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Renormalization Group for nonlinear oscillators in the absence of linear
restoring force
Article in EPL (Europhysics Letters) · October 2010
DOI: 10.1209/0295-5075/91/60004
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Renormalization Group for nonlinear oscillators in the absence of linear restoring force
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September 2010
EPL, 91 (2010) 60004
doi: 10.1209/0295-5075/91/60004
www.epljournal.org
Renormalization Group for nonlinear oscillators in the absence
of linear restoring force
A. Sarkar(a) and J. K. Bhattacharjee
Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences - Salt lake, Kolkata 700098, India
received 7 July 2010; accepted in final form 8 September 2010
published online 14 October 2010
PACS
PACS
PACS
05.10.Cc – Renormalization group methods
47.20.Ky – Nonlinearity, bifurcation, and symmetry breaking
02.30.Mv – Approximations and expansions
Abstract – Perturbative Renormalization Group (RG) has been very useful in probing periodic
orbits in two-dimensional dynamical systems (Sarkar A., Bhattacharjee J. K., Chakraborty
S. and Banerjee D., arXiv:1005.2858v1 (2010)). The method relies on finding a linear center,
around which perturbation analysis is done. However it is not obvious as to how systems devoid
of any linear terms may be approached using this method. We propose here how RG can be done
even in the absence of linear terms. We successfully apply the method to extract correct results
for a variant of the second-order Riccati equation. In this variant the periodic orbit disappears as
a parameter is varied. Our RG captures this disappearance correctly. We have also applied the
technique successfully on the force-free Van der Pol-Duffing oscillator.
c EPLA, 2010
Copyright Renormalization Group in the study of nonlinear
ordinary differential equations was introduced by Chen
et al. [1]. It has been a very effective tool, since it
works by analyzing a direct perturbation expansion.
Being perturbative in nature, it works around a linear
dynamical system which has a center [2] —a family
of initial condition-dependent periodic orbits. We ask
the following question: What if the dynamical system
does not have a linear part? As an example, for the
usual nonlinear oscillator, having the equation of motion
ẍ + ω 2 x + λx3 = 0, the part ẍ + ω 2 x = 0 serves as the
linear system with a center, about which the perturbation
is done. The oscillator ẍ + λx3 = 0, on the other hand, has
no linear part about which perturbation can be carried
out. We first establish how the RG can be used in this
case, where apparently no perturbative technique is feasible. Having done that, we consider a generalization of the
second-order Riccati equation [3,4]. We write this damped
oscillator as ẍ + βλxẋ + λ2 x3 = 0, where λ is a coupling
constant and β a dimensionless number. This differential
equation, with slight variations has occurred widely in
literature under several names and in several studies,
like the generalized Emden equation [5–8], Lienard-type
oscillator [9] and has been the subject of extensive study
by Leach et al. [10], Mahomed and Leach [11], Bouquet
(a) E-mail:
et al. [12], Lemmer et al. [13], etc. The second-order
Riccati equation has β = 3. This oscillator has no linear
part and consequently not amenable to perturbation
theory. For β = 3 the equation is exactly solvable and
does not show oscillatory behavior [14]. For β 1,
however, it is known to have periodic solutions [14].
We use our modified RG to predict that the oscillatory
behavior seen for β 1 disappears through a vanishing of
the frequency of oscillation (time period diverges). This
happens at a critical value βc , for
which a second-order
calculation yields the value βc = 75/11. We carry out a
numerical integration which yields βc = 2.82, in very good
agreement with our predictions. We further show that for
β close to βc , the time period diverges as (β − βc )−1/2 .
The numerical results bear out our analytical results.
We further illustrate the effectiveness of our modified
RG methodology applying it on a variant of the so-called
Van der Pol-Duffing oscillator (ẍ + ẋ(x2 − 1) + λx3 = 0),
which again is devoid of any linear terms. We successfully
apply the modified RG method to extract the correct
dependence of the time period on the parameter λ.
We begin by demonstrating how the perturbative RG
works for the nonlinear oscillator,
ẍ + ω 2 x + λx3 = 0.
