Ultra-Wide-Band Digital Chaotic Circuits
Part II: Characterization and Details
Hugo L. D. de S. Cavalcante, Rui Zhang, Zheng Gao, Joshua S. Socolar, and Daniel J. Gauthier
Department of Physics, Duke University, Box 90305, Durham, NC, 27708
Matthew Adams, and Daniel P. Lathrop
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD, 20742
Introduction
Bifurcation diagram showing bistability
Two close time series
2.5
Voltage (V)
We have observed that a simple autonomous
Boolean-like network with time delays can
display chaos. The network is realized using
three high-speed, commercially available logic
gates that feed back to each other with
independent delays.
Simulated gate – comparison with experimental gate
Input
2
1.5
Experimental
Data
1
0.5
Actual output
0
Simulated output
840
860
880
900
920
Time (ns)
The distance in Boolean space at continuous time
was adapted to
Temporal evolution of the voltage
Output (V)
2
1
0
1400
1450
1500
1550
Time (ns)
1600
1650
1700
For details, see the poster P72 by Rui Zhang et al.
Chaos
For some applications it is important to verify
whether the signals are deterministic chaos or
amplified noise or quasi-periodic oscillations. In
chaotic systems trajectories that are close in phase
space separate at an exponentially increasing rate.
Periodic orbits
Bifurcation with numerical model
We used this definition to estimate the maximum
-1
positive Lyapunov exponent. (λ = 0.3 ns )
Boolean Distance (arb. units)
Network diagram
Schematics diagram
Bifurcation diagram indicating
windows of periodic behavior
Numerical
Model
Evolution of close trajectories
Conclusion
Models
Time (ns)
Bifurcation Diagrams
The delay times change with the supply voltage
used in the circuit. We have used the supply
voltage as a bifurcation parameter to change the
dynamics, for instance, from chaotic to periodic.
Theory of Boolean delay equations [1,2] can not
explain periodic windows in the bifurcation
diagram. Finite response time of the gates and
other non-ideal effects are necessary to have
stable periodic orbits while scanning the relative
values of the delay times. We have used
continuous models for the analog voltages
produced by CMOS gates
A simple and inexpensive electronic circuit can
display complex behavior. The existence of
stable periodic orbits and exponentially
unstable aperiodic ones, depending on system
parameters, is an evidence of deterministic
chaos. The circuit bears similarity with general
networks of strongly nonlinear elements.
References
[1] - D. Dee and M. Ghil, Boolean Difference Equations, I:
Formulation and Dynamical Behavior, SIAM J. Appl. Math. 44,
111 (1984).
[2] - M. Ghil and A. Mullhaupt, Boolean Delay Equations. II.
Periodic and Aperiodic Solutions, J. Stat. Phys. 41, 125 (1985)