1. 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
Abstract
The differential equations suggested by Edward Lorenz explain the reason why it is difficult to predict complex events such as weather change, stock market trends at high accuracy; because even very small difference in the initial condition can greatly affect the result.
To understand this resulting behavior, a Lorenz Attractor circuit was built using three OPAMPs(LM741) and two multipliers(AD633). A typical shape of the output signal of z with respect to x is owl’s face (Figure 1). However it was found that small difference in resistance values can yield completely different output patterns.
The objective of this experiment was to show one of the outputs(solutions), z and the sensitivity to the initial condition. We observed the output actually appears on the scope as expected (owl’s face). Before doing the actual output z measurement, transient simulation was done through LTspice.
Figure 1. A Typical Output of Lorenz Attractor
Application of the circuit
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.
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.
Design methodology
The following three equations describe the Lorenz attractor;
With a=10, b=28, c=8/3, integrating three equations, they become;
Here we can see inverse integrators, summers, and multipliers can model these equations.
Figure 2 shows the block diagram of the circuit:
Figure 2. Block Diagram of Lorenz Attractor
Figure 3. Circuit Diagram of Lorenz Attractor
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.
Figure 4. Ltspice Simulation of Lorenz Attractor
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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 5. Actual Circuit Built
Figure 6. Output Z
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Theory
Lorenz Attractor can be expressed as,
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 sides of the cylinder. z represents the deviation of the system from a linear.
Prandtl number, p is the ratio of the fluid viscosity of a substance to the thermal conductivity. Rayleigh number, r is the temperature difference between the top and bottom of the gaseous system. Finally, b is the ratio of the width to height of the container for the gaseous system.
Using integration on both sides of the equations,
This new equation set allows us to realize Lorenz Attractor with integrators and multipliers with appropriate resistance and capacitor values.
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.
Acknowledgments
The authors wish to thank Linear tech for the OPAMPs, and Analog Devices for the multipliers.
Key References
Andrew Ho, Lorenz’s Attractor <
For further information
Hiroto Kato:
Minsu Bang: