Transcript Demonstrating Chaos with Sprott Circuits

Demonstrating Chaos with
“Sprott Circuits”
Michael Braunstein
Central Washington University
PNACP, April 15 and 16, 2005
University of Portland.
•Non-Linear Dynamics and Chaos is rapidly becoming a
standard component of the undergraduate physics curriculum,
e.g.:
Analytical Mechanics, Fowles and Cassiday, 6th Edition;
Classical Dynamics of Particles and Systems, Marion and
Thornton, 4th edition.
Classes: Computational Physics, Nonlinear Dynamics and
Chaos
Laws, P. W. (2004). "A unit on oscillations, determinism
and chaos for introductory physics students." American
Journal of Physics 72(4): 446-452.
•Physics is an experimental science – hence instruction requires:
Demonstrations
Hands on activities
Laboratory Exercises
“If it’s chaotic, then how can we measure anything?”
Physics Student
Mechanical Chaotic systems
Kinesthetic - less abstract; Typically slower evolving; Transducers to measure;
Mechanical skill level/expense for assembly
Berdahl, J. P. and K. V. Lugt (2001). "Magnetically driven chaotic pendulum."
American Journal of Physics 69(9): 1016-1019.
Nunes, J. E. B. G. and Jr. (1997). "A mechanical Duffing oscillator for the
undergraduate laboratory." American Journal of Physics 65(9): 841-846.
VERSUS
Electronic Chaotic systems
More abstract; Audio frequencies or higher; Voltage output, Electronic skill level for
assembly; Components are inexpensive
Roy, P. K. and A. Basuray (2003). "A high frequency chaotic signal generator: A
demonstration experiment." American Journal of Physics 71(1): 34-37.
Jones, B. K. and G. Trefan (2001). "The Duffing oscillator: A precise electronic
analog chaos demonstrator for the undergraduate laboratory." American Journal of
Physics 69(4): 464-469.
Wiener, R (2005). Controlling Chaos in a Simple Electronic Circuit, PNACP
(unpublished).
•Publications introducing “Sprott” systems:
Sprott, J. C. (2000), "Simple chaotic systems and circuits,"
American Journal of Physics 68(8): 758-763
Kiers, K., D. Schmidt, J.C. Sprott (2004), "Precision
measurements of a simple chaotic circuit," American Journal of
Physics 72(4): 503-509.
OAVR :V  V  0
OACR : i  i  0
C
R
Vin
-
Vout
+
LM741
Vin
i
R
q
 Vout 
C

Vin
V out  
RC

 V out
Vout
i

C
1

Vin dt

RC
OAVR :V  V  0
R
OACR : i  i  0
V1
R1
V2
R
Vout
-
+
LM741
V1
 i1
R
Vout
V2
 i2
R
R1
  V1  V2 
R
 Vout
 i1  i2
R1
Sprott, J. C. (2000), "Simple chaotic systems
and circuits," American Journal of Physics 68(8):
758-763
Sprott, J. C. (2000), "Simple
chaotic systems and circuits,"
American Journal of Physics
68(8): 758-763
Circuit Analysis
Rv
VB
( RC) 2Vx
 (RC)Vx
Vx
R



( RC) Vx  
( RC) 2Vx  ( RC)Vx  Vx  VB
RV
3
x   Ax  x  x 1
BTW: Can be expressed as
three coupled first order ODE’s
Notes: Scaling of time; Rv is the control parameter
(don’t use voltage divider)
V
Diode
=
x
D(x)
X=0
(Wiener, et. al)
X=0
Braunstein
Demonstrating Chaos
Audio Amplifier and
Speaker
Investigating
Chaos
Oscilloscope
 (RC)Vx
Vx
Time Series and Phase Space
Time Series and Phase Space
Investigating
Chaos
Computer
Data
Acquisition
and Analysis
( RC) 2Vx
NI-DAQ
LabVIEW
 (RC)Vx
Vx
Time Series
Time Series
Ph ase Sp ace o f sin g le lo o p
6
4
2
Vx
(v o lts)
0
-2
-4
-1. 00
-5. 00E-1
0. 00
.
RC3Vx (v o lts)
5. 00E-1
1. 00
Ph ase Sp ace o f 8 lo o p
10. 0
8. 0
6. 0
4. 0
Vx (v o lts)
2. 0
0. 0
-2. 0
-4. 0
-1. 00
-5. 00E-1
0. 00
5. 00E-1
.
RC3Vx (v o lts)
1. 00
1. 50
P h ase S p ace o f Ch ao s
10. 0
8. 0
6. 0
4. 0
Vx (v o lts)
2. 0
0. 0
-2. 0
-4. 0
-1. 00
-5. 00E-1
0. 00
5. 00E-1
.
RC3Vx (v o lts)
1. 00
1. 50
Po wer Sp ectru m o f two lo o p
5. 0
L n(Amplitude)
0. 0
-5. 0
-10. 0
-15. 0
-20. 0
-25. 0
0
500
1000
1500
F req u en cy (Hz)
2000
2500
3000
Return or Lorenz
Map
Investigating Chaos
Computer Data
Acquisition and
Analysis with
computer controlled
(voltage controlled)
resistor -LM13700
VB
( RC) 2Vx
NI-DAQ
LabVIEW
 (RC)Vx
Vx
Bifurcation Diagram
What Else ?
•Full Characterization of the strange attractor
•Other Sprott circuits and other chaotic circuits
•Synchronization of chaos (tuning/encryption)
•Control of Chaos
Thanks to the following Central Washington University students:
Sam Rowswell
David Cross
Erika Beam
Ryan Tervo
Alfredo Meza