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STUDY OF MATHIEU EQUATION NEAR
STABILITY BOUNDARY
A Project Report
submitted in partial fulfillment of the
requirements for the degree of
Master of Technology
in
Computational Science
by
Mritunjay Kumar
Supercomputer Education and Research Centre
Indian Institute of Science
Bangalore - 560012
July, 2011
Acknowledgements
I take immense pleasure to express my deep gratitude towards my supervisor Dr.
Atanu Mohanty for his guidance, invaluable suggestions and constant encouragement
throughout the project. His suggestions motivated me to persistently pursue a problem
which apparently looked very difficult. It was a great privilege working with him. Aided
with his guidance, I could complete the project without much trouble and it was really a
smooth ride. I am extremely thankful to him for his untiring efforts.
I thank Prof. Govindrajan, chairman, Supercomputer Education Research Centre, IISc
who has always been very motivating. I thank the faculty members of IISc for their
insightful teaching throughout my coursework, their courses helped me learn many things.
I also thank my colleagues for sharing some nice moments during my stay at IISc. I am
also grateful to the office staff of SERC for helping me with various issues throughout my
stay at SERC.
Finally, I thank my parents and all my family members for their understanding, never
ending blessings, faith and support.
i
Abstract
For small values of the parameter q Dehmelt’s theory provides a very good approxima-
tion to the solution of Mathieu equation but not much study has been done previously for
higher q values. Thus an attempt has been made to study the behaviour of frequency and
amplitude variation for q near the stability boundary (i.e. for values of q close to 0.908).
Fourth order Runge-Kutta method and fast fourier transform have been used to obtain
simulation results which have been further verified by Hill’s method of solution.
ii
Contents
Acknowledgements i
Abstract ii
1 Introduction 1
2 Numerical Method 3
2.1 Classical Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Calculating determinant of a Tridiagonal matrix . . . . . . . . . . . . . . . 6
3 Mathieu equation 7
3.1 Mathieu equation and its solution . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Ion Motion Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.2 Equation of motion of an ion in Paul trap . . . . . . . . . . . . . . 11
3.2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Dehmelt’s approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Study of Mathieu Equation close to Stability boundary 19
4.1 Motivation for studying the behaviour . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Solution for high q values . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 Frequencies present in the spectra . . . . . . . . . . . . . . . . . . . 20
4.2 Floquet’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Hill’s Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Hill’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
iii
4.3.2 Evaluation of Hill’s Determinant . . . . . . . . . . . . . . . . . . . . 26
4.4 Variation of frequency with q . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Variation of amplitude with q . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Concluding Remarks 37
Bibliography 38
iv
List of Tables
4.1 Simulation Results (0.7 ≤ q ≤ 0.75) . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Simulation Results (0.76 ≤ q ≤ 0.83) . . . . . . . . . . . . . . . . . . . . . 33
4.3 Simulation Results (0.84 ≤ q ≤ 0.90) . . . . . . . . . . . . . . . . . . . . . 34
4.4 Simulation Results (0.902 ≤ q ≤ 0.9076) . . . . . . . . . . . . . . . . . . . 35
4.5 Simulation Results (0.9078 ≤ q ≤ 0.90805) . . . . . . . . . . . . . . . . . . 36
v
List of Figures
3.1 Three-dimensional Paul trap elecrtodes . . . . . . . . . . . . . . . . . . . . 10
3.2 Time vs displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Details of the high frequency part . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Stability plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Sample plot of solution to Mathieu equation . . . . . . . . . . . . . . . . . 15
3.6 Dehmelt’s approximation for Z displacement in Figure 3.3 . . . . . . . . . 18
3.7 Dehmelt’s approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 q = 0.908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 q = 0.90807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 q = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 q = 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 q = 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.6 q = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.7 q = 0.904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.8 q vs frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.9 q vs angular frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.10 q vs amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.11 q vs amplitude relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
vi
Chapter 1
Introduction
Mathieu equation is a linear differential equation of second order. It was first discussed
by Emile Leonard Mathieu in 1868 in connection with the problem of vibrations of an ellip-
tic membrane. Mathieu differential equations arise as models in many contexts, including
the stability of railroad rails as trains drive over them, seasonally forced population dy-
namics, ion motion in quadrupole ion traps, propagation of light in periodic media and the
phenomenon of parametric resonance in forced oscillators.
The canonical form for Mathieu differential equation is
y′′(x) + [a − 2q cos 2x]y(x) = 0
where a and q are constants.
Every point except infinity is a regular point of this equation. In certain circumstances
particular solutions of it are called Mathieu functions. The functions which occur in Math-
ematical Physics and which come next in order of complication to functions of hypergeo-
metric type are called Mathieu functions; these functions are also known as the functions
associated with the elliptic cylinder. They arise from the equation of two dimensional
wave motion; in general, the solution of differential equations that are separable in elliptic
cylindrical coordinates.
Closely related is Mathieu modified differential equation
y′′(u) − [a − 2q cosh 2u]y(u) = 0
1
which follows on substituting x = −iu.
The substitution t = cos x transforms Mathieu equation to the algebraic form
(1 − t2)y′′(t) − ty′(t) + [a + 2q(1 − 2t2)]y(t) = 0.
This has two regular singularities at t = −1, 1 and one irregular singularity at infinity,
which implies that in general (unlike many other special functions), the solutions of Mathieu
equation cannot be expressed in terms of hypergeometric functions.
