"modnum" scilab toolbox for the communication systems user's guide
TRANSCRIPT
-
"Modnum"Scilab toolbox
for the communication systemsUsers Guide
Draft Version
IRCOM GroupAlan Layec
February 8, 2006
-
CONTENTS 1
Contents
I Scicos Diagrams 3
1 Chaotic dynamical systems 5
Continous time systems 51.1 Lorentzs system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Discrete Van Der Pols system (trapezoidal mthode) . . . . . . . . . . . . . . . . . . . . . . . 71.3 Discrete forced Van Der Pols system (Euler method) . . . . . . . . . . . . . . . . . . . . . . . 81.4 Duffing-Van Der Pols system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Rsslers system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Duffings oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Chuas system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Sub-system of Chuas circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Master-slave synchronization of Chuas circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Discrete time systems 211.10 2D bifurcation of the logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.11 3D bifurcation of the logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.12 Henons system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.13 Chaotic first order Delta-Sigma modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.14 Freys chaotic encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.15 Second order Infinite Impulse Response filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.16 Master-slave synchronization of second order IIR filter . . . . . . . . . . . . . . . . . . . . . . . 30
2 Oscillators and Phase Locked Loop systems 32
Open loop models of oscillators 322.1 Continous Voltage Controled Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Discrete Voltage Controled Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Integer N Frequency synthesizers 352.3 Simple scicos diagram of an integer-N frequency synthesizer . . . . . . . . . . . . . . . . . . . 352.4 Scattered diagram of a third order integer-N frequency synthesizer . . . . . . . . . . . . . . . . 372.5 Integrated diagram of an integer-N frequency synthesizer . . . . . . . . . . . . . . . . . . . . . 392.6 Scattered diagram of a third order integer-N frequency synthesizer with an interpoled non-linearity
of VCO and with dual-modulus feedback divider . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Fractional N/N+1 Frequency synthesizers 442.7 Modulated fractional frequency synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
-
CONTENTS 2
3 Communication systems 46
PSK/QAM Transmission 463.1 Scattered diagram of base band QPSK transmission chain . . . . . . . . . . . . . . . . . . . . . 463.2 Integrated diagram of base band QPSK transmission chain . . . . . . . . . . . . . . . . . . . . . 493.3 Base band single user QPSK spread-spectrum transmission chain . . . . . . . . . . . . . . . . . 513.4 Scattered diagram of sample-based 16-QAM chain transmission . . . . . . . . . . . . . . . . . . 53
Delta-Sigma Transmission 553.5 Scattered diagram of first order Delta-Sigma modulator . . . . . . . . . . . . . . . . . . . . . . 553.6 Scattered diagramm of second order Delta-Sigma modulator . . . . . . . . . . . . . . . . . . . . 563.7 Integrated diagram of Delta-Sigma modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.8 Delta-sigma modulator with two in-loop integrators . . . . . . . . . . . . . . . . . . . . . . . . 603.9 Delta-sigma modulator with three in-loop integrators . . . . . . . . . . . . . . . . . . . . . . . . 61
FSK chaotic transmission 623.10 Chaotic frequency hopping emitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.11 Chaotic frequency hopping transmission system . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Electrical circuits 674.1 Resitif attenuator/coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
II Simulation Scilab scripts 69
1 Simulations of chaotic systems 711.1 Output probability density of autonomous second order IIR filter . . . . . . . . . . . . . . . . . 711.2 BER estimation of pair chaotic encoder/decoder with introduction of jitter in receiver clock . . . 75
2 Simulations of oscillators and Phase Locked Loops 782.1 Output spectra of Integer N frequency synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . 782.2 Output jitter of integer-N frequency synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . 822.3 Gaussian modulated fractional frequency synthesizer . . . . . . . . . . . . . . . . . . . . . . . 85
3 Simulations of communication systems 883.1 Output spectra of Delta-Sigma modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.2 BER estimation of single user QPSK transmission . . . . . . . . . . . . . . . . . . . . . . . . . 923.3 Rayleigh noise generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
-
I. SCICOS DIAGRAMS 3
Part I
Scicos Diagrams
-
I. SCICOS DIAGRAMS 4
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 5
Chapter 1
Chaotic dynamical systems
1.1 Lorentzs system
1/s
1/s
1/s
aa
bb
11
11
cc
MScopeMScope
1.1.1 Description
The Lorentzs system is defined by the following continuous system :
dx(t)
dt= a (x(t) + y(t)) (1.1)
dy(t)
dt= bx(t) y(t) x(t)y(t) (1.2)
dz(t)
dt= cx(t) + x(t)y(t) (1.3)
The state variables x(t), y(t) and z(t) are temperature of the air, speed of wind and a third caracteristic whichrepresents the variation of temperature in accord to the altitude.
