Biyolojik ve elektronik osilatörlerin analizi için teori ve nümerik yöntemler

Oscillatory behavior is encountered in many types of systems including electronic, optical, mechanical, biological, chemical, financial, social and climatological systems. Carefully designed oscillators are intentionally introduced into many engineered systems to provide essential functionality for system operation. Oscillatory behavior in biological systems is seen in population dynamics models, in neural systems, in the motor system, and in circadian rhythms. Intracellular and intercellular oscillators of various types perform crucial functions in biological systems. Due to their essentialness, and intricate and interesting dynamic behavior, biological oscillations have been a research focus for decades. Genetic oscillators that are responsible for setting up the circadian rhythms have received particular attention. Oscillators in electronic and telecommunication systems are adversely affected by the presence of undesired disturbances such as noise. These have an impact on the spectral and timing properties of the ideally periodic signals generated by oscillators, resulting in power spreading in the spectrum and zero-crossing jitter and phase drift in the time domain. Unlike other systems which contain an implicit or explicit time reference, autonomously oscillating systems respond to noise in a peculiar and somewhat nonintuitive manner. Understanding the behavior of oscillators used in electronic systems in the presence of disturbances and noise has been a preoccupation for researchers for many decades. The behavior of biological oscillators under various types of disturbances has also been the focus of a good deal of research work in the second half of 20th century. The work on oscillator analysis in these two disparate disciplines seem to have progressed independently, without any cross-fertilization in between. In this thesis, we first decipher previous work on oscillator analysis in both biology and electronics by translating them into a common terminology and formalism. We then develop a rigorous, unifying oscillator analysis theory by using results and concepts from both domains in a synergistic manner. In doing so, we fill certain conceptual and theoretical gaps that we identify in oscillator analysis theories that have been developed both in electronics and biology that pertain to phase analysis. We formulate a general phase analysis technique that captures both state and parametric perturbations in a unified manner. This phase analysis technique we develop can be applied to oscillators modeled with mixed differential-algebraic equations as opposed to pure differential ones. By reviewing the numerical methods that have been developed for both electronic and biological oscillator analysis, we show that the numerical techniques currently in use for biological oscillators are superseded by the ones that have been recently developed for electronic oscillator analysis. Oscillator perturbation analysis examples produced using a Matlab oscillator analysis toolbox we have developed are presented.