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Appendix 351

APPENDIX 1: Mathematica�R� (Version 6) notebook for sawtooth power-series solutions for the oscillator �t,t x � xm � 0.

This module builds sawtooth power series solutions of the oscillator.The module is easy to modify on polynomial characteristics of the oscillator.

�� n � the number of iterations ���� m � the exponent ��STSR�n�, m�� :� Module��f, h, RHS, Ε, suc, x, X, H�,f�x�� :� xm;

h �i�0

n

Εi�Hi;

RHS � �Ε h f�i�0

n

Εi�Xi�Ξ�;

suc � Cancel�Coefficient�Normal�Series�RHS, �Ε, 0, n���,Table�Εi, �i, 1, n����;X0�Ξ�� :� A Ξ;Do�Xk�Τ�� � Integrate�suc��k�� ��Ξ Τ�, �Ξ, 0, Τ�,Assumptions � Τ � 0 && Re�m� � �1�, �k, 1, n��;H0 � H0 . Solve� Τ �X0�Τ� X1�Τ�� � 0 . Τ � 1, H0���1��;Do�Hk � Hk . Solve� Τ Xk1�Τ� � 0 . Τ � 1, Hk���1��, �k, 1, n��;�Table�Xk�Τ�, �k, 0, n��, Table�Hk, �k, 0, n � 1���

Successive approximations (execution):

n � 3;

AnalyticalSolution � STSR�n, Α�;Do�Xi�1 � AnalyticalSolution��1����i��, �i, 1, n 1��;Do�Hi�1 � AnalyticalSolution��2����i��, �i, 1, n��;Do�Print�"X", i, "�", Xi Simplify�, �i, 0, n��Do�Print�"H", i, "�", Hi Simplify�, �i, 0, n � 1��

X0�A Τ

X1��A Τ2�Α

2 � Α

X2��A�2 Α Α Τ �A Τ�1�Α �AΑ �3 � 2 Α� � �2 � Α� Τ �A Τ�Α�

2 �2 � Α�2 �3 � 2 Α�

X3��A1�2 Α Α Τ2�Α �A2 Α �1 � Α�2 �4 � 3 Α� � AΑ Α �8 � 10 Α � 3 Α2� Τ �A Τ�Α � ��2 � Α � 3 Α2 � Α3� Τ2 �A Τ�2 Α�

2 �2 � Α�3 �3 � 2 Α� �4 � 3 Α�

352 Appendix

H0�A1�Α �1 � Α�

H1�A1�Α Α �1 � Α�2 �2 � Α�

H2�A1�Α Α �1 � Α�3

2 �2 � Α�2 �3 � 2 Α�Successive approximation truncated series:

x �i�0

n

Xi; h �i�0

n�1

Hi; v �1

h

� Τ x;

Appendix 353

APPENDIX 2: Mathematica�R� (Version 6) notebook for sawtooth power-series expansion of periodic functions.

In[1]:= �� f � f�t� � periodic function of the period T ���� m � "length" of the series; see the example below ���� smoothness � True or False; use smoothness

� True to smooth the series ��TauSeries�f�, T�, m�, smoothness��:� Module��a,s1, s2, NLX, NLY,RE, IM, der, TauSpectrum�,a � T 4;s1 � �t �� a �; s2 � �t � 2�a � a �;

RE �1

2���f . s1� �f . s2��; IM �

1

2���f . s1� � �f . s2��;

der�F�� :� Apply�D, �F, Τ ��;NLX � NestList�der, RE, 2�m 1� . �Τ � 0�; NLY

� NestList�der, IM, 2�m 1� . �Τ � 0�;

Do��KX�2�i � 1� �k�1

i NLX��2�k��Factorial�2�k � 2�

, KX�2�i� �k�1

i NLX��2�k 1��Factorial�2�k � 1�

,

KY�2�i � 1� �k�0

i NLY��2�k��Factorial�2�k � 1�

,

KY�2�i� �k�0

i NLY��2�k 1��Factorial�2�k�

�, �i, �1, m�;