(1)
Perturbation theory begins with the expansion
x = x0 + λx1 + λ2 x2 + · · · .
amarta345@bose.res.in
60004-p1
(2)
A. Sarkar and J. K. Bhattacharjee
Inserting the above series in eq. (1) and equating the which integrate to A = A0 and Θ = 3λA2 τ /8ω + Θ0 .
coefficient of λn to zero for all n, we have
Inserting in eq. (12), we have
O(λ0 ) : ẍ0 + ω 2 x0 = 0,
(3)
3λA30
τ + Θ0
x(t) = A0 cos ωt +
8ω
O(λ1 ) : ẍ1 + ω 2 x1 = −x30 ,
(4)
2
3A
− 0 λ(t − τ ) sin(ωt + Θ)
(5)
O(λ2 ) : ẍ2 + ω 2 x2 = −3x20 x1 .
8ω
3
λA
0
If we take the initial condition as x(t0 ) = A0 and ẋ(t0 ) = 0,
+
{cos 3(ωt + Θ) − cos(ωt + Θ)}. (15)
32ω 2
we have x0 = A0 cos(ωt + Θ0 ) and solving for x1 from
ẍ1 + ω 2 x1 = −A30 cos3 (ωt + Θ0 )
A3
= − 0 (3 cos(ωt + Θ0 ) + cos 3(ωt + Θ0 )) , (6)
4
The final removal of the τ -dependence is done through the
choice τ = t and this gives
λA3 x(t) = A0 cos(Ωt + Θ0 ) +
cos 3(Ωt + Θ0 )
2
32Ω
(16)
− cos(Ωt + Θ0 ) + O(λ2 ),
we get as solutions of the system of eqs. (3) and (4) up to
O(λ)
3
A0
where
x = A0 cos(ωt + Θ0 ) + λ
{cos 3(ωt + Θ0 )
3λA2
32ω 2
Ω
=
ω
+
+ O(λ2 ).
(17)
8ω
3A20
Θ0
− cos(ωt + Θ0 )} −
t+
sin(ωt + Θ0 ) , (7) This is the standard result of amplitude-dependent oscilla8ω
ω
tor frequency [15] for this oscillator, which as shown above
where Θ0 = −ωt0 . The perturbative series solution is easily obtained by perturbative RG calculations.
not important from here
diverges when t t0 . In order to handle this divergence,
The next issue that we need to address is what happens
we introduce an arbitrary time τ and split the interval when ω = 0 and there is no obvious way doing perturbation
t + Θ0 /ω as t − τ + τ + Θ0 /ω. We now exploit the fact theory. A direct integration allows us to obtain the
that for a given trajectory, the present solution is inde- oscillation frequency [15] in this case which turns out to
pendent of where one puts the initial condition. Any be
point on the trajectory could have served as the initial
√ Γ(3/4)
1/4
1/4
condition and accordingly, we take a new initial condition
(λE) 1.18 (λE) ,
(18)
Ω0 = (2 π)
Γ(1/4)
at t = τ , writing
A0 (t0 ) = A(τ )Z1 (τ, t0 ),
(8)
Θ0 (t0 ) = Θ0 + Z2 (τ, t0 ),
(9)
where E is total energy of the oscillating particle. Our aim
is to modify the perturbative RG used in eq. (1) to handle
this situation. To this end, we write eq. (1) (with ω = 0)
as
(19)
ẍ + αλx2 x + λ x3 − αλx2 x = 0,
where the renormalization factors Z1 and Z2 are designed
to remove the divergence coming from the “past”. This
is done perturbatively by expanding the renormalization where x2 is the average of x(t)2 over a cycle. In the
factors as
above equation α is a constant number which we need to
(10) find out. If we ignore the term in parenthesis, we have an
Z1 = 1 + λα1 + λ2 α2 + · · · ,
equivalent linear oscillator with frequency
(11)
Z2 = λβ1 + λ2 β2 + · · · .
The choice of α1 = 0 and β1 = (3A2 τ /8ω + Θ0 /ω) removes
the divergent contribution of the past and we are left with
Ω2 = αλx2 .
