(T) = y(Tx) + y(T2) + 1 = kx + k2 + 1 = n. Q.E.D.



Table 4. All T eT for the equation (5.9) with ord F < 7

ordr = l: .

ordT = 3 :

ordT = 5 :

J

M

ordT = 7: y ^ j^ ¿i ox:

Proposition 23. Let T e T, ord F = 2« + 1. Then there exists a polynomial

Pt with nonnegative integer coefficients such that Pt(0) is a positive integer anda(T) = q»pT(q).

Proof. We recall that each T eT is of odd order, and assume again that, for

« > 1, F is obtained from Fi and T2 as in (5.10). Therefore,

(5.11) XT = XTl+qXT2

and we deduce that XT = I +<f(q), q -^ 0. Moreover,

(5.12) a(F) = (Ar-l)a(F,)a(F2).

The statement of the proposition is true for « = 0 and we continue by

induction. Given « > 1, (5.11) and (5.12) yield

a(T) = (At-, + qXT2)qk'pTl(q)qk2pT2(q) = (XT¡ + qXT2)qn~xPT(q)PT2(q)-

Moreover,

lim q-l(XTi+qXT2-l)= lim q~x((l +cf(q)) + q(l +cf(q)) - 1) = (f(l),q->0+ <?->0+

therefore q~"a(T) =cf(l), q -» 0, and

(5.13) pT(q)= (^^ + XT^PT,(q)PT2(q).

Finally, it is easy to demonstrate that Pr(0) €{1,2,3,...}. Since each XT

is a polynomial in q with nonnegative integer coefficients and Ar, = 1 +cf(l),it follows that

¡¡m(k^l+XT\=^r,«-»0+ \ q 2Jdq

+ 19=0

is a natural number. Hence, and since Pt0 = 1, an induction on (5.13) affirmsthat Pt(0) is itself a natural number for all F e T. D


472 A. ISERLES

Substitution into (5.9), in tandem with the last two propositions, affirms that

1 fdTocodT = -«7-c

Pt \ q

where ord F = 2« + 1 and œ = a2yX/b. Moreover, pr(q) > 1 for all T eT.and we deduce that

.'Idr co\\"(5.14) \dT\<\dTo11

q

Let us denote by D„ all T eT of order 2« + 1. According to Proposition 21,

T = U„ez+ D« • We let Mn stand for the number of elements in D„ . Then, by

(5.14),

(5.15) d*<\dTo\M(\dToco\/q),

whereoo

M(x):=Y,M"x"-n=0

Proposition 24. The Taylor expansion of M converges for all \x\ < \ and

(5.16) gM-'-^-S iln)2x ^n\(n+l)\ '

n=0

Therefore, M„ = (2«)!/(«!(« + 1)!).

Proof. Suppose that F e D„, « > 1 has been obtained from (5.10). Hence

there exists k e {0, 1,... ,n- 1} such that Tx e Bk , T2 e B„_k_x. It followsthat

n-l

Mn = y^MkMn_k_x, « = 1,2,....k=0

Multiplying the last expression by x" , summing up for « = 1,2,... and

adding Mo = 1 readily yields

oo n— 1

M(x) = 1 + E E MkMH_k_xxH = 1 + x¥2(jc).n=l k=0

It follows that M is a solution of the quadratic equation xt2 -1 + 1 = 0. Since

M(0) = 1, it follows that M(x) = (1 - \/l - 4jc)/(2x) , with the radius of con-

vergence \ . The remainder of the lemma is a consequence of a straightforwardTaylor expansion of the square root function. D

Theorem 25. For every \dTQ\ < ^/(4|fe>|) the Dirichlet expansion of the equation(5.9) converges in an interval of the form [0, /*), where t* = oo // Reb < 0.

Proof. A straightforward consequence of our analysis and of §3. D

As before, we can deduce from the last theorem that Re b < 0 implies asymp-

totic stability, whereas Re b = 0, b ^ 0, implies almost periodicity. Likewise,

rephrasing in terms of y(0) rather than t/^ , we can deduce asymptotic stability

or almost periodicity, as the case might be, subject to |y(0)| being sufficientlysmall.



