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