(20)
If A is the amplitude of motion, i.e. ẋ = 0 at x = A, then
E = 14 λA4 . Clearly, our task is to find “α” and we proceed
to do it by treating λ(x3 − αλx2 x) as the perturbation.
The introduction of the term λαx2 x has the advantage
of making the parameter “α” dimensionless and further
makes the perturbation term λ(x3 − αλx2 x) “small”,
Imposing the condition that x(t) has to be independent of in a sense i.e. a certain part of x3 is subtracted. This
τ leads to the flow equations
replacement of x3 by x2 x is generally called equivalent
linearization [15,16]. In most cases it is used as a basis
dA
= 0,
(13) for self-consistent treatment, however, here we use it for a
dτ
RG-based perturbation theory. Now, rewriting eq. (19) as
dΘ 3λA2
=
,
(14)
ẍ + Ω2 x + λ x3 − αλx2 x = 0,
(21)
dτ
8ω
3A3
λ(t − τ ) sin(ωt + Θ)
x(t) = A(τ ) cos(ωt + Θ) −
8ω
λA3
+
{cos 3(ωt + Θ) − cos(ωt + Θ)} .
(12)
32ω 2
60004-p2
Renormalization Group for nonlinear oscillators in the absence of linear restoring force
we expand exactly as in eq. (2) and obtain at different We will now show that carried out to the next order
orders of λ
this method actually gives us an improvement. In the
process, the method of carrying out higher-order calcuẍ0 + Ω2 x0 = 0,
(22) lation becomes apparent. At O(λ2 ), we write eq. (24) as
2
(23)
ẍ1 + Ω2 x1 = − −x30 − αx20 x0 ,
A3
A
2
(1
+
cos
2Φ)
x
=
−
3
(cos 3Φ − cos Φ)
ẍ
+
Ω
2
2
32Ω2
ẍ2 + Ω2 x2 = − 3x20 x1 − αx1 x20 − 2αx0 x1 x0 . (24)
A3
With the initial condition x = A0 and ẋ = 0 at t = t0 , we
−α
(cos 3Φ − cos Φ)
32Ω2
have
A4
(25)
x0 = A0 cos(Ωt + θ0 ).
−2αA cos Φ
(cos
3Φ
−
cos
Φ)
32Ω2
At O(λ), we get
αA5
3A5
αA5
3
=
−
cos
Φ
+
+
cos Φ
3A0
A3
64Ω2
64Ω2 32Ω2
cos(Ωt + θ0 ) + 0 cos 3(Ωt + θ0 )
ẍ1 + Ω2 x1 = −
4
4
+ higher harmonics
αA30
cos(Ωt + θ0 ) ,
−
(26)
2
3(1 − α)A5
=
cos Φ + higher harmonics, (33)
64Ω2
leading to
where Φ = Ωt + θ. It should be clear from the derivation of
A3 3 α
the amplitude equation in eq. (13) that only the resonating
x1 = − 0
−
)
t sin(Ωt + θ0
2Ω 4 2
terms matter and not the higher harmonics. Also noting
2
2
A30
+
[cos(Ωt + θ0 ) − cos 3(Ωt + θ0 )] . (27) that Ω = αλA /2 to the lowest order, we can write the
32Ω2
solution to eq. (33), right up to the second order as
In exact analogy with eq. (1), we can now write x(t), up
3 1−α
A30 3 α
−
−
cos(Ωt
+
θ)
−
λ
x(t)
=
A
0
to O(λ)
2Ω 4 2 32 α
3
)
+
·
·
·
(34)
×
cos(Ωt
+
θ
A
0
{cos(Ωt + θ0 )
x(t) = A0 cos(Ωt + θ0 ) + λ
2
32Ω
which leads to
A3 3 α
θ0
−
− cos 3(Ωt + θ0 )} −
t+
3
1
3
1
2Ω 4 2
ω
+
α2
=
1+
.
(35)
4 32
2
24
(28)
× sin(Ωt + θ0 ) + O(λ2 ).
The actual frequency to this order comes from
The renormalization constants are introduced in an
identical fashion as before and the requirement that x(t)
has to be independent of τ yields
dA
= 0,
dτ
3 α
dθ
=
−
.
dτ
4 2
(29)
(30)
The frequency Ω as defined has to remain unaffected.
