7. INITIAL VALUE PROBLEMS:
THE STRAIGHTFORWARD EXPANSION
We show that the straightforward expansion method, successfully applied
in static problems, does not work in initial value problems.
Example: The Raileigh-Duffing oscillator
 x(t ) + ω 2 x(t ) − µ x (t ) + bx 3 (t ) + cx3 (t ) = 0

 x ( 0 ) = a0 , x ( 0 ) = 0
Rescaling:
µ → εµ , x → ε 1/ 2 x .
Series expansion:
x(t ; ε ) = x0 (t ) + ε x1 (t ) + ε 2 x2 (t ) +
1
Perturbation equations:
2

x
(
t
)
+
ω
x0 (t ) = 0
0  0
ε :
 x0 ( 0 ) = a0 , x0 ( 0 ) = 0
3
3

x
(
t
)
+
x
(
t
)
=
x
(
t
)
bx
(
t
)
−
cx
(t )
ω
µ

1
0
1
0
0
0
1
ε :
 x1 ( 0 ) = 0, x1 ( 0 ) = 0
Generating solution:
x0 = (a0 / 2) eiω t + c.c.
ε -order equation:
 x1 (t ) + ω0 x1 (t ) = f1 eiω t + f3 e3iω t + c.c.

 x1 ( 0 ) = 0, x1 ( 0 ) = 0
where:
1
3
3

f1 :=  i µω - iω 3ba02 - ca02  ,
2
4
4

a02
f3 :=
iω 3b - c
8
(
)
2
Solution to the ε -order equation:
1
1
1
i (ωt +ϑ1 )
iωt
x1 (t ) =
a1 e
−i
f1 t e − 2 f3 e3iωt + c.c.
2 2
ω
8ω
complementary solution
secular term
non-secular term
The secular term diverges in time. The series is not uniformly valid,
since O(ε x1 / x0 ) ≥ 1 , i.e. x1 is not a small correction of x0. Therefore, the
straightforward method has not practical utility in initial value problems.
3
8. THE MULTIPLE SCALE METHOD: BASIC ASPECTS
The MSM is probably the most powerful perturbation method able to
furnish uniformly valid expansions for oscillatory problems.
Basic idea
The response of a weekly nonlinear system, can be considered as a periodic
signal slowly modulated on slower scales. Example in nature: the
temperature in a fixed site varies periodically on a daily-scale, but it is
modulated, in amplitude and phase, on a yearly-scale, and, in turn, on a
century-scale.
4
Introducing independent time-scales
• Rayleigh-Duffing oscillator :
 x(t ) + ω 2 x(t ) − µ x (t ) + bx 3 (t ) + cx3 (t ) = 0

 x ( 0 ) = a0 , x ( 0 ) = 0
We assume that the variable x (t ) depends on several independent time
scales, defined as:
t0 := t , t1 := ε t , t2 := ε 2t , Rules for derivatives:
Since x(t ) = x(t0 (t ), t1 (t ), t2 (t ),) , the chain rule furnishes:
x (t ) =
∂x
∂x
2 ∂x
+ε
+ε
+
∂ t0
∂t1
∂t2
Hence, formally:
∞
D = d 0 + ε d1 + ε d 2 + = ∑ ε d k ,
1
2
k
k =0
d
∂
D := , d k :=
∂tk
dt
5
Similarly, for second-order derivative:
D 2 = (d 0 + ε 1 d1 + ε 2 d 2 + ) 2 = d 02 + 2ε d 0 d1 + ε 2 (d12 + 2 d 0 d 2 ) + Rescaling:
µ → εµ , x → ε 1/ 2 x
Series expansion:
x (t ; ε ) = x0 (t0 , t1 , t2 ,) + ε x1 (t0 , t1 , t2 ,) + 6
Perturbation equations:
2
2

d
+
ω
x
x0 = 0
0  0 0
ε :
 x0 ( 0 ) = a0 , d 0 x0 ( 0 ) = 0
2
2
3
3

d
+
=
−
2
d
d
+
d
−
(d
)
−
x
x
x
x
b
x
cx
ω
µ
 0 1
1
0 1 0
0 0
0 0
0
ε1 : 
 x1 ( 0 ) = 0, d 0 x1 (0) = − d1 x0 ( 0 )
d 02 x2 + ω 2 x2 = −(2 d 0 d 2 x0 + d12 x0 + 2 d 0 d1 x1 )