2
Chapter 2
Numerical Method
The methods for the solution of the initial value problem
u′ = f(t, u), u(t0) = η0, t ∈ [t0, b] (2.1)
can be classified mainly in two types. They are (i) singlestep methods and (ii) multistep
methods. In singlestep methods, the solution at any point is obtained using the solution
at only the previous point. Thus, a general single step method can be written as
uj+1 = uj + hφ(tj+1, tj, uj+1, uj, h) (2.2)
where φ is a function of the arguments tj+1, tj, uj+1, uj, h and also depends on f . This
function φ is called the increment function. If uj+1 can be obtained simply by evaluating
the right hand side of (2.2), then the method is called an explicit method. If the right
hand side of (2.2) depends on uj+1 also, then it is called an implicit method.
Local Truncation Error
The true(exact) value u(tj) satisfies the equation
uj+1 = uj + hφ(tj+1, tj, u(tj+1), u(tj), h) + Tj+1
where Tj+1 is called the local truncation error or discretization error of the method.
3
Order of a Method
The order of a method is the largest integer p for which
∣
∣
∣
1
hTj+1
∣
∣
∣= O(hp).
In the literature of numerical analysis, we have many singlestep methods which have
different increment functions. A few among those is the family of Runge-Kutta methods.
The Runge-Kutta methods are an important family of implicit and explicit iterative meth-
ods for the approximation of solutions of ordinary differential equations. These techniques
were developed around 1900 by the German mathematicians Carl Runge (1856-1927) and
Martin Wilhelm Kutta (1867-1944). One member of the family of Runge-Kutta methods
is so commonly used that it is often referred to as “RK4”, “classical Runge-Kutta method”
or simply as “the Runge-Kutta method”. Below we describe this method in detail.
2.1 Classical Runge-Kutta Method
Consider the following Runge-Kutta method with four slopes
uj+1 = uj + W1K1 + W2K2 + W3K3 + W4K4 (2.3)
where
K1 = hf(tj, uj)
K2 = hf(tj + c2h, uj + a21K1)
K3 = hf(tj + c3h, uj + a31K1 + a32K2)
K4 = hf(tj + c4h, uj + a41K1 + a42K2 + a43K3).
The parameters c2, c3, c4, a21, . . . , a43 and W1, . . . , W4 are chosen to make uj+1 closer to
u(tj+1).
Expanding K2, K3, K4 in Taylor series about tj , substituting in (2.3) and matching the
4
coefficients of h, h2, h3 and h4, we obtain the following system of equations:
c2 = a21
c3 = a31 + a32
c4 = a41 + a42 + a43
W1 + W2 + W3 + W4 = 1
W2c2 + W3c3 + W4c4 =1
2
W2c22 + W3c
23 + W4c
24 =
1
3
W2c32 + W3c
33 + W4c
34 =
1
4
W3c2a32 + W4(c2a42 + c3a43) =1
6
W3c22a32 + W4(c
22a42 + c2
3a43) =1
12
W3c2c3a32 + W4(c2a42 + c3a43)c4 =1
8
W4c2a32a43 =1
24. (2.4)
We have 11 equations in 13 unknowns. Therefore, there are two arbitrary parameters.
Since the terms upto O(h4) are compared, the truncation error is of O(h5) and the order
of the method is 4. The simplest solution of the Equations (2.4) is given by
c2 = c3 =1
2, c4 = 1
W1 = W4 =1
6, W2 = W3 =
1
3
a21 =1
2, a31 = 0, a32 =
1
2, a41 = 0, a42 = 0, a43 = 1.
Thus the fourth order method (2.1) becomes
uj+1 = uj +1
6(K1 + 2K2 + 2K3 + K4)
K1 = hf(tj, uj)
K2 = hf(tj +1
2h, uj +
1
2K1)
K3 = hf(tj +1
2h, uj +
1
2K2)
K4 = hf(tj + h, uj + K3).
5
2.2 Calculating determinant of a Tridiagonal matrix
An =
a11 a12 0 . . . . . . 0
a21 a22 a23 0 . . . 0...
......
......
...
0... 0 a(n−1)(n−2) a(n−1)(n−1) a(n−1)n
0 . . . . . . 0 an(n−1) ann
We denote
dn = det An
Therefore
dn = anndn−1 − an(n−1)a(n−1)ndn−2.
For starting the iteration we need
d0 = 1 and d−1 = 0
Complexity:
Thus for calculating the determinant of a tridiagonal matrix by the above scheme we
perform 3n multiplications and n additions.
6
Chapter 3
Mathieu equation
3.1 Mathieu equation and its solution
u′′(ξ) + (au − 2qu cos 2ξ)u(ξ) = 0 (3.1)
An equation of the form (3.1) is called Mathieu equation. In (3.1) au and qu
are called Mathieu parameters. The general solution of second order homogeneous linear
differential equation can be written as linear combination of two independent solutions. So
the general solution of Mathieu equation can be written as follows
u(ξ) = Auu1(ξ) + Buu2(ξ) (3.2)
where Au, Bu are constants determined by initial conditions of the differential equation.
The coefficients of dependent variable and its derivative in Mathieu equation are periodic
with period π. According to Floquet theory there exists at least one solution of the form
eµξf(ξ) with the property f(ξ + π) = f(ξ) and µ is complex constant of the form αu + iβu.
