Technological applications of deterministic chaos.
Thesis DisciplineElectrical Engineering
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Engineering and technological applications and consequences of deterministic chaos are considered. Four technological areas are investigated: electrical noise, electrical signal sources, data encryption and packet switching. In addition, deterministic chaos in both dissipative and conservative systems is reviewed, the philosophical and practical consequences that stem from deterministic chaos are considered, and the historical development that has led to the present renewal of interest in dynamical systems is presented. The use of a hierarchy of nonlinear recursive equations to generate coloured noise and the tailoring of these equations to generate noise with a specified probability density function and power spectrum is examined. The most immediately striking aspect of the concept of a hierarchy of recursive equations is that it constitutes such an elementary means of generating a noisy process having an excess low frequency character (in particular noise having an almost 1/f power spectrum). This is claimed to constitute a significant addition to the literature on l/f noise. An analysis of the hierarchy in terms of Lyapunov exponents and information theory is presented. It is found that the dynamics of the hierarchy exhibits non uniformity and noise-induced predictability, thereby providing further evidence for the conjecture that non-uniformity is a necessary condition for the occurrence of noise induced predictability. It is claimed that noise-induced predictability may be of use technologically, since it might allow systems and processes which are at present unpredictable (e.g. system reliability, weather), to be made (more) predictable and useful. The extent and under what circumstances deterministic chaos contributes to noise within sinusoidal oscillators is considered. Effects (Le. signal-dependent delay and circuit parameter variations) which tend to be neglected in established approaches to oscillator analysis are included in the oscillator models studied. Conditions sufficient for an oscillator to exhibit deterministic chaos are found to be, first, the existence of a signal delay within the oscillator and, second, certain types of amplifier nonlinearity. It is conjectured that all oscillators may satisfy these conditions, and therefore to some degree exhibit deterministic chaos. A novel discrete time oscillator is developed. It gives insights into the way deterministic chaos arises within a closed loop. Chaotic behaviour in generalisations of an oscillator circuit named after Chua are examined. It is found that relatively minor circuit alterations can inhibit the chaotic behaviour in the circuit. It is indicated how this appears to provide insight into chaotic dynamics in general. The extent to which the seemingly random numbers generated by a chaotic dynamical system are suitable for data encryption is examined. This leads to an examination of the consequences of number quantisation (i.e. the use of finite precision numbers) which inevitably occurs in digital implementation of chaotic systems. Two encryption schemes are considered, termed isolated and influenced chaotic encryption. Isolated chaotic encryption is based on a conventional encryption scheme (the one time tape), while influenced chaotic encryption is a new encryption method. Sequences exhibiting maximal entropy are required for encryption. Computer simulations performed on finite length sequences (limited to allow the computer simulation to run in a reasonable time), generated by the encryption schemes, show that they exhibit maximal entropy. Although this does not confirm that sequences longer than those tested exhibit maximal entropy, it does reveal that deterministic chaos has potential for data encryption and is worthy of further study. The advantage of chaotic encryption schemes over other schemes may lie in their ease of implementation, and the difficulty of breaking the encryption from observation of the encrypted sequence. A new packet switching flow control algorithm, termed cooperative flow control, is examined. The new algorithm is based on a modification to an algorithm which forms the basis of many flow control algorithms used in practice. Cooperative flow control represents an attempt at developing a packet entry flow control algorithm which induces resource efficient self-organising behaviour within a packet switching network, while at the same time ensuring that a specified grade of service is delivered to network users. Under certain traffic arrival patterns cooperative flow control becomes chaotic, demonstrating that the algorithm can induce self-organising behaviour within the network. The studies reported herein suggest that if such an algorithm can be perfected it may provide a considerable advance in communication network design.