And this happens if eq. (30) has a fixed point which is
achieved for α = 3/2. Thus the solution x(t) is A cos Ωt +
O(λ) and hence x2 = A2 /2 = (E/λ)1/2 . The frequency
of the nonlinear oscillator at this order is found from
Ω2 = (3/2)λ(E/λ)1/2 and is given by
1/2
3
(λE)1/4 .
Ω=
2
Ω2 = αλx2 = αλx20 + 2λx0 x1 A2
+ 2αλ2 x0 x1 = αλ
2
A2 αλA2
= αλ
−
λA2
using eqs. (25) and (32)
2
32Ω2
A2 λA2
−
using eq. (20)
αλ
2 16 αλA2
1
1−
2
12
1
1
1/2 3
= (λE)
1+
1−
2
24
12
3
1
+··· ,
(36)
(λE)1/2
1−
2
24
leading to
(31)
Ω=
This is 3% above the exact result shown in eq. (18). As in
eq. (16), the final form of x1 is
A3
x1 =
[cos 3(Ωt + θ0 ) − cos(Ωt + θ0 )].
32Ω2
(32)
3
1
(λE)1/4 1 −
+··· ,
2
48
(37)
within 1% of the actual result. This shows how our
perturbation theory can be systematically improved and
how effective it is in giving reasonable answers.
60004-p3
A. Sarkar and J. K. Bhattacharjee
We now turn to the generalized second-order Riccati equaAs before we define the renormalization constants Z1
tion. This equation has been widely studied by both and Z2 as in eqs. (8) and (9). And a procedure identical
mathematicians and physicists for more than a century. to that which yielded eqs. (13) and (14), now leads to
This equation arises in a variety of mathematical probdA
lems [8,17]. And various mathematicians have studied this
= 0,
(50)
dτ
equation and its variants for specific choices of parame
ters [17–20]. Time and again it has been shown to arise
3 α β 2 A2
dθ
in a number of physical problems —like modeling fusion
=λ
− −
.
(51)
dτ
4 2 12 2Ω
of pellets [21], one-dimensional analogue of Yang-Mill’s
2
boson gauge theory [3], etc. We write down the equation as
The frequency remains at Ω, provided, α = 32 1 − β9 .
(38) By definition α > 0 and hence the restriction β < 3 for a
ẍ + βλxẋ + λ2 x3 = 0
periodic state to exist. This is the lowest non-trivial order.
and carry out the perturbation calculation as we have We need to take into consideration the next order as we
detailed before. The expansion of x is
did for the value of α when β was equal to zero.
For this we turn to eq. (44) and using eqs. (46), (47)
2
3
4
x = x0 + λx1 + λ x2 + λ x3 + λ x4 + · · · .
(39)
and (49) obtain
As done for the earlier example, we introduce a dimenβ(1 − β 2 )A4
βA4 2
sionless parameter α and rewrite eq. (38) as
α
−
3
+
2β
sin
2Φ
+
sin 3Φ
x3 =
36Ω3
32Ω3
3
2
2
2
2
ẍ + βλxẋ + λ αx x + λ x − αx x = 0, or
13β 2
βA4
1
3
2
2
2
+
−
sin 4Φ.
(52)
ẍ + Ω x = −βλxẋ − λ x − αx x ,
(40)
Ω3
144 × 15 80
where Ω2 = λ2 αx2 . We find at different orders
2
ẍ0 + Ω x0 = 0,
ẍ2 + Ω x2 = −
(44)
(45)
In the second term of the RHS of the above equation, we
insert the lowest-order value of λA2 /Ω2 , namely 2/α. With
this substitution and noting that, to the lowest order α is
(9 − β 2 )/6, we finally arrive at
(41)
ẍ1 + Ω2 x1 = −βx0 ẋ0 ,
2
Turning to eq. (45) and finding the coefficient of
cos(Ωt + θ), it is immediately possible to write the RG
phase flow for eq. (38) correct to O(λ2 ) as
3 α β 2 A2
dθ
=λ
− −
dτ
4 2 12 2Ω
λ2 A4 α
37β 2 (23β 2 + 9)(9 − β 2 )
+
−
3+
. (53)
2Ω3 64
9
96 × 18
x30
− αx20 x0
(42)
− βx1 ẋ0 − βx0 ẋ1 ,
(43)
ẍ3 + Ω2 x3 = − 3x20 x1 − αx1 x20 − 2αx0 x1 x0 −β(x2 ẋ0 + ẋ2 x0 + ẋ1 x1 ),
ẍ4 + Ω2 x4 = β(x0 ẋ3 + x1 ẋ2 + x2 ẋ1 + x0 ẋ3 )
+ − 3x0 x21 − 3x20 x2 + αx2 x20 +2αx1 x1 x0 + αx0 x0 x2 .
