Table Of ContentMultiple scales and matched asymptotic expansions for the
discrete logistic equation
6
1
0 Cameron L. Hall∗ and Christopher J. Lustri†
2
n *Mathematical Institute, University of Oxford, Woodstock Rd, Oxford, OX2
a
6GG, United Kingdom. [email protected].
J
†Department of Mathematics and Statistics, Faculty of Science, University of
3
1 Sydney, Sydney, 2006, Australia. [email protected].
]
S
D January 14, 2016
.
h
t
a
m
Abstract
[
In this paper, we combine the method of multiple scales and the method of matched
1 asymptotic expansions to construct uniformly-valid asymptotic solutions to autonomous
v andnon-autonomousdifferenceequationsintheneighbourhoodofaperiod-doublingbifur-
7 cation. Ineachcase, webegin byconstructingmultiplescales approximationsin whichthe
3
slow time scale is treated as a continuum variable, leading to difference-differential equa-
1
tions. The resultant approximations fail to be asymptotic at late time, due to behaviour
3
ontheslowtimescale,itisnecessary toeliminatetheeffectsofthefast timescaleinorder
0
to find thelate time rescaling, but there are then no difficulties with applying the method
.
1 of matched asymptotic expansions. The methods that we develop lead to a general strat-
0
egy for obtaining asymptotic solutions to singularly-perturbed difference equations, and
6
we discuss clear indicators of when multiple scales, matched asymptotic expansions, or a
1
combined approach might beappropriate.
:
v
i
X
1 Introduction
r
a
1.1 Multiple scales for difference equations
The method of multiple scales is extremely well established as an asymptotic method for
analysing differential equations. As early as 1973, Nayfeh [24] was able to describe how the
method of multiple scales had been applied to nonlinear oscillator theory, orbital mechanics,
flight mechanics, buckling analysis, flutter, wave propagation (in both solids and fluids), dis-
persive waves, plasma physics, atmospheric science, and statistical mechanics (amongst other
areas). Themethodofmultiplescaleshascontinuedtobeusedextensivelyintheanalysisofdif-
ferentialequations, and is coveredin most standardtexts andcourses onperturbation methods
(see, for example, [2, 10, 11, 15, 24]).
In contrast, the application of the method of multiple scales to difference equations is much
less well-developed. While Hoppensteadt and Miranker [12] introduced the method of multiple
scales for difference equations as early as 1977, there were relatively few developments in this
areauntil the rediscoveryofmultiple scales (andthe Poincar´e–Lindstedtmethod) for difference
∗Electronicaddress: [email protected]; Correspondingauthor
†Electronicaddress: [email protected]
1
equations by Luongo [17] and Maccari[18] in the 1990s,followed by the innovative recent work
by van Horssen and ter Brake [13], and Rafei and van Horssen [25, 26, 27]. Very few texts
on perturbation methods discuss difference equations in detail ([2] and [11] being two notable
exceptions), and Holmes [11] recently observed that there are still many open problems on the
application of multiple scales to difference equations.
Theessenceofthemethodofmultiplescalesliesinreplacingtheoriginalindependentvariable(or
system of variables) with a new system that reflects the different scales over which important
phenomena occur. In the analysis of the ordinary differential equation (ODE) for a weakly
nonlinear oscillator, for example, the method of multiple scales involves transforming the ODE
intoapartialdifferentialequation(PDE)withtwoindependenttimevariables: onerepresenting
the fasttimescaleonwhichtheoscillationstakeplace,andthe otherrepresentingtheslowtime
scale on which the phase and amplitude of the oscillations evolve.
The method of multiple scales for an ordinary difference equation (O∆E) is essentially the
same in that the original, discrete independent variable is replaced by multiple independent
variables that reflect the different time scales inherent to the problem. However, there are two
different approaches to slow time variables for difference equations in the present literature.
One approach, introduced by Hoppensteadt and Miranker [12], involves treating the slow time
variable as continuous and using Taylor expansions in the continuum variable to transform the
originalO∆E into a difference-differential equation (∆DE). The other approach, introduced by
Subramanian and Krishnan [29] (but corrected by van Horssen and ter Brake [13]), involves
treating the slow time variable as discrete, so that the O∆E is transformed into a partial
difference equation (P∆E). Both approaches continue to be in current use: for applications
involvingacontinuumslowtimevariable,seeforexample[14,17,18,19,21,22];forapplications
involving a discrete slow time variable, see for example [25, 26, 27].