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 6
1.1.2 ContextTsampl=3e-3a=10b=28c=8/3ci=[5.5;5;20]Tfin=20
1.1.3 Scope Results
10 11 12 13 14 15 16 17 18 19 2025.0
12.5
0.0
12.5
25.0
+
10 11 12 13 14 15 16 17 18 19 2025.0
12.5
0.0
12.5
25.0
+
10 11 12 13 14 15 16 17 18 19 200.0
12.5
25.0
37.5
50.0
+
(a) Temporal wave forms of state variables
30 24 18 12 6 0 6 12 18 24 300
5
10
15
20
25
30
35
40
45
50
+
......................................................
...........
..........................................................
.......................................................................................................................................................................
...........
......................................................
..................................................................................................................................................................................
..........
.................................................................................................................................................................................................................................................
............
...........
...............................................
....................................................................................................................................................................................................................................................................
.........
......
..............................................
....................................................................................................................................................................
.............
.........
.......
......
.......
..............................
.................................................................................................................................................................................
.........
........
......
......
.......
....................................................................................................................................................................................................................
............
.......
.......
.....
....
....
....................................................................................................................................................................................................................................................................
............
......
.....
.....
....
....
....
....
.....
.......................
...................................................................................................................................................................................
.........
......................................................
.........................................................................................................................................................................................................
.........
.......
......
........................................
......................................................................................................................................................................................................................................................
.........
......
.....
.....
....
....
....
....
.....
.....................
................... .. ........................................................................................................................................................
..........
......................................................
............................................................................................................................................................................................
.........
................................................................................................................................................................................................................................................................................
...........
.......
.................................................
.................................................................................................................................................................................................
..........
......
......
....
.....
....
....................................................................................................................................................................................................................................................................
...........
........
.....
....
....
.....
....
....
....
.................................................................................................................................................................................................
...........
.........
..................................................
.....................................................................................................................................................................................................
..........
......
......
...........................................................................................................................................................................................................................................................................................................................................................................................
.............
.......
......
....
....
....
...
..
...
..
...........................
.......... . . . . . . . . . . . . . . . . . . .. . . . .. .........................................................................................................................
.............................................................
.................................................................................................................................................................
.......................................................................................................................................................................................................................
..............
....................................................................................................................................................................................................................
...............
...........................................................
...................................................................................................................................................................
...........
....................................................................................................................................................................................................................
..............
..............................................................
.......................................................................................................................................................................
..........
........................................................
...............................................................................................................................................................
(b) Phase plan
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 7
1.2 Discrete Van Der Pols system (trapezoidal mthode)
Trapezoidal
u^au^a
11
11
11
Trapezoidal
MScopeMScope
1.2.1 Description
The Van Der Pols Equation is written with the equation :
x +(
x2 1)
x + x = 0 (1.4)
It is fully described by the following system of state equation :
x = y (1.5)
y =(
1 x2)
y x (1.6)
1.2.2 ContextTe=0.01Tfin=90ci=[-1;0]
1.2.3 Scope Results
1.2.4 Mod_num blocks
TRAPINTEGRAL_f - Single step integrator block by trapezoidal method
1.2.5 See Also
EULERINTEGRAL_f - Single step integrator block by Euler method (Scicos Block)
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 8
0 9 18 27 36 45 54 63 72 81 902
1
0
1
2
+
0 9 18 27 36 45 54 63 72 81 905.0
2.5
0.0
2.5
5.0
+
(a) Time domain wave forms of state variables
2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.54.0
3.2
2.4
1.6
0.8
0.0
0.8
1.6
2.4
3.2
4.0
+
............................................
............................
..........................
.......................
..................
..................
........................
...........................................................................................................................................................................................................................................................................................................................................................................
.........
........
......
......
........
........
..........................................................
......................................................
...................................................
................................
......................
.................
.............
...............
..................
........ ....................................................................................................................................................................................................................................................................................................................................................................................
........
........