Since Mathieu equation remains unchanged even if we change ξ to −ξ, therefore the function
e−µξf(−ξ) is also its solution and is independent of eµξf(ξ) if αu 6= 0 or βu is not an integer
when αu = 0. So the general solution of Mathieu equation can be written as under
u(ξ) = Aueµξf(ξ) + Bue
−µξf(−ξ). (3.3)
7
In (3.3), since f(ξ) is periodic with period π, it can be written in series form using
Fourier theorem as follows
f(ξ) =∞
∑
n=−∞
C2n,uei2nξ. (3.4)
Now substituting f(ξ) from (3.4) in (3.3), we get
u(ξ) = Aueµξ
∞∑
n=−∞
C2n,uei2nξ + Bue
−µξ
∞∑
n=−∞
C2n,ue−i2nξ
(or)
u(ξ) = Aueαuξ
∞∑
n=−∞
C2n,uei(βu+2nξ)
+Bue−αuξ
∞∑
n=−∞
C2n,ue−i(βu+2nξ). (3.5)
If αu 6= 0, the solution of Mathieu equation is unstable otherwise it is stable. In stable
case if βu is irrational then solution is nonperiodic and if βu is rational but not an integer
then solution is periodic. Thus the stable solution of Mathieu equation can be written as
u(ξ) = A′
u
∞∑
n=−∞
C2n,u cos (2n + βu)ξ + B′
u
∞∑
n=−∞
C2n,u sin (2n + βu)ξ (3.6)
where A′
u = Au + Bu ,B′
u = i(Au − Bu) and βu can be found by the following continued
fraction
β2u = a +
q2
(βu + 2)2 − a − q2
(βu+4)2−a−q2
(βu+6)2−a−....
+q2
(βu − 2)2 − a − q2
(βu−4)2−a− q2
(βu−6)2−a−....
(3.7)
Here a, q are real parameters of the Mathieu equation.
8
3.2 Ion Motion Example
3.2.1 Paul trap
Three dimensional Paul trap is an ion trap in which electric field in X, Y and Z
directions varies linearly. We know that electric field is negative gradient of potential. So
to get linearly varying electric field in X, Y and Z directions we need to have potential as
quadratic in variables x, y and z.
Let−→E be electric field corresponding to potential φp then
−→E = −−→∇φ (3.8)
where−→∇ = i
∂
∂x+ j
∂
∂y+ k
∂
∂z.
For linearly varying−→E we should have potential φ as follows
φ = αx2 + βy2 + γz2 (3.9)
where α, β and γ are arbitrary constants.
We know that Gauss law from electrostatics says that divergence of an electric field in
charge free domain is zero.
i.e div−→E =
−→∇ .−→E = 0.
Substituting−→E from (3.8), we get the following
∇2φ = 0 (3.10)
where ∇2 =−→∇.
−→∇ =∂2
∂x2+
∂2
∂y2+
∂2
∂z2.
From (3.9) and (3.10) we get α + β + γ = 0 and commonly α, β and γ are chosen to be
9
1, 1 and −2 respectively. So from (3.9) potential is as follows
φ = x2 + y2 − 2z2
⇒ φ =
(
1
2(x2 + y2) − z2
)
−(
1
2(−x2 − y2) + z2
)
⇒ φ = φ1 − φ2 (3.11)
where φ1 = 12(x2 + y2) − z2 and φ2 = −φ1.
The Equation (3.11) says that we can get potential φ with two electrodes having po-
tentials φ1 and φ2 respectively on them. So the electrodes can be taken as surface given by
φ1 = k1 which is hyperboloid of one sheet and surface given by φ2 = k2 which is hyperboloid
of two sheets. Here k1, k2 are constants. The following figure shows the view of Paul trap
in which central electrode is hyperboloid of one sheet and upper end cap, lower end cap
together gives hyperboloid of two sheet.
Figure 3.1: Three-dimensional Paul trap elecrtodes
With the help of electrodes as shown in Figure 3.1 we can get potential of the form
Φ(x, y, z) = A + B(x2 + y2 − 2z2) (3.12)
where A, B are constants determined by boundary conditions applied to electrodes.
In Equation (3.12), let x = ρ cos θ and y = ρ sin θ then x2 + y2 = ρ2 and
Φ(ρ, z) = A + B(ρ2 − 2z2). (3.13)
10
The absence of θ in the above equation means that potential does not vary with respect
to theta.
In Paul trap, generally, the end cap electrodes are grounded and central electrode is
kept at required potential say Φ0. Here Φ0 = constant gives electrostatic field inside the
trap with which we can not trap ions because Earnshaw’s theorem states that collection of
point charges can not be maintained in a stable stationary equilibrium configuration with
the help of electrostatic field alone. So we take Φ0 = U +V cos Ωt which gives time varying
electric field. Where U is direct current voltage, V is alternating current voltage amplitude
and Ω is angular frequency. Let ρ0 be radius of central electrode in z = 0 plane and z0
is distance of either of end caps from z = 0 plane then from (3.13) above said boundary
conditions become
Φ(ρ0, 0) = Φ0 (3.14)
Φ(0, z0) = 0. (3.15)
Solving system of Equations (3.14) and (3.15) together gives
A =2z2
0Φ0
ρ20 + 2z2
0
, B =Φ0
ρ20 + 2z2
0
.