Foraweaklynonlinearmultiplescalesproblem,theleading-orderproblemintheslowtimevari-
able willoften be nonlinear,evenif the leading-orderproblemin the fast time variable is linear.
Since a wider range of tools exist for solving nonlinear differential equations than nonlinear dif-
ference equations,it wouldseemthat there areclearadvantagesto the Hoppensteadt–Miranker
approach of treating the slow time variable as continuous. However, difference equations –
especially nonlinear difference equations – can behave very differently from their analogous dif-
ferentialequations(see[13]forexamples),andimportantfeaturesofadifferenceequationmight
be lost if a continuum slow time variable is introduced. This motivates van Horssen and ter
Brake [13] to avoid continuum variables altogether, instead introducing a discrete slow time
scale so that the discrete character of the problem can be maintained throughout.
In this paper, we demonstrate by examples that a combinationof the method of multiple scales
with the method of matched asymptotic expansions can capture the subtleties of a nonlinear
O∆E without the needfor a discrete slow time variableandthe challengesthat this introduces.
In[13],vanHorssenandterBrakearecorrecttoobservethattheO∆Eobtainedfromdiscretising
an ODE may behave very differently from the original ODE (e.g. the ODE may exhibit finite-
time blow-upwhilethe O∆Ehasasolutionforalltime). However,suchdiscrepanciesmustalso
be associated with an asymptotic failure of the original multiple scales expansion, suggesting
that they could be correctedusing the method of matched asymptotic expansions. (For further
discussion of this point, see Section 4.1.)
Weillustrateourmethodofcombiningthemethodofmultiplescaleswiththemethodofmatched
asymptotic expansions by investigating the discrete logistic equation in the neighbourhood of a
perioddoublingbifurcation,althoughthe methods thatwe developaremorebroadlyapplicable
to singularly perturbed difference equations.
2
Figure1: Bifurcationdiagramshowingthe stableequilibriaof thediscretelogisticequation (1)whenthebifur-
cationparameter isvariedbetweenλ=2.45andλ=3.8.
1.2 The discrete logistic equation
The discrete logistic equation takes the form
x(n+1)=λx(n) 1 x(n) , 0<x(0)<1, (1)
−
(cid:2) (cid:3)
where λ is a dimensionless parameter with 0 < λ 4. The discrete logistic equation is well-
≤
known as a model of population dynamics with discrete generations, and it is commonly used
as an archetypal example of a nonlinear difference equation that exhibits period-doubling to
chaos as the bifurcation parameter, λ, is increased towards 4. For a detailed discussion of the
bifurcation structure of the discrete logistic equation, see, for example, [20, 23, 28].
For all λ>1, the discrete logistic equation has an equilibrium solution
λ 1
x˜(λ)= − . (2)
λ
When 1 < λ < 3, this equilibrium is stable. However, as λ increases through 3, this 1-periodic
equilibrium becomes unstable in favour of a 2-periodic cycle that alternates between the values
λ+1 √λ2 2λ 3
x˜˜±(λ)= ± − − . (3)
2λ
For 3 < λ < 1+√6, this 2-periodic cycle is stable, and all trajectories with 0 < x(0) < 1
and x(0) = x˜(λ) converge to a cycle which alternates between x˜˜+ and x˜˜− as n . At
6 → ∞
λ=1+√6, the 2-periodic cycle becomes unstable as a stable 4-periodic cycle appears, and the
period doubling process continues as shown in Figure 1.
In general, it is difficult to analyse solution trajectories of the discrete logistic equation using
asymptoticmethodssince(1)willbestronglynonlinearformostchoicesofλandx(0). However,
3
by picking x(0) to be close to x˜(λ) (or indeed, any periodic solution), (1) can be rescaled as a
weakly nonlinear difference equation that is amenable to the method of multiple scales.