If�smoothness, �X � �RE . �Τ � 0�� i�1

2�m

KX�i��Τi

i�Τi2

i 2,

Y �i�0

2�m

KY�i���Τi � Τi2��, �X ��k�1

2�m NLX��k���Τk�1Factorial�k � 1�

,

Y ��k�1

2�m NLY��k���Τk�1Factorial�k � 1�

�

In[2]:= �� ������������� Example of usage ������������ ��

f�t�� :� Sin�Рt

23

T � 4;

In[4]:= �� � Expansions with no smoothing procedure �� ��fnosmooth�t�, m�� :� ��X Y e� . TauSeries�f�t�, T, m, False��

. �Τ � Τ�4�t T�, e � e�4�t T��

fnosmooth�t, 1�fnosmooth�t, 2�

354 Appendix

In[10]:= �� ��� Expansions with smoothing procedure ��� ��fsmooth�t�, m�� :� ��X Y e� . TauSeries�f�t�, T, m, True��

. �Τ � Τ�4�t T�, e � e�4�t T��fsmooth�t, 1�fsmooth�t, 2�fsmooth�t, 3�fsmooth�t, 4�

Out[11]= 0

Out[12]=3

8Π3

Τ�t�33

�Τ�t�55

Out[13]=3

8Π3

Τ�t�33

�Τ�t�55

�3 Π3

8�5 Π5

64

Τ�t�55

�Τ�t�77

Out[14]=3

8Π3

Τ�t�33

�Τ�t�55

�3 Π3

8�5 Π5

64

Τ�t�55

�Τ�t�77

�3 Π3

8�5 Π5

64�

91 Π7

15 360

Τ�t�77

�Τ�t�99

In[15]:= �� ������ Defining the basis functions ������ ��

��� :�2

Π�ArcSin� Sin�

Π

2�Ξ;

e�Ξ�� :� Sign� Cos�Π

2�Ξ;

Out[7]=1

8Π3 Τ�t�3 �

1

64Π5 Τ�t�5

Out[8]=1

8Π3 Τ�t�3 �

1

64Π5 Τ�t�5 �

13 Π7 Τ�t�715 360

Out[9]=1

8Π3 Τ�t�3 �

1

64Π5 Τ�t�5 �

13 Π7 Τ�t�715 360

�41 Π9 Τ�t�91 548288

Out[5]= 0

Out[6]=1

8Π3 Τ�t�3

fnosmooth�t, 3�fnosmooth�t, 4�fnosmooth�t, 5�

Appendix 355

In[18]:= �� ���������� Convergence with smoothing �������� ��

Plot�Evaluate��f�t�, fsmooth�t, 2�, fsmooth�t, 3�,fsmooth�t, 4��, �t, 0, 2 T��,

PlotStyle � ��Thickness�.005�, Color � Black�,�Thickness�.002�, Dashing��0.01, 0.01��, Color � Black�,�Thickness�.002�, Color � Black�,�Thickness�.003�, Color � Black��, AxesLabel � �"t", "x"�,

PlotRange � All�

Out[18]=

2 4 6 8t

�1.5

�1.0

�0.5

0.5

1.0

1.5

x

Out[17]=

2 4 6 8t

�4

�2

2

4

x

In[17]:= �� ��������� Convergence with no smoothing ������� ��

Plot�Evaluate��f�t�, fnosmooth�t, 2�, fnosmooth�t, 3�,fnosmooth�t, 4�, fnosmooth�t, 5��,�t, 0, 2 T��,PlotStyle � ��Thickness�.005�, Color � Black�,�Thickness�.002�, Dashing��0.01, 0.01��, Color � Black�,�Thickness�.002�,Color � Black�,�Thickness�.002�,Color � Black�,�Thickness�.003�, Color � Black��,

AxesLabel � �"t", "x"�, PlotRange � All�

Appendix 357

APPENDIX 3: Mathematica�R�(Version 4) notebook.