We can now write down the solutions at the first two orders
as
x0 = A0 cos(Ωt + θ0 ),
x1 = −β
(46)
A20
[sin 2(Ωt + θ0 ) − 2 sin(Ωt + θ0 )].
(47)
6Ω
The divergence comes at the next order
3 α β2
ẍ2 + Ω2 x2 = A30 − + +
cos(Ωt + θ0 )
4 2 12
+ non-resonating terms.
(48)
dθ λA2 9 − β 2 − 6α 23β 2 + 9
3
37β 2
=
−
−
+
. (54)
dτ
2Ω
12
144
32
288
The critical value βc of β up to which
a positive α can
be found to give dθ/dτ = 0 is βc = 75/11 2.61. We
thus conclude from a calculation good to the fifth order in
amplitude, that a periodic state will be found in eq. (38)
for β < βc (= 2.61) and that the time period will start
to diverge as βc is approached. From the fixed point of
eq. (54) we find α = (225 − 33β 2 )/144 and hence a time
period of
To order O(λ2 ), we have
λβA20
[sin 2(Ωt + θ0 )
6Ω λ2 A30
3 α β2
− 2 sin(Ωt + θ0 )] +
− + +
2Ω
4 2 12
θ0
× t+
(49)
sin(Ωt + θ0 ).
Ω
2π
2π
=
∝
T=
1/2
Ω
[αλx2 ]
x = A0 cos(Ωt + θ0 ) −
75
−β
11
−1/2
,
(55)
which diverges in a characteristic fashion, (βc − β)−1/2 , as
β approaches βc .
We have carried out the numerical simulation of eq. (38)
for different values of β. As β is increased, the time period
60004-p4
Renormalization Group for nonlinear oscillators in the absence of linear restoring force
0.2
Now we consider the so-called Van der Pol-Duffing oscillator without the forcing term, given by the equation
β=0.65
β=1.00
β=1.30
β=1.95
β=2.60
0.15
ẍ + ẋ(x2 − 1) + λx3 = 0,
Y
0.1
0.05
0
-0.05
-0.1
-1.5
-1
-0.5
0
0.5
1
1.5
X
(56)
where and λ are both small. This is often referred to
as the Van der Pol-Duffing oscillator [22], in our case
though there is no forcing term. This equation is yet
another example of an oscillator without any linear term.
As outlined above, to carry out perturbative RG we have
to rewrite eq. (56) as
ẍ + ẋ(x2 − 1) + αλx2 x + λ x3 − αx2 x = 0, or
ẍ + Ω2 x = −ẋ(x2 − 1) − λ x3 − αx2 x ,
(57)
where Ω is given by
Fig. 1: Solutions in phase space for different values of β.
Ω2 = λαx2 .
(58)
We expand x as in eq. (39) and find at different orders
increases as well and, as can be concluded from fig. 1,
it becomes extremely large as β approaches the critical
value, 2.61. Figure 1 shows the phase space evolution of
the oscillator for five different values of β. We notice that
the smaller the value of β, the larger is the amplitude of
the periodic orbit. For ẋ < 0, i.e. the lower part of the
orbit has a convexity, the tangent to which approaches
asymptotically towards the x-axis as the value of β
approaches the critical value βc = 2.61. As β approaches
βc the velocity becomes slower and slower while crossing
the y-axis (x = 0), and this ultimately leads to the time
period divergence, through which the periodic solution
vanishes.