In Section 2, we take λ=3+ǫ, and select x(0) near the unstable equilibrium (2). We find that
the ‘plain’ method of multiple scale only sees the escape from the unstable equilibrium at x˜,
and that matched asymptotic expansions are required in order to describe the approach to the
stable cycle at x˜˜±. Then, in Section 3 we broaden our analysis to consider a non-autonomous
nonlinear difference equation: the discrete logistic equation with a slowly varying bifurcation
parameter.
In [1], Baesens uses the method of renormalisation to perform a comprehensive analysis of the
discrete logistic equation with a slowly varying bifurcation parameter. Baesens considers a
numberofcomplicationsthatarebeyondthescopeofthepresentwork: [1]dealswithbackward
sweep as well as forward sweep, gives results for the number of observable doublings through
the entire period-doubling cascade, and contains a detailed analysis of the critical effects of
noise. Althoughwewilldiscusstheresultsin[1]anddrawonsomeimportantideas(suchasthe
stable and unstable adiabatic manifolds), our approach stands independently, focusing on how
the asymptotic methods of multiple scales and matched asymptotic expansions can be applied
to non-autonomous difference equations.
1.3 Outline of paper
In this paper, we proceed by considering two examples of the discrete logistic equation where
multiple scales and matched asymptotic expansions can be used. These illustrative examples
will be of increasing complexity, and they show clearly how to deal with many of the chal-
lengesassociatedwithapplyingmultiple scalesandmatchedasymptoicexpansionstosingularly
perturbed difference equations.
InSection2,wedemonstrateourmethodofcombiningmultiplescaleswithmatchedasymptotic
expansions by analysing the discrete logistic equation with λ = 3 +ǫ and x(0) = 2, where
3
0 < ǫ 1. Rescaling x(n) in (1) by introducing x(n) = 2 + ǫX(n), we obtain a weakly
≪ 3
nonlinear difference equation that gives a singular perturbation problem as ǫ 0+.
→
Using the method of multiple scales with a continuum slow time variable, this leads to ap-
proximate solutions where x(n) grows exponentially. The solution breaks down for large n as
the dominant balance in the rescaled difference equation changes and a new scaling for x is
required. The latetime solutionbasedonthis new scalingcanthenbe connectedwiththe early
time solution using the method of matched asymptotic expansions.
Then, in Section 3, we demonstrate that our methods can be applied to non-autonomous dif-
ference equations by analysing the case where λ is taken to be a slowly varying function of n:
λ(n) = 3+ǫ2n. This introduces a number of subtleties, especially with regard to finding the
appropriate asymptotic scalings for x and n in the early time and late time regimes. However,
themethodsdevelopedinSection2canstillbeused,andwefindthatwecanobtainauniformly
valid composite asymptotic solution.
ThroughoutSections2 and3, wefocus solelyonthe discrete logisticequation. Then, inSection
4,wedescribehowthemethodsdevelopedinthesesectionscanbeappliedtoamuchwiderclass
ofdifferenceequations,andwediscussthedifferentcombinationsofmultiplescalesandmatched
asymptotic expansions that could be used as a general strategy for the asymptotic analysis of
difference equations.
4
Figure2: Numerical solution to the discrete logistic equation in the neighbourhood of its first period doubling
bifurcation. Blackcirclesindicatethesolutionx(n)to(4)whereǫ=0.02andx(0)= 2. Asnincreases,
3
weseethatx(n)moves awayfromtheunstable equilibriumat x˜= 2+ǫ ≈0.6689, beforesettlinginto
3+ǫ
theoscillatorypatterngivenbyequation(6),withtheperiod-twomanifoldillustratedbydashedcurves.
2 The discrete logistic equation with λ = 3+ǫ and x(0) = 2
3
2.1 Early time scaling
In Section 2, we will apply the combined method of multiple scales with matched asymptotic
expansions to the discrete logistic equation, (1), with λ=3+ǫ, where 0<ǫ 1:
≪
x(n+1)=(3+ǫ)x(n) 1 x(n) . (4)
−
(cid:2) (cid:3)
Using (2) and (3), 1-periodic and 2-periodic equilibria of (4) are asymptotically given by
2+ǫ 2 ǫ
x˜(3+ǫ)= = + + (ǫ2), (5)
3+ǫ 3 9 O
x˜˜±(3+ǫ)= 2 √ǫ ǫ + (ǫ32), (6)
3 ± 3 − 18 O
as ǫ 0+.
