Qualitative Reasoning for Automated Exploration for Chaos

Toyoaki Nishida

Chaos is ubiquitous in our everyday life and even a simple system may manifest chaotic behaviors. Chaos has been a challenge to the methodology of qualitative reasoning as well as classic science and engineering, due to unpredictability and complexity of behavior. In this paper, I claim that associating continuous domain with symbolic representation, a basic principle of qualitative reasoning, is vital for automating analysis of chaos, as long as it is properly formalized. As an empirical support to this claim, I present a computer program called PSX3 that can semi-automatically explore for chaotic behavior of a given system of piecewise linear ordinary differential equations with three unknown functions. The power of PSX3 originates from an ability of reasoning about smooth surfaces that implicitly exist in the phase space. PSX3 is im lemented using Common Lisp and Mathematica TM.

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