NSTT & Shooting method for periodic solutions of the oscillator

d2 �x

dt2� Ζ dx

dt� Ε x3 � B e� t

a�.

p�x�, v�, t�� :� Ζ v Ε x3;a �

Π

2 Ω;

�� Substitution x�t� � X��t a��Y��t a���e�t a�applied to the equation of motion: ��

eqX �1

a2� Τ,ΤX�Τ�

1

2� p�X�Τ� Y�Τ�,

1

a�� Τ X�Τ� Τ Y�Τ��, a Τ

p�X�Τ� � Y�Τ�,1

a��� Τ X�Τ� Τ Y�Τ��, 2 a � a Τ � 0 Simplify;

eqY �1

a2� Τ,ΤY�Τ�

1

2� p�X�Τ� Y�Τ�,

1

a�� Τ X�Τ� Τ Y�Τ��, a Τ

� p�X�Τ� � Y�Τ�,1

a��� Τ X�Τ� Τ Y�Τ��, 2 a � a Τ � B Simplify;

�� Parameters: ��Ω � 1.0; Ζ � 0.05; Ε � 1.0; B � 7.4;

Clear�g, h�; dg � 3; dh � 20;sol�g�, h�� :� NDSolve��eqX, eqY, X��1� � g, X'��1� � 0, Y��1� �� 0,Y'��1� � h�, �X, Y�, �Τ, �1, 1�, MaxSteps � Infinity�;Xn�g�, h�� :� X'�1� . sol�g, h���1��;Yn�g�, h�� :� Y�1� . sol�g, h���1��;

plx � ContourPlot�Xn�g, h�, �g, �dg, dg�, �h, �dh, dh�,Contours � �0�, ContourShading � False,FrameLabel � �"g", "h"�,PlotPoints � 100,RotateLabel � False,DisplayFunction � Identity�;

ply � ContourPlot�Yn�g, h�,�g,�dg, dg�,�h, �dh, dh�,Contours � �0�,ContourShading � False, FrameLabel � �"g", "h"�, PlotPoints � 100,

RotateLabel � False,

ContourStyle � �Dashing��0.01, 0.01���, DisplayFunction � Identity�;pxy � Show�plx, ply, DisplayFunction � $DisplayFunction�;

358 Appendix

-3 -2 -1 0 1 2 3g

-20

-15

-10

-5

0

5

10

15

h

�� Magnified portion of the diagram: ��

Show�pxy, PlotRange � ���0.3, 0�, ��5, �4���;

-0.25 -0.2 -0.15 -0.1 -0.05 0g

-4.8

-4.6

-4.4

-4.2

-4

h

Appendix 359

�� Find one of the roots: ��

�g, h� � �g, h� . FindRoot��Xn�g, h� � 0, Yn�g, h� � 0�, �g, �0.3, 0�,�h, �5, �4��

�0.113351, �4.51957�� Check precision: ��

sln � NDSolve��eqX, eqY, X��1� � g, X'��1� � 0, Y��1� �� 0, Y'��1� � h�,�X, Y�, �Τ, �1, 1��;

�Y�1�, X'�1�� . sln��1��

��2.09795�10�9, �1.47496�10�9��� Introduce the saw�tooth sine and the rectangular cosine: ��

�t�� :�2

Π�ArcSin� Sin�

Π

2�t;

e�t�� :� Sign� Cos�Π

2�t;

�� Graphic output compared to solution of the related Cauchy problem: ��

x�t�� :� �X��t a�� Y��t a���e�t a�� . sln��1��;

v�t�� :� 1

a��Y'��t a�� X'��t a���e�t a�� . sln��1��;

xxtt � Plot�Evaluate�x�t�, �t, 0, 4 a��, PlotRange � All,

AxesLabel � �"t", "x"�, PlotStyle � ��Thickness�.009���,TextStyle � �FontSize � 14�, DisplayFunction � Identity�;

xxvv � ParametricPlot�Evaluate��x�t�, v�t��, �t, 0, 4�a��, PlotRange � All,

AxesLabel � �"x", "v"�, Frame � True, PlotStyle � ��Thickness�.009���,TextStyle � �FontSize � 14�, DisplayFunction � Identity�;

Clear�y�;Tmax � 4�a; dirsol � NDSolve�� t,t y�t� � t y�t� Šy�t�3 � B e�t a�,

y�0� � x�0�, y'�0� � v�0��, y, �t, 0, Tmax�,MaxSteps � Infinity�; y � y . dirsol��1��;

yyvv � ParametricPlot�Evaluate��y�t�, y'�t��, �t, 0, Tmax��,PlotRange � All, AxesLabel � �"x", "v"�,PlotStyle � ��Thickness�.001���, TextStyle � �FontSize � 14�,PlotPoints � 500, Frame � True, DisplayFunction � Identity�;

Show�GraphicsArray���xxtt�, �xxvv�, �yyvv����;

360 Appendix

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

x

v

-3 -2 -1 0 1 2 3

-4

-2

0

2

4

x

v

1 2 3 4 5 6t

-3-2-1

123

x

Appendix 361

APPENDIX 4: Mathematica�R� notebook.Conducts NSTT of the T-periodic differential equation of the form

�t,t x � f �x, �t x, t� � 0.