2
+ µ (d1 x0 + d 0 x1 ) − 3b(d 0 x0 ) 2 (d1 x0 + d 0 x1 ) − 3cx02 x1
ε :

x2 ( 0 ) = 0, d 0 x2 (0) = − d1 x1 ( 0 ) − d 2 x0 ( 0 )

where xk (0) is a shortening for xk (0, 0,) .
Note: the perturbation equations are partial differential equations,
although the original equations are ordinary differential equations.
7
Introducing a time-dependent amplitude
Generating solution:
x0 = a (t1 , t2 ,) cos(ω t0 + θ (t1 , t2 ,))
= A ( t1 , t2 ,...) eiω t0 + c.c.
where:
1
A ( t1 , t2 ,...) := a ( t1 , t2 ,...) eiθ ( t1 ,t2 ,...)
2
Note: x0 is periodic on the fast t0- scale, and modulated on the slower
scales by a complex quantity A or, equivalently, by two real unknowns,
a and θ.
By enforcing the initial conditions, it follows:
a (0) = a0 , θ (0) = 0
8
ε-order equations:
Since:
x0 = A ( t1 , t2 ,...) eiωt0 + c.c.
d 0 d1 x0 = iω d1 A eiωt0 + c.c.,
x03 = ( A eiωt0 + A e − iωt0 )3 = A3 e3iωt0 + 3 A2 A eiωt0 + c.c.
(d 0 x0 )3 = (iω A eiωt0 − iω A e − iωt0 )3
= −iω 3 A3 e3iωt0 + 3iω 3 A2 A eiωt0 + c.c.
then:
d 02 x1 + ω 2 x1 = f1 eiω t0 + f3 e3iω t0 + c.c.

 x1 ( 0 ) = 0, d 0 x1 (0) = − d1 A(0) + c.c.
where:
(
)
f1 := −2iω d1 A + iω µ A − 3 iω 3b + c A2 A,
(
)
f3 := iω 3b − c A3
9
Eliminating secular terms:
The resonant forcing-term of frequency-ω would lead to secular terms
t0 exp(iω t0 ) to appear in the solution. To remove them, f1 = 0 must be
enforced, i.e.:
1
3 2
c 2
d1 A = µ A +  −ω b + i  A A
ω
2
2
This is a nonlinear differential equation governing the modulation on the
t1 -scale; it is called the (first-order) Amplitude Modulation Equation
(AME).
Real form of AME:
Since d1 A = (1/ 2) ( d1 a + ia d1 ϑ ) eiθ , by separating real and imaginary
parts, the complex AME furnishes two real equations:
10

1 
3 2 2
d
a
=
a
µ
−
ω ba 

 1
2 
4


a d ϑ = 3 c a 3
 1
8ω
to be sided by a (0) = a0 , θ (0) = 0 .
Coming back to the original variables:
By truncating the analysis at this order (first-order perturbation
solution):
o The dependence on t2 , t3 , must be ignored, i.e. a = a (t1 ), θ = θ (t1 ) .
3/ 2
o By multiplying the real AME’s by ε and using
ε 1/ 2 a → a, εµ → µ , ε d1 → D , the perturbation parameter is
reabsorbed and return to the true time t is performed.
11
Reduced dynamical system:
The two real AME can be solved in sequence; first, the amplitude a(t) is
drawn by integrating:
1 
3