Substituting A, B, and Φ0 in (3.13) we get the following time varying potential
Φ(ρ, z) =
[
U + V cos Ωt
ρ20 + 2z2
0
]
(
ρ2 − 2(z2 − z20)
)
(3.16)
(or)
Φ(x, y, z) =
[
U + V cos Ωt
ρ20 + 2z2
0
]
(
x2 + y2 − 2(z2 − z20)
)
. (3.17)
3.2.2 Equation of motion of an ion in Paul trap
Let−→F be the force acting on an ion in Paul trap. Then it is given by Q
−→E , where Q is
charge on ion and−→E electric field at the point where the ion is located inside the trap. Let
~r = xi + yj + zk be the position of ion, then electric field at ~r is given by ∇Φ evaluated at
11
~r. So the force acting on ion is given by
−→F = Q
−→E (3.18)
where−→E =
[
U + V cos Ωt
ρ20 + 2z2
0
]
(
−2xi − 2yj + 4zk)
(3.19)
−→F = Q
[
U + V cos Ωt
ρ20 + 2z2
0
]
(
−2xi − 2yj + 4zk)
. (3.20)
According to Newton’s second law of motion
−→F = m
d2−→rdt2
(3.21)
where m is mass of ion. Now (3.20) and (3.21) together implies that
md2−→rdt2
= Q
[
U + V cos Ωt
ρ20 + 2z2
0
]
(−2xi − 2yj + 4zk). (3.22)
In Equation (3.22), let 2ξ = Ωt. Then
d~r
dt=
d~r
dξ
dξ
dt=
Ω
2
d~r
dξ
d2~r
dt=
Ω2
4
d2~r
d2ξ=
Ω2
4
d2
d2ξ(xi + yj + zk). (3.23)
Substituting the value ofd2~r
dtfrom (3.23) in (3.22) and comparing respective compo-
nents, we get
u′′(ξ) + (au − 2qu cos 2ξ)u(ξ) = 0 (3.24)
where u = x or y or z,
ax = ay =−az
2=
8QU
mΩ2(ρ20 + 2z2
0)and qx = qy =
−qz
2=
−4QV
mΩ2(ρ20 + 2z2
0).
The two figures shown below are examples of Ion motion.
12
0 0.2 0.4 0.6 0.8 1 1.2
x 10−4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−5
Time
Dis
pla
cem
en
t
X displacementZ displacement
Figure 3.2: Time vs displacement
0.8 1 1.2 1.4 1.6 1.8 2 2.2
x 10−5
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
x 10−6
Time
Dis
plac
emen
t
Z displacement
Figure 3.3: Details of the high frequency part
13
3.2.3 Stability Analysis
Figure 3.4: Stability plot
• Stability depends on the parameters a and q.
• For fixed values r0, w, U and V all ions with same M/e have the same operating
point in the stability diagram.
• a
q=
2U
Vand does not depend on m, all masses lie along the operating line a/q =
constant.
• On the q-axis (when DC=0 then a = 0) the stability is in 0 < q < 0.908.
• If U and V are changed proportionately in a way that a/q remains constant then
various masses can be brought in the stability region.
14
3.3 Dehmelt’s approximation
The solution of Equation (3.24) with a = −0.015, q = 0.2 looks like the following:
0 10 20 30 40 50 60 70 80 90 100−20
−15
−10
−5
0
5
10
15
20
Time
Am
plitu
de
Figure 3.5: Sample plot of solution to Mathieu equation
In the figure (3.5) we can see that there are two kinds of oscillating quantities, one
is slowly oscillating with high amplitude and another is rapidly oscillating quantity with
small amplitude. According to Dehmelt’s theory for small a and q the solution of Equation
(3.24) can be written as follows
u(ξ) = U(ξ) + ζ(ξ) (3.25)
where U(ξ) is a slowly varying quantity and ζ(ξ) is rapidly varying quantity.
Here we assume that ζ ′(ξ) U ′(ξ) and ζ ′′(ξ) U ′′(ξ). Averaging ζ(ξ) and its deriva-
tive over a cycle of ζ leaves zero but U(ξ) and its derivative does not vanish. Substituting
u(ξ) from (3.25) in (3.24) we get
U ′′(ξ) + ζ ′′(ξ) + (a − 2q cos 2ξ)(U(ξ) + ζ(ξ)) = 0. (3.26)
15
In (3.26) firstly we consider the significant rapidly oscillating parts ζ ′′(ξ) and (a −
2q cos 2ξ)U(ξ). The quantities (a − 2q cos 2ξ)ζ(ξ) and U ′′(ξ) are assumed to be small,
hence they are not significant to consider. Thus we get the following equation
ζ ′′(ξ) + (a − 2q cos 2ξ)U(ξ) = 0. (3.27)
Since U(ξ) changes slowly as compared to ζ(ξ), therefore in (3.27) we assume that U(ξ)
is constant and then after integrating (3.27) twice, we get rapidly oscillating quantity as
follows
ζ(ξ) = −q
2cos 2ξU(ξ). (3.28)
Substituting ζ(ξ) from (3.28) in (3.26), we get
U ′′(ξ) + aU(ξ) − 1
2aq cos 2ξU(ξ) + q2 cos2 2ξU(ξ) = 0. (3.29)
Over a cycle of ζ average of cosine is zero and cosine square is1
2. So averaging the
Equation (3.29) over a cycle of ζ we get
U ′′(ξ) + (a +q2
2)U(ξ) = 0 (3.30)
The solution of (3.30) is as follows
U(ξ) = U(0) cos ωsξ +U ′(0)
ωs
sin ωsξ (3.31)
where
ωs =
√
a +q2
2(3.32)
is Dehmelt slow frequency.
