Transcript Lorenz_Attractor_Poster_(Minsu_Bang__Hiroto_Kato)x

A Simple Circuit Implementation of Op-Amp Based Lorenz
Attractor
Hiroto Kato, Minsu Bang
Department of Electrical Engineering, San Jose State University, San Jose, California 95192
Abstract
Application of the circuit
The differential equations suggested by Edward
Lorenz explain the reason why it is difficult to predi
ct complex events such as weather change, stock ma
rket trends at high accuracy; because even very smal
l difference in the initial condition can greatly affect
the result.
Since the circuit is a model of differential equations,
the system is deterministic. Also, the system is known to
produce pseudo random signals, which only occupies
confined regions: attractor.
To understand this resulting behavior, a Lorenz
Attractor circuit was built using three OPAMPs(LM
741) and two multipliers(AD633). A typical shape o
f the output signal of z with respect to x is owl’s fac
e (Figure 1). However it was found that small differ
ence in resistance values can yield completely differ
ent output patterns.
The objective of this experiment was to show on
e of the outputs(solutions), z and the sensitivity to th
e initial condition. We observed the output actually a
ppears on the scope as expected (owl’s face). Before
doing the actual output z measurement, transient sim
ulation was done through LTspice.
With these characteristics, Lorenz attractor are used in
experiments on studies of synchronizations including
secure communication systems. Other application
includes the signals being used for parts of avant-garde
music in order to produce specific effects being desired
by the artists.
Where, C1=c2=c3=0.1uF, R1=R2=100kohm,
R5=35.7kohm, R3=10kohm, R4=1Megohm,
R6=10kohm, and R7=374kohm in order to satisfy
the conditions: a=10, b=28, c=8/3. This circuit was
then simulated in LT-Spice prior to physical
implementation. Since the designed values for R5,
R7, and Capacitors were not found, we used
R5=38.9kohm, R7=504kohm, and C=0.07uF
instead. Simulation in LT-Spice was performed
every time the value was changed.
Design methodology
The following three equations describe the Lorenz
attractor;
dx
 a ( y  x),
dt
dy
 bx  y  xz,
dt
dz
 xy  cz
dt
Figure 4. Ltspice Simulation of Lorenz Attractor
<http://youtu.be/3kq31jvjKpw>
With a=10, b=28, c=8/3, integrating three equations, they
become;
x  a ydt  a xdt,


y  b  xdt   ydt   xzdt,
z   xydt  c  zdt
Here we can see inverse integrators, summers, and
multipliers can model these equations.
Figure 2 shows the block diagram of the circuit:
Though the shape produced by Lorenz attractor
is three dimensional, the test was verified by
examining the result on x-z plane in both LT-Spice
and the oscilloscope since neither can produce three
dimensional representation of the trace made by the
Lorenz attractor. In the end, we observed successful
result by producing the trajectory of the Lorenz
attractor.
Figure 1. A Typical Output of Lorenz Attractor
Theory
Lorenz Attractor can be expressed as,
Figure 5. Actual Circuit Built
dx/dt = p(y-x)
dy/dt = rx-y-xz
dz/dt = xy-bz
x is for the rate of rotation of the cylinder, and y
represents the temperature difference at opposite sid
es of the cylinder. z represents the deviation of the s
ystem from a linear.
Figure 2. Block Diagram of Lorenz Attractor
Prandtl number, p is the ratio of the fluid viscosi
ty of a substance to the thermal conductivity. Raylei
gh number, r is the temperature difference between t
he top and bottom of the gaseous system. Finally, b i
s the ratio of the width to height of the container for
the gaseous system.
Figure 6. Output Z
<http://youtu.be/UFr5kkwOMZU>
Using integration on both sides of the equations
,
Conclusion
We have observed completely different
trajectories with small change in our resistor values
which represent the complexity of the simple circuit
as a model of attractor. If the result were presented
in three dimension, Vx vs Vy vs Vx, the output must
appear to be even more complex.
This new equation set allows us to realize
Lorenz Attractor with integrators and multipliers
with appropriate resistance and capacitor values.
Acknowledgments
Figure 3. Circuit Diagram of Lorenz Attractor
The authors wish to thank Linear tech for the
OPAMPs, and Analog Devices for the multipliers.
Key References
For further information
1.
http://frank.harvard.edu/~paulh/misc/lorenz.htm
Hiroto Kato: [email protected]
2.
Andrew Ho, Lorenz’s Attractor <http://www.zeuscat.com/andrew/chaos/lorenz.html>
Minsu Bang: [email protected]