{"title":"Fractal control boundaries of driven oscillators and their relevance to safe engineering design","authors":"J. Thompson, M. Soliman","doi":"10.1098/rspa.1990.0022","DOIUrl":null,"url":null,"abstract":"When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0),ẋ(0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), ẋ)(0), F, ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = ẋ(0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"11 1","pages":"1 - 13"},"PeriodicalIF":0.0000,"publicationDate":"1990-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"87","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1098/rspa.1990.0022","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 87
Abstract
When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0),ẋ(0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), ẋ)(0), F, ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = ẋ(0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.