Kevin M. Short

Reconstructing the keystream from a chaotic encryption scheme

A new technique of secure communication was developed which uses a synchronizing chaotic communication channel to transmit an encrypted message. The important feature of this system is that the keystream used in the encryption stage is not transmitted but may be dynamically reconstructed by the intended receiver. Since an additional level of encryption is employed, it was claimed that an eavesdropper may have limited success using nonlinear dynamic (NLD) forecasting techniques to extract the message. In this paper we show that there may be sufficient geometric information in the transmission to extract an estimate of the keystream (although it could require a difficult search) as well as characteristics of the encrypting function. Consequently, it may be possible to recover the hidden message with good accuracy.

Controlled transitions between cupolets of chaotic systems

We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the systems initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstras shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems.

On the Potential for Entangled States Between Chaotic Systems

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

A Dynamical Systems Approach to Stability Tracking of Treadmill Running

Traditional analysis of running gait utilizes averaged biomechanical data from several strides to generate a mean curve. This curve is then used to define the average picture of a runners gait. However, such measures are frequently accompanied by time normalization, which results in a loss of temporal variations in the gait patterns. An examination of stability requires analysis of both time and position, therefore loss of such information makes stability analysis difficult. On the contrary, the use of a dynamical systems approach for gait analysis allows for a better understanding of how variations in gait pattern change over time. In the current study runners ran on a treadmill, with both a flat and uneven surface, at a self selected speed. Three-dimensional position data was captured for 11 different anatomical locations at a frequency of 120 Hz using a Qualysis motion capture system. The data was first shifted to a lumbar coordinate system to account for low frequency drift attributed to the subjects’ drift on the treadmill. Since all of the markers were rigidly connected, via the subject, the movements and variations of certain components of the 33-dimensional measurements were not independent. As a result, it was possible to reduce the dimensionality of the transformed data using singular value decomposition techniques. The primary components were then analyzed using the method of delay embeddings to extract geometric information, revealing the natural structure found in the data as a result of the periodicity of each running stride. A nearest neighbor mean stride orbit was then computed to create a reference orbit, so that deviations from the mean stride orbit can be measured. The expectation was that a more stable running configuration would lead to smaller deviations from the mean stride orbit. On-going work that will be reported includes: (i) analysis of running stability related to the reference stride comparator, (ii) compensation of lumbar centroid dynamics, (iii) reconstructions using one dimension from the lumbar centroid transformed data, and (iv) consideration of transients, fatigue, adaptation, etc.Copyright