......
......
.......
.......
.............................................................
........................................................
..................................................
................................
.....................
..................
..............
..............
..................
........ ....................................................................................................................................................................................................................................................................................................................................................................................
.........
.......
......
.......
......
........
.....................................................................
..........................................................
..............................................
..............................
.....................
...............
...............
...............
.....................................................................................................................................................................................................................................................................................................................................................................................................
...........
........
.......
......
......
.......
.........................................................................
............................................................
.............................................
.............................
.....................
................
.............
...............
.......................................................................................................................................................................................................................................................................................................................................................................................................
.........
........
......
......
......
........
........
.............................................................
..........................................................
.................................................
..............................
....................
.................
...............
..............
......................... ..............................................................................................................................................................................................................................................................................................................................................................................
...........
........
......
.......
......
........
...............................................................
.........................................................
...................................................
..............................
.....................
..................
...............
...............
......................................................................................................................................................................................................................................................................................................................................................................................................
...........
.......
.......
......
......
........
.........................................................................
............................................................
.............................................
.............................
....................
................
..............
...............
....................... ................................................................................................................................................................................................................................................................................................................................................................................
.........
.........
......
......
.......
.......
...........................................................................
...........................................................
.............................................
............................
....................
................
..............
..............
...................... .... .............................................................................................................................................................................................................................................................................................................................................................................
.........
.........
......
.......
......
......
.........
...........................................................
.......................................................
...................................................
..............................
.....................
..................
..............
..............
....................
...........................................................................................................................................................................................................................................................................................................................................................................................
........
......
.......
......
......
.........
.....................................................................
............................................................
.............................................
.............................
....................
................
.............
...............
.................... ..
... ....................................................................................................................................................................................................................................................................................................................................................................................
........
........
......
......
.......
........
..........................................................................
...........................................................
.............................................
...........................
....................
................
...............
..............
......................... ...........................................................................................................................................................................................................................................................................................................................................................................
.........
.........
......
.......
......
......
.........
.................................................................
............................................................
..............................................
.............................
....................
...............
..............
...............
...................
..........................................................................................................................................................................................................................................................................................................................................................................................
........
.......
.......
......
.......
.........
...........................................................
.........................................................
.................................................
................................
......................
................
..............
...............
...................
.... .... ....................................................................................................................
(b) Phase plan
1.3 Discrete forced Van Der Pols system (Euler method)
u^au^a
11
11
11
MScopeMScope
Euler
Euler
Euler
wowo cos Trig. Function cos Trig. Function
AA
1.3.1 Description
This system illustrates the use of a single step integrator in order to resolve continuous system. Here the system isresolved with the backward Euler method. The forced Van Der Pols oscillator is described with the system of stateequation :
x = y (1.7)
y =(
1 x2)
y x + A cos (z) (1.8)
z =2
T(1.9)
where parameters T is the period of the driven oscillator and A its amplitude.
1.3.2 ContextTe=0.01
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 9
Tfin=90ci=[-1;0]A=0.5;wo=1.1;To=2*%pi/wo;
1.3.3 Scope Results
0 9 18 27 36 45 54 63 72 81 902.50
1.25
0.00
1.25
2.50
+
0 9 18 27 36 45 54 63 72 81 905.0
2.5
0.0
2.5
5.0
+
0 9 18 27 36 45 54 63 72 81 901.0
0.5
0.0
0.5
1.0
+
(a) Time domain wave forms of state variables
2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.54.0
3.2
2.4
1.6
0.8
0.0
0.8
1.6
2.4
3.2
4.0
+
......................................................................
...........................................................
..............................................................................................................................................................................................................................................................................................
..........
........
......
........
........
.......................................................................
......................................
.................................
........................
.................
..............
.............
..........
...........
. ...... .. . . . .
.. . . . . . . . . . . . . . . . . . . .. . . . . ..... ......................................................................................................................................................................................................................................................................................................................................
........
......
......
.......
......
......
.........................................................
..........................................
..........................................
...............................
.....................
................
............
............
.............
. ... ........ ..
. .. . .. . .. .. . . . .. . . ..... .... .............................................................................................................................................................................................................................................................................................................................
.........
......
.......
......
......
......
............................................................
........................................
..........................................
...........................
....................
................
.............
.............
............ ....
... ... ....... ...... ... ..... .............................................................................................................................................................................................................................................................................................................................
