CNCS Graduate Certificate Recipient
Jonathan Neal Blakely
Thesis Title: Experimental Control of a Fast Chaotic Time-Delay
Opto-Electronic Device
Ph.D. Final Defense Date: 28 April 2003
Ph.D. Dissertation Committee:
Dr. Daniel J. Gauthier, Supervisor
Dr. Robert P. Behringer
Dr. Stephen W. Teitsworth
Dr. Joshua E. S. Socolar
Dr. William T. Joines
Abstract:
The focus of this thesis is the experimental investigation of
the dynamics and control of a new type of fast chaotic opto-electronic device:
an active interferometer with electronic bandpass filtered delayed feedback
displaying chaotic oscillations with a fundamental frequency as high as
100 MHz. To stabilize the system, I introduce a new form of delayed feedback
control suitable for fast time-delay systems. The method provides a new
tool for the fundamental study of fast dynamical systems as well as for
technological exploitation of chaos.
The new opto-electronic device consists of a semiconductor laser,
a Mach-Zehnder interferometer, and an electronic feedback loop. The device
offers a high degree of design flexibility at a much lower cost than other
known sources of fast optical chaos. Both the nonlinearity and the timescale
of the oscillations are easily manipulated experimentally. To characterize
the dynamics of the system, I observe experimentally its behavior in the
time and frequency domains as the feedback-loop gain is varied. The system
displays a route to chaos that begins with a Hopf bifurcation from a steady
state to a periodic oscillation at the so-called fundamental frequency.
Further bifurcations give rise to a chaotic regime with a broad, flattened
power spectrum. I develop a mathematical model of the device that shows
very good agreement with the observed dynamics.
To control chaos in the device, I introduce a new control method
suitable for fast time-delay systems, in particular. The method is a modification
of a well known control approach called time-delay autosynchronization
(TDAS) in which the control perturbation is formed by comparing the current
value of a system variable to its value at a time in the past equal to
the period of the orbit to be stabilized. The current state of a time-delay
dynamical system retains a memory of the state of the system one feedback
delay time in the past. As a result, the past state of the system can be
used to predict the current state. In order to take advantage of this effect,
the new control method forms a perturbation according to the TDAS scheme
but delays actuation of the control perturbation by a time equal to the
feedback delay time of the system to be controlled. This effectively sets
the control-loop latency equal to the feedback delay time of the uncontrolled
system. I demonstrate this control method experimentally by stabilizing
a periodic orbit of the active interferometer. I quantify the effectiveness
of the controller by measuring the range of feedback loop gains over which
the orbit can be stabilized. The stabilized orbit, which oscillates with
a frequency of 51.8 MHz, is the fastest unstable periodic orbit in a chaotic
system controlled experimentally to date.
Application of the new control method requires the adjustment
of two time delays in the controller. The first, the control delay time,
should equal the period of the orbit to be controlled, while the second,
the control loop latency, should equal the feedback delay time of the system
to be controlled. I investigate, through experiments and simulations, the
sensitivity of the method to errors in setting these time delays. I find
that the control delay time must be set exactly equal to the period of the
orbit to minimize the control perturbations when the orbit is stabilized.
In contrast, the control loop latency may vary within a finite range without
affecting the performance of the controller.