→
Anumericalsolutionof (4)wherethe initialconditioniscloseto x˜ ispresentedinFigure2. We
notethatthesolutiongivestheappearanceofasmoothenvelopearoundasystemofoscillations
away from the unstable steady state at x˜ towards the 2-periodic equilibrium given by (6).
Consider the case where we wish to solve(4) subject to the initial conditionx(0)= 2. As given
3
in (4), this difference equation is strongly nonlinear. However,by introducing the rescaling
x(n)= 2 +ǫX(n) (7)
3
we obtain the weakly nonlinear problem,
2 X(n)
X(n+1)+X(n)= ǫ +3X(n)2 ǫ2X(n)2, (8)
9 − (cid:20) 3 (cid:21)−
with X(0)=0.
2.2 Early time solution
We begin by introducing t=ǫn as a slow time variable alongside the fast time variable, n, and
using the method of multiple scales where t is treated as a continuum variable. Our multiple
5
scales ansatz takes the form X(n) X(n,t), where X(n,t) will be expanded as an asymptotic
≡
series in powers of ǫ.
Following Hoppensteadt and Miranker [12], X(n+1) can be expanded as a Taylor series of the
form
∞
ǫj djX(n+1,t)
X(n+1,t+ǫ)= . (9)
j! dtj
Xj=0
Substituting into (8), we therefore find that the equation to solve is
2 X(n,t) dX(n+1,t)
X(n+1,t)+X(n,t)= ǫ +3X2(n,t)+
9 − (cid:18) 3 dt (cid:19)
1d2X(n+1,t)
ǫ2 X2(n,t)+ + (ǫ3), (10)
− (cid:18) 2 dt2 (cid:19) O
subjecttothe conditionthatX(0,0)=0. This isnowa∆DE expressedintermsofthe discrete
variable n and the continuous variable t.
We expand X(n,t) as an asymptotic series in the limit ǫ 0+:
→
∞
X(n,t) ǫkX (n,t). (11)
k
∼
Xk=0
Applying this series to (10), we collect terms of the same order in ǫ to obtain a system of
difference equations for the functions X(n,t). Collecting terms of (1) gives
O
(1): X (n+1,t)+X (n,t)= 2, (12)
O 0 0 9
with X (0,0)=0;
0
Solving (12), we find that
X (n,t)= 1[1 ( 1)nA(t)], A(0)=1. (13)
0 9 − −
where A(t) is some arbitrary smooth function of t. Collecting terms at (ǫ) gives an equation
O
for the next term in the series,
X (n+1,t)+X (n,t)= 2 1 [A(t)]2 1( 1)n(A′(t) A(t)). (14)
1 1 −27 − 27 − 9 − −
In order to remove the secular ( 1)n terms from the right hand side of (14), as is standard
in the method of multiple scales,−we impose A′(t) A(t) = 0. Combined with the boundary
−
condition in (13), this gives A(t)=et, and hence X (n,t)= 1[1 ( 1)net].
0 9 − −
As described in Appendix A, this process can be repeated methodically, leading to the result
that the solutions X (n,t) may be written in the form
r
X (n,t)=f (t)+g (t)( 1)n, (15)
r r r
−
where f (t) is given in (74) and g (t) is obtained by solving (76).
r r
Using these expressions to calculate X (n,t), we find that
1
X(n,t)= 1[1 ( 1)net] ǫ 6+3e2t ( 1)n 8et+9tet+e3t + (ǫ2). (16)
9 − − − 162 − − O
(cid:2) (cid:0) (cid:1)(cid:3)
Figure 3 compares a numericalsolution of (8) with asymptotic solutions based on the one term
approximation and the two term approximation, obtained by truncating (16). We see that in
bothcasesthe approximationobtainedfrom(16) accuratelydescribes the solutiontrajectoryin
its earlystages;eventually,however,bothasymptotic series approximationsbecome inaccurate.
Furthermore,using(15)togeneratesubsequenttermsintheseriesdoesnotimprovetheaccuracy
beyond this point.