In[1]:= �� see examples below for the meaning of inputs ��

TauTransform�equation�, function�,argument�, period�, basis�� :�Module��sub, eqn, RE, IM�,sub � �function � X�� Y���e, argument function � �Y'�� X'���e� a, argument,argument function � �X''�� Y''���e� � a2�;

Q � Part�equation, 1� . sub;

eqn �1

2���Q . t �� a � �Q . t � 2�a � a ��

1

2���Q . t �� a � � �Q . t � 2�a � a �� e . a � period 4;

EQNX � Simplify�1

2���eqn . e � 1� �eqn . e � �1��;

EQNY � Simplify�1

2���eqn . e � 1� � �eqn . e � �1��;

subid � �X�� ��1

2��X�� X����, Y�� ��

1

2��X�� � X����,

X'�� ��1

2�� Τ X�Τ� Τ X��Τ��,Y'�Τ� ��

1

2�� Τ X�Τ� � Τ X��Τ��,

X''�� ��1

2� Τ,Τ�X�Τ� X��Τ��,Y''�Τ� ��

1

2� Τ,Τ�X�Τ�� X��Τ���;

IDP � Simplify��EQNX EQNY� . subid�;IDM � Simplify��EQNX � EQNY� . subid�;If�basis � 1, ��EQNX � 0, EQNY � 0�, �Y�1� � 0, Y��1� � 0, X'�1�� 0, X'��1� � 0��, ��IDP � 0, IDM � 0�,�X�1� � X��1� � 0,

X��1� � X���1� � 0, X'�1� X�'�1� � 0, X'��1� X�'��1� � 0���

EXAMPLES

In[2]:= �� If basis � 1, then the output is created in the standard

basis �1,e�, otherwise the output is in the idempotent basis

�e,e�����1e� 2,�1�e� 2� ��

In[3]:= TauTransform�x''�t� x�t�3 � 0, x�t�, t, 4�a, 1�

Out[3]= X��3 � 3 X�� Y��2 �X����a2

0, 3 X��2 Y�� � Y��3 �Y����a2

0�,

Y�1� 0, Y��1� 0, X��1� 0, X���1� 0�

362 Appendix

In[5]:= TauTransform�x''�t� x�t�3 � P Sin�t� � 0, x�t�, t, 2�Π, 0�

Out[5]= �P Sin�Π Τ

2� � X���3 �

4 X�����

Π2 0,�P Sin�Π Τ

2� � X���3 �

4 X�����

Π2 0�,

�X��1� � X��1� 0, �X���1� � X���1� 0, X���1� � X�

��1� 0, X�

���1� � X����1� 0�

In[6]:= TauTransform�x''�t� x�t�3 � P Sin�t� � 0, x�t�, t, 2�Π, 1�

Out[6]= �P Sin�Π Τ

2� � X�Τ�3 � 3 X�Τ� Y�Τ�2 �

4 X���Τ�Π2

0, 3 X�Τ�2 Y�Τ�

� Y��3 �4 Y����

Π2 0�,Y�1� 0, Y��1� 0, X��1� 0, X���1� 0�

In[7]:= �� If present, the function e�e�4t T� must be shown with

no argument; derivatives de dt are not allowed by this code,

but necessary generalizations are easy to implement ��

In[8]:= TauTransform�x''�t� �1 Α e ��x�t� Β x�t�3 � 0, x�t�, t, 4�a, 1�

Out[8]= X�Τ� � Β X�Τ�3 � Α Y�Τ� � 3 Β X�Τ� Y�Τ�2 �X���Τ�a2

0,

Α X�Τ� � Y�Τ� � 3 Β X�Τ�2 Y�Τ� � Β Y�Τ�3 �Y���Τ�a2

0�, Y�1� 0,

Y��1� 0, X��1� 0, X���1� 0�In[9]:= TauTransform�x''�t� �1 Α e ��x�t� Β x�t�3 � 0, x�t�, t, 4�a, 0�

Out[9]= �1 � Α� X��Τ� � Β X��Τ�3 �X�

����a2

0, ���1 � Α� X��Τ� � Β X��Τ�3

�X�

����a2

0�, �X��1� � X��1� 0, �X���1� � X���1� 0, X���1�

� X���1� 0, X�

���1� � X����1� 0�

PROJECT: Using the idempotent basis, find T-periodic solution of the linear oscillator under external and parametric rectangular wave periodic excitation

of the period T=4,

In[4]:= TauTransform�x''�t� x�t�3 � 0, x�t�, t, 4�a, 0�

Out[4]= X���3 �X�

����a2

0, X���3 �X�

����a2

0�, �X��1� � X��1� 0,

�X���1� � X���1� 0, X���1� � X�

��1� 0, X����1� � X�

���1� 0�

Appendix 363

In[12]:= �� Solves the boundary value problem analytically: ��

sol � DSolve�bvp, �X��, X����, � Simplify

Out[12]= X��� �

2 p 3 Cos� Ω

2� Sin� �3

2Ω� � Cos� �3

2Ω� � 4 Cos� �3

2Τ Ω� Sin� Ω

2�

3 Ω2 3 Cos� Ω

2� Sin� �3

2Ω� � Cos� �3

2Ω� Sin� Ω

2�

,

X��Τ� � � 2 p 3 3 Cos� Ω

2� Sin� 3

2Ω�� 4 3 Cos� Τ Ω

2� Sin� 3

2Ω�

� 3 Cos� 3

2Ω� Sin� Ω

2� � 3 Ω2 3 Cos� Ω

2� Sin� 3

2Ω�

� Cos� 3

2Ω� Sin� Ω

2� ��

In[13]:= �� Extracting two components of the solution: ��

X�� � X�� . sol��1��;X��� � X��� . sol��1��;

In[15]:= �� Back to the original x�t variables: ��

x � X���1

2��1 e� X����

1

2��1 � e� . �Τ � Τ�t�, e � e�t��;

v �1

T 4� Τ X�Τ��

1

2��1 e�� Τ X��Τ��

1

2��1 � e� .�Τ � Τ�t�,e � e�t��;

bvp � TauTransform�x''�t� Ω2� 1 1

2e �x�t� � p e � 0, x�t�, t, T, 0

Out[11]= �p �3

2Ω2 X��Τ� � X�

���� 0, p �1

2Ω2 X��Τ� � X�

���� 0�,�X��1� � X��1� 0, �X���1� � X���1� 0, X�

��1� � X���1� 0,

X����1� � X�

���1� 0�

�t,t x � �1� 12�e�t���Ω2�x � pe�t�.

In[10]:=�� Formulates boundary value problem in the idempotent basis: ��

T � 4;

In[17]:= �� Basis functions: ��

364 Appendix

In[19]:= �� Parameters and graphic output: ��٠� 20.0;

p � 1.0;

Plot�Evaluate�x, �t, 0, 8��, PlotRange � All,

AxesLabel � �"t", "x"�, PlotStyle � ��Thickness�.005����

Out[21]=2 4 6 8

t

�0.010

�0.005

0.005

0.010

x

In[22]:= �� Validating the code by durect numerical solution: ��x0 � x . t � 0;

v0 � v . t � 0;

dirsol � NDSolve��y''�t� Ω2� 1 1

2e �t� �y�t��p e�t��0,y�0��x0,

y'�0� � v0�, y, �t, 0, 50�, MaxSteps � Infinity;y � y . dirsol��1��;

In[26]:= Plot�Evaluate�y�t�, �t, 0, 8��, PlotRange � All,

AxesLabel � �"x", "v"�, PlotStyle � ��Thickness�.003����

Out[26]=2 4 6 8

x

�0.010

�0.005

0.005

0.010

v

�t�� :�2

Π�ArcSin� Sin�

Π

2�t;

e�t�� :� Sign� Cos�Π

2�t;