........
.......
......
......
.......
......
.......
......................................................
........................................
......................................
............................
.....................
.................
...............
...............
.................... ... ...................................................................................................................................................................................................................................................................................................................................
............
.......
......
.......
......
......
...........................................................
...................................
.........................................
................................
.......................
..................
...............
.................
................................................................................................................................................................................................................................................................................................................................................................
.........
.........
.......
.......
.........
......................................................
...................................
............................................
.................................
........................
...................
..................
...............................................................................................................................................................................................................................................................................................................................................................................
...........
..........
........
........
.........
.........
...................................................
........................................
..................................................
....................................
...........................
......................
.................................................................................................................................................................................................................................................................................................................................................................................................................
..............
............
...........
..........
............
...................................................................
.......................................................................
......................................................
.................................
.......................
....................
...................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........
.........
......
......
.......
.......
........
.............................................................
.....................................................................
................................................
..............................
.....................
................
............
...........
..........
.......... .
.... ... . .. ..
.. . . . . . . . . . . . . . . . . . . .. .. ..... ...................................................................................................................................................................................................................................................................................................................................................................
.......
......
.......
.....
......
.......
......
....................................................
....................................................
...............................................
...............................
.....................
................
............
............
...........
.. .. .... .. .
. . . . . . ... . . . . . . . . . . . . . . . . . . .. . . ..... .. ................................................................................................................................................................................................................................................................................................................................
........
.......
......
......
.......
......
......
....................................................
.........................................
...........................................
...............................
......................
................
.............
............
............
.... ......... .. . ..
. .. ... . .. . . .. .. ..... ...............................................................................................................................................................................................................................................................................................................................
.........
......
......
.......
......
......
..............................................................
.........................................
.......................................
............................
....................
................
............
.............
...............
. .. .. .... ..... .... .... .. .................................................................................................................................................................................................................................................................................................................................
........
.......
......
......
......
........
....................................................
...................................
..........................................
...............................
......................
..................
...............
..............
...............................................................................................................................................................................................................................................................................................................................................................
..........
.........
......
......
.......
.......
................................................................
.......................................
........................................
.............................
.....................
..................
...............
.........................
...................................................................................................................................................................................................................
(b) Phase plan
1.3.4 Mod_num blocks
EULERINTEGRAL_f - Single step integrator block by Euler method
1.3.5 See Also
EULERINTEGRAL_f - Single step integrator block by Euler method (Scicos Block)
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 10
1.4 Duffing-Van Der Pols system
11
1/s
cos Trig. Function cos Trig. Function
u^au^a
11
11 u^au^amumu
1/s
ww
vv
MScopeMScope
Te
To
1.4.1 Description
Add here a paragraph of the function description.
1.4.2 ContextTe = 5e-3mu=0.2v=17w=4To=2*%pi/wTfin = 40
1.4.3 Scope Results
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 11
0 4 8 12 16 20 24 28 32 36 404
2
0
2
4
+
0 4 8 12 16 20 24 28 32 36 4012
6
0
6
12
+
0 4 8 12 16 20 24 28 32 36 401.10
0.55
0.00
0.55
1.10
+
Scope results
4.0 3.2 2.4 1.6 0.8 0.0 0.8 1.6 2.4 3.2 4.012.0
9.6
7.2
4.8
2.4
0.0
2.4
4.8
7.2
9.6
12.0
+
Scope results
1.5 Rsslers system
bb
1/s
11
MScopeMScope
1/s
aa
1/s
ggScopeXYZScopeXYZ
1.5.1 Description
The Rsslers system is defined with the following system of state equations :
X = (Y + Z) (1.10)
Y = X + Y (1.11)
Z = + Z (X ) (1.12)
1.5.2 ContextTe=0.01
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 12
a=0.2b=0.2g=-5.7ci=[-5;5;0.1]Tfin = 150
1.5.3 Scope Results
0 15 30 45 60 75 90 105 120 135 15012
6
0
6
12
+
0 15 30 45 60 75 90 105 120 135 15012
6
0
6
12
+
0 15 30 45 60 75 90 105 120 135 1500.0
6.2
12.5
18.8
25.0
+
(a) Time domain wave forms of state variables
10.0 7.8 5.6 3.4 1.2 1.0 3.2 5.4 7.6 9.8 12.012.0
9.8
7.6
5.4
3.2
1.0
1.2
3.4
5.6
7.8
10.0
+
.......................................................................................................................................................................................................................................................................................................