a = a  µ − ω 2ba 2 
2 
4

which represents a one-dimensional reduced dynamical system, of type
a = F (a, µ ) .
Successively, the phase θ(t) is determined by integrating:
aϑ =
3c 3
a
8ω
Note: the real amplitude-equation captures the essential dynamics of
the system; the phase equation describes a complementary aspect.
12
Steady-solutions
The AME admit two steady solutions a (t ) = const =: as :
o Trivial solution:
as = 0 ∀µ , ∀ϑ
which describes the trivial equilibrium path x=0 ∀ µ.
o Periodic solution:
3
4
µ = ω 2bas2 , ϑ = κ t
3
8
κ =: cas2
which describes the limit cycle x(t ) = a0 cos (ω + κ ) t  , where κ is
the frequency correction, and Ω = ω + κ the nonlinear frequency.
Note: The MSM filters the fast dynamics, so that a periodic x-motion
appears as an equilibrium a-position
13
Stability of steady-state solutions
Introducing a perturbation:
To analyze stability of the steady solutions, we put:
a (t ) = as + δ a (t )
with δ a (t ) a small perturbation superimposed to the steady amplitude.
Variational equation:
By linearizing the equation in δ a (t ) , the variational equation follows:
1
2
9
4
δ a (t ) = ( µ − ω 2bas2 ) δ a(t )
whose solution is:
1
2
9
4
δ a(t ) = δ a(0) exp[ ( µ − ω 2bas2 )t ]
14
• Stability of the trivial solution:
By substituting as = 0 in the solution of the variational equation, it follows:
1
δ a(t ) = δ a(0) exp[ µ t ]
2
When t → ∞ :
δ a (t ) → δ a (0) if µ < 0 , i.e. the equilibrium is (asymptotically) stable;
δ a (t ) → ∞ if µ > 0 , i.e. the equilibrium is unstable.
Bifurcation diagrams and orbits for (a) supercritical and (b) subcritical Hopf bifurcations.
15
• Stability of the periodic solution:
By substituting as = 4µ /(3bω 2 ) in the solution of the variational equation, it
follows:
δ a (t ) = δ a (0) exp( − µ ) .
Hence, the limit-cycle is stable if the bifurcation is supercritical (a) and
unstable if the bifurcation is subcritical (b).
µ<0
µ=0
x2
µ>0
x2
x2
x
a)
x1
x2
x1
x1
x(t ; µ)
x2
x2
R
N
µ
b)
x1
x1
x1
16
9. THE MULTIPLE SCALE METHOD: ADVANCED
TOPICS
Usually, a first-order solution is sufficient to describe limit-cycles and their
stability. However, there exist problems in which a higher-order solution is
necessary to describe qualitatively new aspects. We illustrate how to get a
second-order solution for the Rayleigh-Duffing oscillator.
Moving to higher-orders
ε-order perturbation equation:
d 02 x1 + ω 2 x1 = f1 eiω t0 + f3 e3iω t0 + c.c.

 x1 ( 0 ) = 0, d 0 x1 (0) = − d1 A(0) + c.c.
where f1 = 0 to avoid secular terms. By solving it:
17
1 c

x1 = B ( t1 , t2 ,...) e +  2 − iω b  A3 e3iωt0 + c.c.
8ω

where B ( t1 , t2 ,...) is an arbitrary function of the slower scales, constrained
to satisfy the initial condition.
iω t0
o To simplify the analysis, we ignore this arbitrary function, by letting
iωt
B=0. Indeed, B ( t1 , t2 ,...) e 0 +c.c. repeats the generating solution.
o Since the initial conditions cannot be enforced at any order, we will
enforce them, as a whole, on the final solution (although this is an
inconsistent method).
ε2-order perturbation equation:
d 02 x2 + ω 2 x2 = −(2 d 0 d 2 x0 + d12 x0 + 2 d 0 d1 x1 )
+ µ (d1 x0 + d 0 x1 ) − 3b(d 0 x0 ) 2 (d1 x0 + d 0 x1 ) − 3cx02 x1
18
By ignoring the non-resonant terms (NRT), the various contributions are:
d 0 d 2 x0 = iω d 2 A eiωt0 + c.c. + NRT , d12 x0 = d12 A eiωt0 + c.c. + NRT ,
d 0 d1 x1 = NRT ,
d1 x0 = d1 A eiωt0 + c.c. + NRT , d 0 x1 = + NRT ,
d 0 x0 = iω A eiωt0 + c.c.,
(d 0 x0 ) 2 d1 x0 = ω 2 (2 AA d1 A − A2 d1 A) eiωt0 + c.c. + NRT ,
3 c