Now substituting the expression for ζ(ξ) from (3.28) in (3.25), we get
u(ξ) =[
1 − q
2cos 2ξ
]
U(ξ) (3.33)
16
In particular
U(0) =u(0)
[
1 − q
2
] . (3.34)
Differentiating (3.33), gives
u′(ξ) =[
1 − q
2cos 2ξ
]
U ′(ξ) + [q sin 2ξ]U(ξ). (3.35)
Setting ξ = 0 in (3.35), we get
U ′(0) =u′(0)
[
1 − q
2]. (3.36)
Putting U(0) from (3.34) and U ′(0) from (3.36) in (3.31), we get
U(ξ) =u(0)
[
1 − q
2]cos ωsξ +
u′(0)[
1 − q
2]ωs
sin 2ξ (3.37)
and finally putting this value of U(ξ) in (3.33), we get
u(ξ) =
[
1 − q
2cos 2ξ
]
[
1 − q
2]
[
u(0) cosωsξ +u′(0)
ωs
sin 2ξ]
. (3.38)
In the following two figures the green line shows Dehmelt’s approximation with wrong
denominator. In Figure 3.6 the green line is a better fit to the numerical plot obtained (in
blue). However in Figure 3.7, which has been obtained for different initial conditions, the
green line is a worse fit than the original Dehmelt’s approximation (in red).
Results for secular frequency obtained at 10V
omega Dehmelt = 5.4186e + 04 omega numerical = 5.4157e + 04
17
0 1 2 3 4 5 6
x 10−5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1x 10
−5
Time
Dis
plac
emen
t
Numerical SolutionDehmelt ApproximationDehmelt with wrong Denominator
Figure 3.6: Dehmelt’s approximation for Z displacement in Figure 3.3
0 1 2 3 4 5 6
x 10−5
−6
−4
−2
0
2
4
6x 10
−3
Time
Dis
plac
emen
t
Numerical SolutionDehmelt ApproximationDehmelt with wrong Denominator
Figure 3.7: Dehmelt’s approximation
18
Chapter 4
Study of Mathieu Equation close to
Stability boundary
4.1 Motivation for studying the behaviour
4.1.1 Solution for high q values
Below we have figures which describe the solution of Mathieu equation at q = 0.908
and q = 0.90807.
0 100 200 300 400 500 600 700 800 900
−800
−600
−400
−200
0
200
400
600
800
Time
Disp
lace
men
t
displacement vs Time
Figure 4.1: q = 0.908
19
Figure 4.1 clearly exhibits the Beat phenomena.
0 50 100 150 200 250 300 350 400−2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
2500
Time
Dis
plac
emen
t
Displacement vs Time
Figure 4.2: q = 0.90807
In Figure 4.2 we can see that the solution becomes unstable.
4.1.2 Frequencies present in the spectra
In Figure 4.3, i.e. for q = 0.05 we can see that there are only two frequencies present.
However as we move towards higher values of q, i.e, for q = 0.6, 0.8 (Figure 4.4 and 4.5)
there are many frequencies present and the solution is too complex to study in this region.
Again for q = 0.9, 0.904 (Figure 4.6 and 4.7) and still higher values primarily only two
frequencies are present thus making it easier to study the behaviour here.
20
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
5
10
15
20
25
30
X: 0.005341Y: 25.29
Frequency (Hz)
X: 0.3128Y: 0.378
X: 0.3242Y: 0.2827
Am
plitu
de
Amplitude Vs Frequency
Figure 4.3: q = 0.05
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
X: 0.07324Y: 7.544
Frequency (Hz)
Am
plitu
de
X: 0.7095Y: 0.02445
X: 0.563Y: 0.083
X: 0.2441Y: 1.814
X: 0.3922Y: 0.7362
Amplitude vs Frequency
Figure 4.4: q = 0.6
21
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
X: 0.4272Y: 1.118
Frequency (Hz)
Am
plitu
de
X: 0.5264Y: 0.3413 X: 0.7462
Y: 0.04829
X: 0.209Y: 5.309
X: 0.1099Y: 11.05 Amplitude vs Frequency
Figure 4.5: q = 0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
5
10
15
20
25
30X: 0.1465Y: 29.83
Frequency (Hz)
X: 0.4639Y: 3.547
Am
plit
ud
e
X: 0.4913Y: 2.522
X: 0.1724Y: 28.38
Figure 4.6: q = 0.9
22
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
X: 0.1495Y: 46.83
Frequency (Hz)
X: 0.1694Y: 35.75
Am
plitu
de
X: 0.4684Y: 4.514
X: 0.4868Y: 3.833
Figure 4.7: q = 0.904
4.2 Floquet’s Theory
This method is applicable to any linear equation with periodic coefficients which are
one-valued functions of the independent variables.
Let g(z), h(z) be a fundamental system of solution of Mathieu Equation (or indeed of
any linear equation in which the coefficients have period 2π), then if F (z) be any other
integral of such an equation, we must have
F (z) = Ag(z) + Bh(z), A, B are constants.
The solutions may not be identical with g(z), h(z) respectively as the solution of an
equation with periodic coefficients is not necessarily periodic. For example,
u′(z) = (1 + cot z)u(z)
⇒ u = ez sin z.