..................
.............
............
..........
..........
..................................................
........
.........
..........
..........
............
...............
.........................
...................................................................................................................................................................................................................................................................................................................................................................................................
..................
...............
............
...........
..........................................................
...........
..........
..........
...........
.............
...............
......................
...........................................................................................................................................................................................................................................................................................................................................................................................
.....................
...............
............
...........
..........
...............................................
........
.........
........
..........
............
...............
............................
.............................................................................................................................................................................................................................................................................................................................................................................................
....................
................
.............
.......................................................................
..........
..........
............
............
............
..............
.................
......................
........................................................................................................................................................................................................................................................................................................................................................................................
....................
.............
............
..........
..........
........
................................
........
........
.........
............
.................
.............................................................................................................................................................................................................................................................................................................................................................................
...............................
.......................................................................................
.....................
.........................
.............................
........................................................................................................................................................................................................................................................................................................................................................................................................................
...........................
...................
................
............................................................................
............
............
..............
..............
................
....................
...........................
...............................................................................................................................................................................................................................................................................................................................................................................................
................
.............
...........
..........
..........
........................................
........
........
........
..........
.............
.................................
...................................................................................................................................................................................................................................................................................................................................................................................
........................
....................
..................................................................................
..............
................
................
...................
.......................
.........................................................................................................................................................................................................................................................................................................................................................................................................
.......................
...............
.............
...........
..........
......................................................
.........
.........
..........
..........
............
................
........................
..................................................................................................................................................................................................................................................................................................................................................................................................
...................
..............
............
...........
...........................................................
........
..........
..........
..........
.............
................
...........................
...............................................................................................................................................................................................................................................................................................................................................................................................
..................
...............
............
...........
........................................................
.........
.........
..........
..........
............
...............
.....................
..............................................................................................................................................................................................................................................................................................................................................................................................
......................
...............
.............
............
..........
.....................................................
........
.........
..........
...........
.............
.................
...........................................
..........................................................................................................................................................................................................................................................................................................................................................................
..................
...............
.............
............
...........................................................
..........
..........
..........
...........
............
.............
..................
.................................
..................................................................................................................................................................................................................................................................................................................................................................................
.................
.............
...........
...........
.........
.........................................
........
........
..........
...........
..............
................................................................................................................................................................................................................................................................................................................................................................................................
..............................
.....................
...........................................................................................
..............
...............
................
..................
.....................
........................................
..................................................................................................................................................................................................................................................................................................................................................................................
.................
.............
............
...........
..........
.................................................
........
.........
..........
..........
............
...............
.........................
...................................................................................................................................................................................................................................................................................................................................................................................................
..................
...............
............
...........
........................................................
.........
..........
..........
..........
...........
.............
..................
................................
.....................................................................................................................................................................................................................................................................................................................................................................................
.................
.............
............
..........
..........
...........................................
........
........
.........
..........
.............
..................
....................................................................................................................................................................................................................................................................................................................................................................................................
.........................
.................
...............
................................................................................
...........
...........
............
............
..............
...............
..................
..................................
...............................................................................................................................................................................................................................................................................................................................................................................
................
.............
..........
..........
........
....................................
.........
........
.........
...........
................
...................................................................................................................................................................................................................................................................................................................................................................................
.............................
......................................................................................
.....................
.......................
.............................
................................................
..............................................................................................................................................................................................................................................................................................................................................................................................
.....................
.................
...............
....................................................................
............
............
............
...............
...............
..................
.........................
.........................................................................................................................................................................................................................................................................................................................................................................................
....................
..............
............
..........
..........
.............................................
........
........
.........
...........
.................
.....................................................................................................................................................................................................................................................................................................................................................................................
.................................
.......................
...................................................................................
................
.................
...................
.....................
................................
.........................................................................................................................................................................................................................................................................................................................................................................................................
..................
...............
.............
.........
