Synchronization of Chaotic Flows using Variable Replacement

ABSTRACT
Sending sensitive information between remote locations can be accomplished using coupled chaotic oscillators to mask and  retrieve the information. The variable replacement method of Pecora and Carroll provides a simple method to synchronize transmitting and receiving systems. This paper examines variable replacement within the Lorenz equations as a method for achieving synchronization of multi-dimension allows. We discuss stability considerations imperative to system designers and show that Lyapunov exponents provide a ready litmus test to di.erentiate between synchronizing and uncoupled systems. To complete this survey, we study how synchronization depends on system parameters and initial conditions and present practical design considerations that motivate this analysis.

COLLABORATORS
Martina Balestra, Olin '10


PROJECT BACKGROUND
To fulfill my final mathematics requirement at Olin, I took a class on Nonlinear Dynamics and Chaos with Prof. John Geddes. The class focused on interactively exploring basic concepts in the field, covering discrete and continuous theory. The course's only deliverables were four-page research reports in the style of Physical Review Letters highlighting results of a multi-week investigation on a topic of the student's choosing (related to the current topic, of course). This page highlights my final such report, which focused on the topic of advanced topics in continuous chaotic flows.

My partner and I reviewed online and library literature to find an exciting application (synchronization!) and then refined our goals to understanding one way for how synchronization arises (first reported in [1]) and what kinds of governing equations yield synchronization in this method. We developed a proof-of-concept simulation in Matlab and used it to conduct experiments studying which modifications of the Lorenz equations might yield synchronization.  I learned a lot about calculating Lyapunov exponents and thinking practically about chaotic systems.


FULL TEXT
Available as attached PDF for viewing and download below.


INTRODUCTION

The ability to mask signals containing information has important applications in communications and signal processing. One possible method to safely transmit a message between remote locations involves chaotic oscillators producing multi-dimensional flows as a source of pseudo-randomness for masking information. A challenge in this setting is to obtain the same masking signal at both sending and receiving ends to allow successful encryption and decryption. This process of achieving an identical multi-dimensional masking signal at both sending and receiving systems given only a partial, lower-dimensional masking signal shared between both is called synchronization, and remains a hot topic of research in dynamic systems.

The synchronization process starts with the transmitting chaotic oscillator, which we call the "driver" because it produces the pseudo-random masking signal used for encryption. The driver emits both the masked message and a partial component of the masking signal, which travel to the receiving system. Here, the partial masking signal flows into the \response" oscillator governed by an identical system of equations as the driver. This input synchronizes the response oscillator, causing it to produce the complete original masking signal for decrypting the message. In practice, the governing equations behind each oscillator can be physically achieved as electrical circuits, where the equation parameters correspond to the values of circuit components (e.g. capacitors, resistors) [4].

Designing a communications system based on synchronized chaotic oscillators is a challenge for many reasons. First, the mathematics behind coupling chaotic oscillators are nontrivial. Additionally, practical concerns related to implementing equations in circuitry are numerous. Finally, achieving a protocol robust to perturbations is particularly important because long-term or sensitive installations need to recover from the introduction of noise into the system. Thus, considering all the possible design considerations involved is compulsory for successful system design. This report intends to provide an introductory overview in pursuit of this goal.


EXAMPLE COMPARISON: STABLE vs. UNSTABLE SYSTEMS


Some variable replacements in response system governing equations can produce synchronization, while others cannot.  The litmus test to determine the difference is to calculate the Lyapunov exponents.  If all converge to values below zero, synchronization is possible.  If a system yields any exponents above zero, it will not synchronize.  Side-by-side comparisons of two possible response systems based on Lorenz equations illustrate this difference.
SYNCHRONIZING SYSTEM

Consider the governing equations shown below.


Note that the driving variable y from the driver oscillator has been replaced in the top equation.




We can see in the background plot of that this response system has all Lyapunov exponents converge to negative asymptotic values, indicating convergence and stability.

This means synchronization is possible, as evidenced by the phase plots below all converging to the y=x line after an initial transient phase.



UNSTABLE SYSTEM

Alternatively, consider the response system governing equations shown below.

Note that the driving variable y from the driver oscillator has been replaced in the second equation.




We can see in the background plot of the Lyapunov exponents that the asymptotic value of the largest exponent converges to a value well above 0, indicating divergence and a lack of synchronization.

This makes synchronization impossible.  Notice how the phase plots below do not converge to a y=x line.




Ċ
Mike Hughes,
May 20, 2010, 1:47 PM
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