(d 0 x0 ) d 0 x1 = iω [−  2 − iωb  A3 A 2 ]eiωt0 + c.c. + NRT ,
8ω

2
3
1 c

x x = [  2 − iωb  A3 A 2 + ]eiωt0 + c.c. + NRT
8ω

2
0 1
2
where d1 A is known from the first-order AME, and d1 A ≡ d1 (d1 A) is
evaluated by differentiation:
19
1
3 2
c 2
d1 A = µ A +  −ω b + i  A A
2
2
ω
2
1
c
9
c
d12 A = µ 2 A − 3µ (bω 2 − i ) A2 A + (3b 2ω 4 − 2 − 4ibcω ) A3 A2
4
ω
4
ω
The ε 2 -order perturbation equation reads:
1
3 c
d 02 x2 + ω 2 x2 = [−2iω d 2 A + µ 2 A − i µ A2 A
4
2 ω
45 c 2 9 2 4
iω t0
3 2
+(
−
b
−
3
ibc
)
A
A
]e
+ c.c. + NRT
ω
ω
2
24 ω
8
Elimination of the secular terms:
1 2
3 c
45 c 2
9 2 3 3
2
3 2
d 2 A = −i
µ A−
µ
A
A
+
(
−
i
+
i
b
ω
−
bc
)
A
A
2
3
8ω
4ω
48 ω
16
2
which governs the evolution of A on the t2-scale.
20
The reconstitution method
To come back to the true time t, the t1- and t2-derivatives of A are
recombined as follows:
A = ε d A + ε 2 d A + 1
2
1
3
c
= ε [ µ A +  −ω 2b + i  A2 A]
ω
2
2
2
c
c
1
3
45
9 2 3 3
2
3 2
µ2A−
µ
ω
+ε 2 [ −i
A
A
+
(
−
i
+
i
b
−
bc
)
A
A ] +
2
3
8ω
4ω
48 ω
16
2
This is the second-order AME. By multiplying both members by ε 1/ 2 and
using ε 1/ 2 A → A, εµ → µ , the perturbation parameter is reabsorbed. Finally,
by letting A = a / 2exp(iθ ) , the real form follows:

1
3 2 2
3 c
1 2 2 3
 a = 2  µ − 4 ω ba  a − 16 ω 2 ( µ + 2 ω ba )a




2
2
aϑ = 3 c a 3 − 1 µ a + 3 (3b 2ω 3 − 5 c )a 5

ω3
8ω
8 ω
256
21
The response
Once the AME have been solved, the solution to the Rayleigh-Duffing
equations reads:
x(t ) = a(t ) cos(ω t + θ (t )) +
1 3
c
a (t ){ 2 cos[3(ω t + θ (t )] + ω b sin[3(ω t + θ (t )]} +
32
ω
where a(0),ϑ (0) follow from the initial conditions x ( 0 ) = a0 , x ( 0 ) = 0 .
22
Limit cycles and their stability
• Limit-cycles
The limit-cycles are the fixed points a(t ) = const =: as of the (real) AME.
They satisfy the following algebraic equation:
1
3 2 2
3 c
1 2 2 3
( µ + ω ba )a = 0
 µ − ω ba  a −
2
2
4
16 ω
2

which implicitly defines a curve on the ( µ , a ) -plane, for given values of
the auxiliary parameters. By solving it:
3 4bω 2 + bcas2 2 2
µ=
ω as
2
2
2 8ω − 3cas
or, by expanding for small amplitudes:
3 2 2 15
µ = bω as + bcas4 +
4
32
23
Numerical results
The most interesting case is c < 0 (soft elastic nonlinearities), in which
the second-order term subtracts in modulus from the first-order term.
Since equilibrium of the Raileigh-Duffing oscillator is governed by:
ω 2 xE + cxE3 = 0
two nontrivial equilibrium points xE := ±ω / c exist in this case.
The two points are connected by separatrices which prevent the limit
cycle to increase unboundedly (as instead occurs in the case c > 0 ).
These properties are not captured by the reduced dynamical system,
since xE = O(1) is too large for the asymptotic analysis ( x = O(ε 1/2 ) ) to
hold.
24
• Bifurcation diagram
Bifurcation diagram for ω = 1, b = 1, c = −10 ; --- exact second-order solution, --- expanded
second-order solution,
first-order solution, + numerical solutions;
heteroclinic bifurcation: a=0.316
25
• Numerical integrations of the original system:
Time-histories and orbits of the original system; ω = 1, b = 1, c = −10 : (a) µ = 0.02 ; (b)
µ = 0.035
Note: when the limit cycle collides with the non-trivial equilibrium
points, it disappears. Such a phenomenon is called a heteroclinic
bifurcation.
26