(b) Phase plan
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
25.0
12.5
0.0
ZAXES
12
0
12YAXES 12
0
12
XAXES
(c) 3d phase plan
1.5.4 Mod_num blocks
SCOPXYZ_f - 3D trajectory scope block
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 13
1.6 Duffings oscillator
ww1/s
gg
u^au^a
cos Trig. Function cos Trig. Function
aa
1/s
1/s
11
MScopeMScope
X1 X1
X2 X2
X3X3
Te
To
1.6.1 Description
The Duffings equation is described by this non-linear differential equation :
y (t)
dt= x (t) x3 (t) y (t) + cos (t) (1.13)
where and are parameters and a pulsation.This forced oscillator should be written as a three dimensional state equation system :
x1 = x2 (1.14)
x2 = x1 x31 y + cos (x3) (1.15)
x3 =2
T(1.16)
1.6.2 ContextTe=0.02Tfin=200g=0.3a=0.15w=1To=2*%pi/wci1=0.1ci2=0.1ci3=0
1.6.3 Scope Results
-
I. 1. CHAOTIC DYNAMICAL SYSTEMS 14
100 110 120 130 140 150 160 170 180 190 2003.0
1.5
0.0
1.5
3.0
+
100 110 120 130 140 150 160 170 180 190 2002
1
0
1
2
+
100 110 120 130 140 150 160 170 180 190 2000.50
0.25
0.00
0.25
0.50
+
(a) Time domain wave forms
2.0 1.6 1.2 0.8 0.4 0.0 0.4 0.8 1.2 1.6 2.01.80
1.44
1.08
0.72
0.36
0.00
0.36
0.72
1.08
1.44
1.80
+
.........................
................
................
.................
.........................................................................................................................................
...................
....................
.......................
.......................
....................................................................
....................................
...........................................................................................................................................................................................
...............................................................................................................................................................................................................
.......................................
.............................
...........................................................
............
...........
............
........................................................................................................................
.........................................................................................................
........
.....
....
.....
.................................
.......
.............................................................................
................. .. .. ... ... .. .. ............................................................................................................
................................................................................
.........
......
.....
...................................
.........
..................................................................................................................................................................................................................
......................
...............
.............
............
............
.......................................................................................................................
............................
...........................................................................................
......
....
....
.................................
.......
............... .... .........................................................
............. .. ... ..
...... ..... ....... ... .....................................................................................................
..........................................................................
......
.....
....
.................................
.......
.....................................................................................................
..............................................................................................................................
.............
............
............
............
................................................
............
............
................
.......................................................................................................................................
.................................................................................................
.......
......
......
................................
..........
........................................................................................................................................................................................................................
.......
......
...................................
.........
...................................................................................................................
..............................................................................................................................................
.............
............
..............................................
.............
..................................................................................................................................
..............................................................................................................................
.........
......
......
....................................
............
........................................................................................................................................................................................................................................................................................................................................................................................................................
..............
..............................................
............................................................................................................................................................................................................................................................................................................
............................................
..................................................................................................................................................................................................................................................................................
.........
......
......................................
.............
......................................................................................................................
...................
..................
.....................................................................................................................................
...........................
.................................................................................................
............
..........
.......................................
...........................................................................................................................................
.............................................
.....................
.......................................................................................................................................................................................................................................................................
...................
................
......................
........................................................................................................................
.....................
..................................
..................................................................................................................................
.........................................................................
..............................................
.....................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................
.............................................................
..............
....................
............................................................................................................................
..............................
................................................................................................................................................................
..........
........
...................................
.............................................................................................................................................................
.........
.......
...................................
..........
.........................................................................................................
.......................
........................................................................................................................................
....................
.................
.....................
........................
.............................
.......................................................................................................................................................................................................
.........
......
..................................
.......
....................................................................................
..................
............... ....
. .. .. .. ..................................................................................................................
.......................................................................
......
....
....
....
...............................
.......
.......... .... .. ... .. ............................................................
.............................................................................................................................
...................
.......................................................................................
...............
.........
........
.......................................
..............................................................................................................................................
...........
..........................................
................
...........................................................................................................................................
.......................................................................................................................................................
.............
...................................................
.............
...................................................................................................................................
.........................................................................................................................
(b) Phase plan
1.60 1.29 0.98 0.67 0.36 0.05 0.26 0.57 0.88 1.19 1.501.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
+
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
..
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.. .
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
..
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
..
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
...
.
.
.
.
..
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
..
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.. .
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
..
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
...
..
.
.
.
.
.
.
.
.
.
.
.
.
..
..
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
.
..
.
.
.
.
.
..
.
.
.
.
..
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
..
.
.
.
..
.
..
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
..
.
.
.
.
.
..
. .
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.
..
. .
.
..
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.