The fuzzy shadowing property

Addresses the modeling of chaotic systems using fuzzy functional approximations. In numerical simulations of chaos, local modeling errors are cumulative. A natural question, therefore, arises in examining a numerically generated chaotic trajectory: is there a “true” trajectory of the chaotic system, starting from a nearby set of initial conditions, that shadows, for a certain ε distance, the numerically generated one? This paper extends the literature on fuzzy chaos by introducing concepts from chaos theory, such as δ-pseudo-orbits and ε-shadowing, to the language of fuzzy approximation theory. It presents corollaries to the fuzzy approximation theorem and the SAM theorem (B. Kosko, 1997) that describe sufficiency conditions for additive fuzzy logic systems to generate δ-pseudo-orbits that are ε-shadowed by true orbits of the dynamic process