23
Therefore g(z + 2π) = α1g(z) + α2h(z) and h(z + 2π) = β1g(z) + β2h(z), then
F (z + 2π) = (Aα1 + Bβ1)g(z) + (Aα2 + Bβ2)h(z).
Consequently F (z + 2π) = kF (z), where k is a constant. If k is chosen so that
Aα1 + Bβ1 = kA (4.1)
Aα2 + Bβ2 = kB (4.2)
has a solution other than A = B = 0 iff
∣
∣
∣
∣
∣
∣
α1 − k β1
α2 β2 − k
∣
∣
∣
∣
∣
∣
= 0
and if k is taken to be either root of this equation, the function F (z) can be constructed
so as to be a solution of the differential equation such that
F (z + 2π) = kF (z).
Defining µ by the equation k = e2πµ and writing φ(z) for e−µzF (z), we see that
φ(z + 2π) = e−µ(z+2π)F (z) = φ(z)
and hence the differential equation has a particular solution of the form eµzφ(z), where
φ(z) is periodic with period 2π.
For Hill’s or Mathieu General Equation
u(z) = c1eµzφ(z) + c2e
−µzφ(−z)
and µ is a definite function of a and q.
24
4.3 Hill’s Method of Solution
4.3.1 Hill’s Equation
u′′(z) + J(z)u(z) = 0
where J(z) is an even function of z with period π.
CASE I: z is real and for all z, J(z) can be expanded in the form
J(z) = θ0 + 2θ1 cos 2z + 2θ2 cos 4z + 2θ3 cos 6z + . . .
the constants, θn are known and
∞∑
n=0
θn converges absolutely.
CASE II: z is complex and J(z) is analytic in a strip of the plane (containing the real
axis) whose sides are parallel to the real axis. The expansion of J(z) in Fourier series
θ0 + 2
∞∑
n=1
θn cos 2nz is then valid throughout the interior of the strip and as before
∞∑
n=0
θn converges absolutely.
Defining θ−n = θn, we assume u(z) = eµz
∞∑
n=−∞
bne2nzi as a solution of Hill’s equation.
Substituting in the equation, we get
∞∑
n=−∞
(µ + 2ni)2bne(µ+2ni)z +∞
∑
n=−∞
θne2nzi
∞∑
n=−∞
bne(µ+2ni)z = 0
and equating the coefficients of powers of e2zi to zero, we get
(µ + 2ni)2bn +
∞∑
m=−∞
θmbn−m = 0 (n = . . . ,−2,−1, 0, 1, 2, . . .).
If we eliminate the coefficients bn (after dividing the typical equation by θ0 − 4n2 to
secure convergence) we obtain Hill’s determinantal equation
25
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
......
......
......
...
. . . (iµ+4)2−θ0
42−θ0
−θ1
42−θ0
−θ2
42−θ0
−θ3
42−θ0
−θ4
42−θ0
. . .
. . . −θ1
22−θ0
(iµ+2)2−θ0
22−θ0
−θ1
22−θ0
−θ2
22−θ0
−θ3
22−θ0
. . .
. . . −θ2
02−θ0
−θ1
02−θ0
(iµ)2−θ0
02−θ0
−θ1
02−θ0
−θ2
02−θ0
. . .
. . . −θ3
22−θ0
−θ2
22−θ0
−θ1
22−θ0
(iµ−2)2−θ0
22−θ0
−θ1
22−θ0
. . .
. . . −θ4
42−θ0
−θ3
42−θ0
−θ2
42−θ0
−θ1
42−θ0
(iµ−4)2−θ0
42−θ0
. . ....
......
......
......
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
= 0.
We write 4(iµ) for the determinant, so the equation determining µ is 4(iµ) = 0. For
Mathieu equation θ0 = a,−θ1 = q and θn = 0 for all n ≥ 2. Substituting these values in
above determinant, we get
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
......
......
......
...
. . . (iµ+4)2−a
42−a
q
42−a
0 0 0 . . .
. . . q
22−a
(iµ+2)2−a
22−a
q
22−a
0 0 . . .
. . . 0 q
02−a
(iµ)2−a
02−a
q
02−a
0 . . .
. . . 0 0 q
22−a
(iµ−2)2−a
22−a
q
22−a
. . .
. . . 0 0 0 q
42−a
(iµ−4)2−a
42−a
. . ....
......
......
......
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
= 0.
4.3.2 Evaluation of Hill’s Determinant
We shall obtain an extremely simple expression for Hill’s determinant, namely
4(iµ) ≡ 4(0) − sin2(π
2iµ
)
csc2(π
2
√
θ0
)
.
4(iµ) is conditionally convergent since the product of the principal diagonal elements
does not converge absolutely. We can however obtain an absolutely convergent determinant
26
41(iµ) by dividing by θ0 − (iµ − 2n)2 instead of θ0 − 4n2 and
41(iµ) = Bm,n
where Bm,m = 1 and Bm,n =−θm−n
(2m − iµ)2 − θ0, if m 6= n.
The absolute convergence of
∞∑
n=0
θn secures the convergence of the determinant [Bm,n]
except when µ has such a value that the denominator of one of the expression Bm,n vanishes.
