Resonance and Fractal Geometry
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Resonance and Fractal Geometry Henk W. Broer
Received: 25 September 2011 / Accepted: 9 January 2012 / Published online: 27 January 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com
Abstract The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples. Keywords Resonance · Resonance tongue · Subharmonic bifurcation · Covering space · Cantor set · Fractal geometry · Devil’s staircase · Lyapunov diagram
1 What Is Resonance? A heuristic definition of resonance considers a dynamical system, usually depending on parameters, with several oscillatory subsystems having a rational ratio of frequencies and a resulting combined and compatible motion that may be amplified as well. Often the latter motion is also periodic, but it can be more complicated as will be shown below. We shall take a rather eclectic point of view, discussing several examples first. Later we shall turn to a number of universal cases, these are context-free models that occur generically in any system of sufficiently high-dimensional state and parameter space. Among the examples are the famous problem of Huygens’s synchronizing clocks and that of the Botafumeiro in the Cathedral of Santiago de Compostela, but also we briefly touch on tidal resonances in the planetary system. As universal models we shall deal with the Hopf–Ne˘ımark–Sacker bifurcation and the Hopf saddle-node bifurcation for mappings. The latter two examples form ‘next cases’ in the development of generic bifurcation theory. The term universal refers to the context independence of their occurrence: in any with certain H.W. Broer () Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands e-mail: [email protected] url: http://www.rug.nl/~broer
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Fig. 1 Amplitude-response diagrams in the (ω, R)-plane. Left: the periodically driven harmonic oscillator (1) and right: the driven Duffing oscillator (2)
generic specifications these bifurcations occur in a persistent way. We witness an increase in complexity in the sense that in the parameter space the resonant phenomena correspond to an open & dense subset union of tongues, while in the complement of this a nowhere dense set of positive measure exists, corresponding to multi- or quasi-periodic dynamics. This nowhere dense set has a fractal geometry in a sense that will be explained later. From the above it follows that this global array of resonance tongues and fractal geometry has a universal character. As we shall see, both locally and globally Singularity Theor
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