The last example does not fit into the pattern of Lemma 18 and, indeed, it

displays different behaviour altogether—Dirichlet series converge regardless of

the value of q > 0, as long, of course, as \dr0 | is sufficiently small.

The radius of convergence of M (after an obvious change of variables) is

more generous than of the function M from §4. This should come as no

surprise, since M has been derived by counting all members of W. In the caseof (5.9) all the ak¡s, save for one, vanish and T forms a small subset of W.

6. A RATIONAL EQUATION

A natural step in generalizing (5.4) consists of replacing a bivariate polyno-

mial in y(t) and y(qt) by a bivariate rational function. In the present section

we present a preliminary result of that kind. For the sake of clarity we restrictour analysis to the simple equation

where q > 0 and a, b ^ 0. Letting again

y(t) = YJdTebkTt, t>0,

Tei

we derive upon substitution into (6.1) the equation

E(l-Ar)¿rexp{6Ar/}(6-2) TeT ^

= a 22 Ar, ¿r, dTl exp{ b (XTi + gXT2 )}.Ti, T2ei

It is easy to deduce that the composition rules for the equation (5.9) are valid,

hence the set T is the same as in the latter part of §5 and Table 4. However,

the recurrence for ¿fY is different and it reads

dr = -.—?— dT, ¿T;IT{_ ,-,,-,,,

F having been derived from Fi and F2 by (5.10).Bearing in mind Proposition 21, we can prove, by revisiting the technique of

the previous section, that

(6.3) dT = ^.[^\ ordF = 2«+l,

where

(6.4) rT = rT(q) := -^ + q~X^ ~ l)rTl(q)rT2(q).Ar,

Since Xt = 1 +cf(q), q -> 0, it follows that rT is a rational function in q and

that |/y| has nonnegative coefficients. Table 5 displays the functions Pt (from

§5) and rr for ord F < 7.An examination of Table 5 is suggestive of the assertion that Pt is the nu-

merator of rT ■ This can be easily proved, dividing (5.13) by (6.4),

$)-M%)(p_j\


474

order

A. ISERLES

Table 5. Functions Pt and rT for ord F < 7

tree | pr | T

J -1

¿ 1 + 9 "(1 + 9)

•—* 1 + 9

(1 + 9)(1 + 9 + 92)

■J/ 2(1 + 2ç)

(1 + 9K1 + 9 + 92)

2(1 + 2?)

Ji/ 2 + g

1 + 9

2-f-g

1 + 9

.VI (l + g)(2 + 9)

•—*—t

(l + g)(2 + 9)

1 + 9 + 92

6

(1 + 9)(1 + 2?)



Since PT0/rT0 = 1, it follows by induction that pr/rr is a polynomial, con-sequently pt is indeed a numerator of rT. We conclude that, since the denom-

inator of rT has nonnegative coefficients, necessarily |rr| < pT . Therefore, if

the Dirichlet series for (6.1 ) converges, then so does the Dirichlet series for (5.9)(with a replaced by co). This, however, is of limited significance, since we al-

ready possess in Theorem 25 a convergence condition for Dirichlet expansionsto (5.9).

Let us suppose that q > 1 and set

F(x,y)-= A x_x-y + VTherefore,

(6.5) \r-x\ = F(XTl,XT2)\rT-yx\\r-x\,

provided that F has been composed from Fi and F2 . It is straightforward to

verify that q > 1 implies that dF/dx > 0, dF/dy < 0 for all x, y > 1.

Proposition 26. Provided that q > I, every T e D„ obeys the inequality

(6.6) l + nq <XT< l+q + --- + q".

Moreover, there exist Un,V„e D„ such that

Xv„ = l + nq, XVn = l+q + --- + qn.