6
(a)One-termapproximation (b)Two-termapproximation
Figure3: Numericalsolutionx(n)to(8)withx(0)= 2 comparedwith(a)theone-termearly-timeapproximation
3
and(b) the two-term early-timeapproximation, bothobtained from(16), inthe casewhereǫ=0.02.
In each figure, the exact solution is represented by black circles, while the asymptotic approximation
is represented by white circles. The point n= 1log1 is depicted as a thick line; this represents the
2ǫ ǫ
approximate time at which the series ceases to be asymptotic, as discussed in Section 2.3. In both
cases,theapproximationfailswhencontinued beyondthispoint.
2.3 Failure of the early time expansion
Themethodofmultiplescalesisusefulforconsideringtheslowaccumulationofsmalldeviations;
however,forchangesinthedominantbalanceofanequation,themethodofmatchedasymptotics
is necessary [16].
This is not straightforward in the present problem. Considering the asymptotic sizes of the
varioustermsin(8)asX ,wenotethattheX(n+1)andX(n)termswillalwaysdominate
→∞
the 2 term, and that the 3ǫX(n)2 term will dominate all of the other terms. Hence, a na¨ıve
inspe9ction of (8) would su−ggest that a new dominant balance is attained when X = ord ǫ−1 ,
which is incorrect. In fact, subsequent analysis in this section will show that the failure (cid:2)of th(cid:3)e
asymptotic series occurs earlier, when X =ord ǫ−12 .
(cid:2) (cid:3)
This error occurs because the failure of the early time asymptotic expansion is not associated
with the discrete-scale behaviour of solutions to (8), but rather due to the exponential growth
of the continuous slow time envelope. This growth causes the differential equations associated
with the multiple scales formulation to undergo a change in dominant balance. To determine
when this occurs, we can ignore discrete-scale variation, and reformulate (8) purely in terms of
the slow time variable, t.
The most natural way is to remove the effects of the discrete scale by considering the doubled
map, as it is clear from the early-time analysis that the behaviour on the discrete scale is
2-periodic. Applying (8) twice, we find that
X(n+2) X(n)= ǫ 2X(n) 2 +ǫ2 18X(n)3+3X(n)2+X(n) 4
− − 9 − − 81
+ǫ3(cid:2) 27X(n)(cid:3)4 18(cid:2)X(n)3+2X(n)2+ 4 X(n) (cid:3)
− − 27
+ǫ4 (cid:2) 27X(n)4 6X(n)3+ 1X(n)2 (cid:3)
− − 3
ǫ5 (cid:2)9X(n)4+ 2X(n)3 ǫ6X(n)4.(cid:3) (17)
− 3 −
(cid:2) (cid:3)
Thus, we could skip the method of multiple scales entirely by proposing the continuum ansatz
7
X(n) χ(t) in (17) with t=ǫn, leading to the infinite-order ODE
≡
dχ ∞ ǫj−12j djχ
2 + = 2χ(t) 2 +ǫ 18χ(t)3+3χ(t)2+χ(t) 4
d① t Xj=2 j! dtj (cid:20) ② − 9(cid:21) (cid:20)− ③ − 81(cid:21)
+ǫ2 27χ(t)4 18χ(t)3+2χ(t)2+ 4 χ(t)
− − 27
+ǫ3 (cid:2) 27χ(t)4 6χ(t)3+ 1χ(t)2 (cid:3)
− − 3
ǫ4 (cid:2)9χ(t)4+ 2χ(t)3 ǫ5χ(t)4.(cid:3) (18)
− 3 −
(cid:2) (cid:3)
This is now in a form where we can apply the usual methods for finding the new dominant
balance; we can propose rescalings for χ and t and seek a new dominant balance in (18) that is
consistentwithlongtimebehaviouroftheleadingordersolution(13). Notethattheexponential
growthoftheleadingordersolutionX (n,t)meansthatitwillbenecessarytoproposeanaffine
0
rescaling of t rather than a linear rescaling.
Specifically,wemaketherescalingsχ=δξ andt=K +K s,whereξ andsare (1)variables,
0 1
O
and where δ, K , and K depend only on ǫ with δ 1, K 1 and K K as ǫ 0+.