Leach et al. [10] have considered the Riccati equation in the form ẍ + xẋ + αx3 = 0 and have carried out
extensive studies on its solution for different values of α.
They have used a self-similar transformation to obtain
a different form of the equation of motion in the new
phase space (ζ = xt; η = ẋt2 ), where the potential (V (ζ) =
αζ 4 /4 − ζ 3 /3 + ζ 2 ) is much more transparent to analysis.
Further, for a value of α = 1/8, V (ζ) is seen to have a
stationary point of inflection which corresponds to the
critical value where periodic solution vanishes. Recently,
Chandrasekhar et al. [8] have found the general solutions
to the second-order Riccati equation, ẍ + αxẋ + βx3 = 0,
for arbitrary values of α and β. Choosing suitable canonical transformations for each of the three cases, α2 < 8β,
α2 = 8β, and α2 > 8β, they write down their respective
general solutions. From their results as well, it is apparent that the transition from periodic to non-periodic solutions occurs at α2 = 8β. Noting that α as defined
by
Leach et al. is related to β defined here as β = 1/α,
we see that the critical value according
to both them
√
and Chandrasekhar et al. is βc = 8 = 2.83. Numerical
simulations put the critical value of β at 2.82. Keeping
in mind that our approach is perturbative, the value of
βc = 2.61 obtained through perturbative RG up to O(λ2 )
is in fairly good agreement with previous literature and
numerics.
O(0 λ0 ) :
ẍ0 + Ω2 x0 = 0,
O(1 λ0 ) :
ẍ1 + Ω2 x1 = −ẋ0 (x20 − 1),
0 1
O( λ ) :
2
ẍ2 + Ω x2 = −
(59)
x30
− αx20 x0
(60)
.
(61)
Now we can write down the solutions to the above set of
equations and up to first order in perturbation parameters
we have
3
A0
(sin 3Ωt − 3 sin Ωt)
x(t) = A0 cos Ωt − 32Ω
3
A0
A0 A20
+
−1 +λ
(cos 3Ωt − 3 cos Ωt)
2
4
32Ω2
A3 3 α
−
− 0
t sin Ωt .
(62)
2Ω 4 2
Proceeding from now on as earlier, we arrive at the RG
flow equations, right up to first order in the perturbation
parameters and λ, given by
A0 A20
dA
= −
−1 ,
(63)
dτ
2
4
dθ
A30 3 α
=λ
−
.
(64)
dτ
2Ω 4 2
The frequency remains at Ω, provided the RHS of eq. (64)
vanishes and we have, α = 3/2. From the amplitude flow,
eq. (64), it can be concluded that the oscillator exhibits
limit cycle oscillations
with √radius 2. So from eq. (58),
we have Ω = λα(A2 /2) = 3λ. Numerical simulations
of the oscillator confirm the above results as is quite
evident from fig. 2. The figure shows time periods for
different values of λ. And it is clear that the simulated data
compares extremely well with the 1st-order perturbative
RG result.
To conclude, we again emphasize that although doing
perturbative RG in systems with linear terms is obvious, its not so in cases without any linear terms. We
60004-p5
A. Sarkar and J. K. Bhattacharjee
60
Simulated Data
Theoritical Data
55
T (in arbitrary units)
50
45
40
35
30
25
20
15
10
0
0.01
0.02
0.03
0.04
0.05
λ
0.06
0.07
0.08
0.09
0.1
Fig. 2: Dependence of time period on λ. Comparison between
RG result and simulations.
have in this paper borrowed ideas from equivalent linearization [14–16,23] and combined with RG approach to
successfully reproduce results in few such cases. We have
shown that the disappearance of the periodic solution
to the generalized second-order Riccati equation is satisfactorily explained with the results obtained from the
RG analysis. We also capture the correct behavior of the
limit cycle appearing in a force-free Van der Pol-Duffing
oscillator. Here it can be said that this technique of
doing perturbative RG in the absence of linear terms
may find applications in a wide range of problems where
the absence of linear terms makes them inaccessible to
perturbative techniques.
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