Proof. We use induction on « . (6.6) is certainly true for « = 0. Otherwisethere exists k e {0, I, ... , n - 1} such that

XT = Ar, + qXTl F, e Bk , T2 e D„_*_,

k n-k—l n n—k

<$>' + ? E qi = YjqJ-(q-l)Y.qi-7=0 j=0 ;=0 ;'=1

Since q > 1, the upper bound is true. Lower bound can be proved in anidentical fashion.

To prove that both bounds are attainable, we specify Un and V„ explicitly,

Un = ,11-I—I and Vn = JV

v>.> n times

n times

It is trivial to prove by induction that Un,Vne Bn and that they obey the lower

and the upper bound, respectively, since Un is composed from U„-X and F0 ,

whilst V„ is composed from F0 and F„_[. D

Corollary. For every n > 1 and k e {0, 1, ... , n - 1} it is true that

(6.7) F(XT], XTl) < F(XVk, Xu^), TxeBk,T2e ©„_,_,.

Proof. Follows at once from (6.6), since F increases monotonically in its firstargument and decreases in the second. D


476 A. ISERLES

Let

hk :=F(XVk,Xu^k_i)

ak+i _ i

qk + (n - k - l)q2 - (n - k - 2)q - 2 'k = 0, 1,..., n — 1.

It is an elementary exercise to prove that «a^i > hk, k = 0, 1, ... , « - 2.

Consequently,

q" - 1hk < «„_i = —-.-~<q, k = 0, 1, ... , « - 1.k - n l qn_x +(¡ _2 - ■* >

It follows from (6.7) that F(XT¡,XTl) <q and we deduce from (6.5) that

(6.8) kf1|<9kr11lr7:21l-

Let us assume that \rj}\ < qf^ , j = 1,2. Then, according to the inequality

(6.8), |rf l\ < qv(TMv(Ti)+\. We conclude that |rf l\ < q^1^ , where y/(Ta) = 0

and y/(T) = y/(Tx) + ^(F2) + 1. Thus, y/ obeys exactly the same recurrence

and initial condition as y and, by Proposition 22,

(6.9) \rTl\<qn, TeBn-

Theorem 27. Let q > 1. Then, for every \dT0\ < l/(4|a|), the Dirichlet expan-sion of the equation (6.1) converges in an interval of the form [0, t*), where

t* = oo //Ree<0.

Proof. Similar to the proof of Theorem 25, except that, by virtue of (6.9), (5.15)

need be replaced by

\dT\<\dT0\\dToa\n, «eO„.

As before, the existence of a Dirichlet expansion, in tandem with the material

of §3, go a long way towards explaining the asymptotic behaviour of y as t » 0

in the case Re b < 0.The remaining case, q e (0, 1), is considerably easier.

Theorem 28. Provided that q e (0, 1), the Dirichlet expansion of the equation

(6.1) converges in an interval of the form [0, t*), where t* = oo if Reb < 0,

provided that \dTo\ < q/(4\a\).

Proof. Given Tx, T2 e T, set

Ar,G(t) :=

XT2 + t-l(XTt -I)

Since Xt¡ , Xt2 > 1, G is monotone in t and



0<G(t)<G(l) = T-^--<1.Ar, + Ar2 - i

Therefore we deduce from (6.4) that \r^x\ < \r^x\ \r^.x\. The remainder of the

proof follows along identical lines to that of Theorems 25 and 27. D

REFERENCES

1. C. Berge, The theory of graphs and its applications, Wiley, New York, 1962.

2. F. Harary, Graph theory, Addison-Wesley, Reading, Mass., 1969.

3. G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Univ. Press,

Cambridge, 1915.

4. A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl.

Math. 4(1992), 1-38.

5. T. Kato and J. B. McLeod, The functional-differential equation y'(x) = ay(kx) + by(x),Bull. Amer. Math. Soc. 77 (1971), 891-937.

6. T. W. Körner, Fourier analysis, Cambridge Univ. Press, Cambridge, 1988.

Department of Applied Mathematics and Theoretical Physics, University of Cam-bridge, Cambridge, England

E-mail address : A. IserlesQamtp. cam .ac.uk


on nonlinear delay differential equations€¦ · on nonlinear delay differential equations a....

Documents