0 1 0 0 1
≫ ≫ ≫ →
Substituting into the leading order solution from (16), we find that
δξ =ord eK0eK1s , (19)
(cid:2) (cid:3)
where ord is used to represent ‘strict order’ as in [10], so that f = ord(g) is equivalent to the
case where f = (g) but f =o(g).
O 6
Since (19) must hold whenever ξ and s are both ord(1), we find that it is appropriateto choose
K =1andK =logδ. Substituting intothe doubledmap(18),wefindthatthe termslabelled
1 0
① and② alwaysremainpartofthe dominantbalance,andthe firsttermtogrowtobethe same
sizeas① and② istheonelabelled③ . Thisoccurswhenǫδ3 =ord(δ);hence,itisappropriateto
take δ =ǫ−12 and K0 = 12log 1ǫ . This suggests that the asymptotic series breaks down due to
a change in dominant balance(cid:0)w(cid:1)hen X =ord[ǫ−21], which occurs at t=ǫn= 1log 1 + (1).
2 ǫ O
(cid:0) (cid:1)
2.4 Late time solution and matched aysmptotic expansions
Using the doubledmap, we determinedthe scalingofX andt, andhence xand n,atwhich the
early-time expansion loses asymptoticity. Hence, we apply the rescalings
x(n)= 2 +ǫ1/2ξ(m,s), (20)
3
t= 1log(1)+s, (21)
2 ǫ
n= 1 log(1) γ+m, (22)
2ǫ ǫ −
where γ is a constant chosen so that 0 γ < 2, and m n 1 (mod 2). The latter condition
≤ − ≡
is chosenfor conveniencein subsequentanalysis. Applying these rescalingsto (8) andusing the
method of multiple scales with discrete fast time variable m and continuum slow time variable
s, we recover
ξ(m+1,s)+ξ(m,s)=ǫ21 2 3ξ(m,s)2
(cid:20)9 − (cid:21)
ǫ 1ξ(m,s)+ dξ(m+1,s) ǫ32 ξ(m,s)2+ (ǫ2) (23)
− (cid:20)3 ds (cid:21)− O
1
Since (23) involves powers of ǫ2, we introduce a series expansion for ξ(m,s) in half-powers of ǫ
of the form
∞
k
ξ(m,s) ǫ2 ξk(m,s). (24)
∼
kX=0
8
Collecting terms of the same order in ǫ, we obtain
(1): ξ (m+1,s)+ξ (m,s)=0, (25)
0 0
O
O(ǫ12): ξ1(m+1,s)+ξ1(m,s)= 92 −3ξ0(m,s)2, (26)
and so on.
At leading order, we can solve (25) easily to find that ξ(0)(m,s) = ( 1)mP(s), where P(s) is
1 −
anarbitrarysmoothly-varyingfunction ins. At (ǫ2), we do notobtainasecularitycondition.
At the next order, however, we find the seculOarity condition for P(s) is given by P′(s) =
P(s) 9P3(s). Solving this yields
−
es
P(s)= , (27)
3 e2s+κ
p
p
where κ is an arbitrary constant, and we recall that ξ =( 1)mP(s).
p 0
−
Inordertodeterminethevalueoftheconstantκ ,weapplyVanDyke’smatchingcriterion(see,
p
forexample,[10])tomatchthebehaviourofthe solutioninthe late-timeregionwiththe known
behaviour in the early-time region.
Van Dyke’s matching criterion involves expanding the early time solution in late time variables
andequatingthiswithanappropriateexpansionoftheearlytimesolutioninlatetimevariables.
To determine κ , we express the one-term late time solution ξ (m,s) in terms of early time
p 0
variables n and t, and equate the leading-order behaviour of this expression with the leading-
order behaviour of the one-term early time solution X (n,t) in terms of late time variables m
0
and s. This gives the matching condition
( 1)mes ( 1)net
− − , (28)
9√ǫ ≡− 3√κp
which,using (20)-(22)andexploitingthe simplifyingassumptionthatm n 1 (mod 2),gives
− ≡
κ =9.