From the definition of infinite determinant
4(iµ) = 41(iµ) limp→∞
p∏
n=−p
[θ0 − (iµ − 2n)2
θ0 − 4n2
]
and hence
4(iµ) = −41(iµ)sin π
2(iµ −
√θ0) sin π
2(iµ +
√θ0)
sin2(π2
√θ0)
Also
1. 41(iµ) is an even periodic function of µ with a period 2i,
2. 41(iµ) is an analytic function of µ (except at its obvious simple poles) which tends
to unity as real part of µ → ±∞
If we choose k such that the function D(µ) defined by
D(µ) = 41(iµ) − k[
cotπ
2(iµ +
√
θ0) − cotπ
2(iµ −
√
θ0)]
has no poles at the point µ = i√
θ0, then since D(µ) is an even periodic function of µ it
follows that D(µ) has no pole at any of the points 2ni ± i√
θ0, n ∈ I.
Therefore D(µ) is a periodic function of µ (with period 2i) having no poles and which
is obviously bounded as R(µ) → ±∞. Therefore by Liouville’s theorem D(µ) is a constant
and letting µ → ∞ we have
D(µ) = 1.
27
Therefore,
41(µ) = 1 + k[
cotπ
2(iµ +
√
θ0) − cotπ
2(iµ −
√
θ0)]
and hence
4(iµ) = −sin π
2(iµ −
√θ0) sin π
2(iµ +
√θ0)
sin2(π2
√θ0)
+ 2k cot(π
2
√
θ0
)
.
For µ = 0,
4(0) = 1 + 2k cot(π
2
√
θ0
)
,
which implies that
4(iµ) = 4(0) −sin2
(
π2iµ
)
sin2(
π2
√θ0
) .
The roots of Hill’s determinantal equation are therefore the roots of the equation
sin2(π
2iµ
)
= 4(0) sin2(π
2
√
θ0
)
For Mathieu equation,
4(0) =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
......
......
......
...
. . . 1 q
42−a
0 0 0 . . .
. . . q
22−a
1 q
22−a
0 0 . . .
. . . 0 q
02−a1 q
02−a0 . . .
. . . 0 0 q
22−a
1 q
22−a
. . .
. . . 0 0 0 q
42−a
1 . . ....
......
......
......
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
.
Therefore µ is given by
sin2(π
2iµ
)
= 4(0) sin2(π
2
√a)
,
28
For a = 0, we define 42 as
42 =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
......
......
......
...
. . . 1 q
42−a
0 0 0 . . .
. . . q
22−a
1 q
22−a
0 0 . . .
. . . 0 q −a q 0 . . .
. . . 0 0 q
22−a
1 q
22−a
. . .
. . . 0 0 0 q
42−a
1 . . ....
......
......
......
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
.
Hence, µ is given by
sin2(π
2iµ
)
= lima→0
4(0) sin2(π
2
√a)
= −(π
2
)2
42
0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.920.1
0.11
0.12
0.13
0.14
0.15
0.16
q
freq
uenc
y
Hill’s VerificationNumerical Variation
Figure 4.8: q vs frequency
29
4.4 Variation of frequency with q
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
sqrt(qc−q)
2(1−
beta
)
y = 1.9*x − 0.0028
beta vs sqrt(qc−q) linear
Figure 4.9: q vs angular frequency
From the above figure we can see that 2(1−β) is found to vary with√
qc − q. Therefore
we have
2(1 − β) ∝√
qc − q.
where β is angular frequency and qc is critical value of q.
Also from the Figure 4.9 we can see that the numerical data and the linear fit do not
agree in the left lower portion of the graph. This can be attributed to the truncation errors
arising out of truncating the data to 4 decimal places.
4.5 Variation of amplitude with q
30
0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.920
100
200
300
400
500
600
700
800
Am
plitu
de
q
q vs Amplitude
Figure 4.10: q vs amplitude
0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3
q
1 / (
Am
plitu
de
)2
y = − 0.027338*x + 0.024821
numerical data linear
Figure 4.11: q vs amplitude relation
31
From the above data we can see that(in the stability region)
1
A2= mq + k (4.3)
where m and k and are constants. Alternatively,
A ∝ 1√qc − q
.
