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Theoretical Studies of Intermingled Basin Structure and Pattern Formation in Fir-Wave Regeneration based on Solvable Models

石川 大海 大阪府立大学 DOI:info:doi/10.24729/00016968

2020.07.07

概要

In this thesis we investigate, based on solvable models, intermingled basin structure (part I) and pattern formation in fir-wave regeneration (part II), both of which are multistable nonlinear sys- tems. We obtain some analytical results, which give deeper understandings of the phenomena.

In Part I, we investigate fat fractal basins, namely, riddled basins and intermingled basins, in several solvable models. These basins can be considered as an extreme obstruction to the determinism, since any finite error in the initial condition can lead to a completely di↵erent final state from that of the errorless initial condition. The complexity of a basin structure can be quantitatively characterized by the singularity spectrum, by which we analyze the multifrac-tal basin structure of riddled and intermingled basins, a particular emphasis put on the role of the transient dynamics. We show that the singularity spectra of riddled or intermingled basins usually exhibit a phase transition, as a result of a mixture of the transient and asymptotic dy- namics. We also introduce a generalized notion of the basin boundary, based on the final state sensitivity, to show that the fractal dimension of the boundary coincides with the left end point of the spectrum, and that the uncertainty exponent, a characteristic of great importance from the viewpoint of the prediction accuracy improvement, is directly expressed by the dimension.

In Part II, we investigate the pattern formation in fir-wave regeneration, where the dieback tree bands along the contour in subalpine forests forms a traveling wave like patterns migrating slowly to the leeward direction of the prevailing wind. We model the phenomenon by a con- tinuous time lattice system with time lag, and show that there exist multiple coexisting stable patterns, and that the velocity at which the wave travels depends monotonically on the wave- length of the pattern, explaining the regular pattern formation mechanism. We also develop a detailed stability analysis, by identifying the linear stability dynamics with a finite ergodic Markov chain.

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