p
Bycontinuingtomatchpowersofǫ,weobtainaformforξ (m,s),withanappropriatesecularity
1
condition. Van Dyke’s matching condition can also be used at higher orders, and hence we are
abletodeterminesubsequenttermsinthe asymptoticseries,withthetwo-termexpansiongiven
by
ξ(m,s)=( 1)m es + 18−e2s ǫ12 + (ǫ). (29)
− 3√9+e2s 162+18e2s O
Since the one-term early time approximation from (16), and the two-term late time approxi-
mation, (29), are both accurate up to (ǫ) in x, we can combine them to obtain a composite
O
asymptotic approximation that will also be uniformly valid to (ǫ). This is obtained by con-
O
verting both approximations into equivalent variables, adding them together, and subtracting
thematchingtermassociatedwithVanDyke’scriterion. Inthis case,wefindthatthe matching
termisidenticaltotheone-termearlytimeapproximation,andhencewerecovertheresultthat
(29) is also the uniformly-valid composite approximation.
In classicalboundarylayerproblems,it is unusualfor the uniformly-validcomposite solutionto
be completely identicalto one ofthe twosolutions involvedin the matching,asis the case here.
It is important to note that this does not imply that it would have been possible to obtain the
uniformly-valid solution without considering the early time and late time cases separately.
Knowing that X(0) = 0, we see that the initial condition in this case should take the form
ξ(0,0) = 0. However, if we take the late time solution from (29) and consider what happens
when n=0, and hence s= 1log(1) and ( 1)m = 1, we find that (29) becomes
−2 ǫ − −
√ǫ 18 ǫ 1
ξinitial =−3√9+ǫ ·1+ 162+−18ǫ ·ǫ2 +“O(ǫ)”. (30)
9
Figure4: Comparisonbetween the numerical solution x(n) and the two-term composite asymptotic expansion,
whichinthiscaseisthelate-timeexpansiongivenin(31),forǫ=0.02. Theexactsolutionisindicated
by filled circles, while the approximation is indicated by white circles. The point n = 1 log(1) is
2ǫ 2ǫ
indicatedbyathickline. Itisatthispointthat theearly-timeapproximationbreaks down;however,
thecompositeapproximationclearlyprovidesanaccurateapproximationevenoncethispointispassed.
Thisisneitheravalidasymptoticexpansion(sincebothtermsarethesamesize),noranobvious
way of representing the true initial condition of ξ(0,0) = 0. While the two-term late time
asymptotic expansion is equivalent to the composite approximation that is uniformly valid up
to (ǫ), it is not a Poincar´e expansion when s = 1log(1)+ (1). As a result, there is no
O −2 ǫ O
powerseriesansatz by which(29) couldbe obtaineddirectly fromthe initialcondition; we need
to use the method of matched asymptotic expansions to obtain the late time solution based
on matching with the early time solution, and then combine the two to obtain a composite
approximation.
Converting back into the variables x and n of the original problem, the composite expansion
based on (29) becomes
2 ( 1)nǫeǫn 18ǫ ǫ2e2ǫn 3
x(n)= − + − + (ǫ2). (31)
3 − 3√9+ǫe2ǫn 162+18ǫe2ǫn O
This composite approximation is compared with the numerical solution to the full problem in
Figure 4 for the case where ǫ = 0.02. While Figure 3 shows that the early time asymptotic
approximationsbecome invalid as n grows large,here we see that the composite approximation
remains valid over the entire domain.
Figure5illustratestheapproximationerrorofthecompositeexpansiongivenin(31)forarange
ofǫvalues,wherethe erroris quantifiedasthe infinity-normofthe difference betweenthe exact
solutionx(n)andthecompositeexpansion. FromFigure5,weseethatthe errorisproportional
3
to ǫ2 as ǫ 0, which is consistent with the form of the asymptotic expansion (31).
→
3 Discrete logistic equation with λ = 3 + ǫ2n and x(0) = 2
3
3.1 Consequences of introducing a slowly varying bifurcation param-
eter
In Section 2, we consideredthe behaviour of the discrete logistic equation when the bifurcation
parameter was taken to be a constant, λ = 3+ǫ. In this section, we consider the dynamic
problem described in 1.2, where λ is taken to be a slowly varying function of n. Specifically,
we will concentrate on a slow forward sweep through the bifurcation at λ = 3, defining our
10