4.6 Simulation Results
FFT Dataq Frequency Amplitude Graphical Amplitude
0.7 0.09003 8.473 13.000.2289 3.0560.4074 0.88150.5478 0.16070.7263 0.03393
0.71 0.09155 8.96 13.430.2274 3.0280.4105 0.82420.5447 0.17390.7294 0.0256
0.72 0.09308 9.171 13.810.2258 2.9430.412 0.92040.5432 0.2036
0.73 0.0946 8.889 14.200.2228 3.3160.4135 0.99190.5417 0.2051
0.74 0.09766 8.314 14.600.2213 3.7680.415 1.0320.5402 0.2325
0.75 0.09918 9.557 15.050.2197 3.8270.4166 0.94730.5371 0.2277
Table 4.1: Simulation Results (0.7 ≤ q ≤ 0.75)
32
FFT Dataq Frequency Amplitude Graphical Amplitude
0.76 0.1007 9.937 15.600.2167 3.610.4196 1.0260.5356 0.26690.737 0.04077
0.77 0.1022 8.957 16.220.2151 4.4140.4211 1.1310.5341 0.29160.7401 0.038690.8514 0.01534
0.78 0.1053 10.45 16.850.2136 4.50.4227 0.99650.531 0.29520.7416 0.031470.8484 0.02052
0.79 0.1068 10.24 17.580.2106 4.7870.4257 1.1750.5295 0.34110.7431 0.042770.8469 0.01941
0.80 0.1099 11.05 18.390.209 5.3090.4272 1.1180.5264 0.34130.7462 0.04829
0.81 0.1114 10.63 19.300.206 5.7410.4303 1.2910.5249 0.43040.7492 0.04344
0.82 0.1144 12.2 20.450.2045 5.6430.4333 1.2370.5219 0.46610.7507 0.05277
0.83 0.1175 12.42 21.750.2014 6.8260.4349 1.2850.5188 0.45490.7538 0.05306
Table 4.2: Simulation Results (0.76 ≤ q ≤ 0.83)33
FFT Dataq Frequency Amplitude Graphical Amplitude
0.84 0.1205 12.41 23.300.1984 7.5630.4379 1.4140.5173 0.52080.7553 0.04375
0.85 0.1236 13.22 25.250.1953 8.5480.441 1.550.5142 0.62930.7599 0.05992
0.86 0.1266 15.06 27.800.1923 9.3770.444 1.5280.5096 0.70550.7629 0.04578
0.87 0.1297 16.75 31.300.1877 9.8940.4486 1.7940.5066 0.91870.766 0.062140.824 0.05017
0.88 0.1343 19.37 36.500.1846 12.750.4532 18.360.502 1.140.7706 0.069270.8209 0.04538
0.89 0.1389 23.12 45.500.1785 15.750.4578 2.4730.4974 1.6030.7767 0.079480.8148 0.05703
0.90 0.1465 29.83 68.300.1724 28.380.4639 3.5470.4913 2.5220.7813 0.12760.8087 0.06967
Table 4.3: Simulation Results (0.84 ≤ q ≤ 0.90)
34
FFT Dataq Frequency Amplitude Graphical Amplitude
0.902 0.148 36.17 78.700.1709 31.850.4654 3.7070.4883 2.620.7828 0.1180.8072 0.1311
0.904 0.1495 46.83 96.200.1694 35.750.4684 4.5140.4868 3.8330.7858 0.20650.8041 0.1456
0.906 0.1526 62.61 135.00.1663 55.170.47 5.5920.4837 5.080.7874 0.20590.8011 0.2182
0.907 0.1541 87.92 188.00.1633 77.150.473 9.0860.4822 8.40.7904 0.34310.8011 0.2453
0.9072 0.1541 87.07 208.00.1633 90.140.473 9.7920.4822 8.6850.7904 0.40870.7996 0.3448
0.9074 0.1556 108.9 237.50.1633 96.760.4745 9.930.4807 9.9220.7919 0.35380.7996 0.4084
0.9076 0.1556 131.8 283.50.1617 116.90.4745 13.250.4807 12.680.7919 0.58750.798 0.4666
Table 4.4: Simulation Results (0.902 ≤ q ≤ 0.9076)
35
FFT Dataq Frequency Amplitude Graphical Amplitude
0.9078 0.1572 182.4 374.50.1617 164.60.4761 16.290.4791 17.080.7935 0.69410.798 0.6033
0.90782 0.1572 194.8 3890.4761 17.50.4791 18.30.7935 0.7737
0.90784 0.1572 206.1 4060.4761 18.580.4791 19.380.7935 0.8451
0.90786 0.1572 215.6 4250.4792 20.27
0.90788 0.1572 222.7 4470.4791 20.93
0.9079 0.1572 226.6 472.40.4791 21.35
0.90792 0.1572 226.8 502.90.4791 21.6
0.90794 0.1572 223.6 540.70.4791 21.8
0.90796 0.1587 238.2 587.80.4761 24.31
0.90798 0.1587 316.1 650.40.4766 32.68
0.908 0.1587 417.1 739.30.4776 43.32
0.90801 0.1587 477.4 799.20.4776 49.54
0.90802 0.1587 545 877.20.4776 56.44
0.90803 0.1587 620.4 981.50.4776 64.06
0.90804 0.1587 704.3 11400.4776 72.47
0.90805 0.1587 797.2 14080.4776 81.73
Table 4.5: Simulation Results (0.9078 ≤ q ≤ 0.90805)
36
Chapter 5
Concluding Remarks
• Frequencies present in the spectra lead to the beat phenomena being observed in the
solution.
• sin(−π2f) and q vary linearly.
• The variation of q vs frequency follows the behaviour as predicted by Hill’s method.
• 2(1 − β) is directly proportional to√
qc − q.
• A is directly proportional to1√
qc − q.
• The constant m in the Equation (4.3) obtained does not vary with parameter a.
• The solution around q = 0.908 seems to be comprising of two quantities which are
slowly oscillating with high amplitude and two quantities which are rapidly
oscillating with small amplitude (based on the information obtained from Fourier
Transformation).
37
Bibliography
[1] Foot, C. J., Atomic Physics, Oxford University Press, 2005.
[2] Griffiths, D. J., Introduction to electrodynamics, Prentice Hall, 1999.
[3] http://ab-initio.mit.edu/wiki/index.php/Harminv
[4] Landau, L. D. & Lifshitz, E. M., Electrodynamics Of Continuous Media, Perga-
mon Press, Oxford 1984.
[5] Mandelshtam, V. A. & Taylor, H. S., Harmonic inversion of time signals, J.
Chem. Phys. 107 (17), 6756-6769, 1997. Erratum, ibid. 109 (10), 4128, 1998.
[6] Paul, W., Electromagnetic Traps For Charged and Neutral Particles, Nobel Lecture,
Dec 8, 1989.
[7] Whittaker, E. T. & Watson, G. N., A Course of Modern Analysis, Fourth Edition,
Cambridge University Press, 2002.
38