Discontinuous Spirals of Stable Periodic Oscillations

OPEN SUBJECT AREAS: APPLIED MATHEMATICS Discontinuous Spirals of Stable Periodic Oscillations Achim Sack1, Joana G. Freire1,2,3, Erik Lindberg4, Thorsten Pöschel1,2 & Jason A. C. Gallas1,2,3,5 PHYSICAL SCIENCES 1 Received 11 October 2013 Accepted 7 November 2013 Published 27 November 2013 Correspondence and requests for materials should be addressed to J.A.C.G. (jason. ) Institute for Multiscale Simulations, Friedrich-Alexander Universität, D-91052 Erlangen, Germany, 2Departamento de Fı́sica, Universidade Federal da Paraı́ba, 58051-970 João Pessoa, Brazil, 3Centro de Estruturas Lineares e Combinatórias, Faculdade de Ciências, Universidade de Lisboa, 1649-003 Lisboa, Portugal, 4DTU Elektro Department, 348 Technical University of Denmark, DK2800 Lyngby, Denmark, 5Instituto de Altos Estudos da Paraı́ba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil. We report the experimental discovery of a remarkable organization of the set of self-generated periodic oscillations in the parameter space of a nonlinear electronic circuit. When control parameters are suitably tuned, the wave pattern complexity of the periodic oscillations is found to increase orderly without bound. Such complex patterns emerge forming self-similar discontinuous phases that combine in an artful way to produce large discontinuous spirals of stability. This unanticipated discrete accumulation of stability phases was detected experimentally and numerically in a Duffing-like proxy specially designed to bypass noisy spectra conspicuously present in driven oscillators. Discontinuous spirals organize the dynamics over extended parameter intervals around a focal point. They are useful to optimize locking into desired oscillatory modes and to control complex systems. The organization of oscillations into discontinuous spirals is expected to be generic for a class of nonlinear oscillators. F ully characterizing self-oscillations in nonlinear oscillators demands knowledge both of its phase-space dynamics, and the corresponding dual dynamics seen in stability (phase) diagrams in the system control parameter space. While phase-space dynamics has been extensively studied ever since the pioneering works of Poincaré and others well over a century ago, the associated description of stability diagrams had to wait for the advent of modern computers, particularly to characterize the intricate chaotic phases and complex phases corresponding to periodic motions other than stationary or the simple nonstationary periodic oscillations which appear immediately after Hopf bifurcations. Although it is not difficult to calculate Hopf boundary curves, no prescription exist for delimiting stability domains of the large succession of oscillatory phases that normally follow Hopf bifurcations. For a number of years, the study of bifurcation boundaries was conducted mainly for systems governed by maps, since they are much easier to deal with both analytically and computationally. Bifurcation boundaries in phase diagrams computed for maps were discussed recently by Lorenz in what turned out to be his last paper1. But the situation is evolving fast. Presently, detailed stability diagrams resulting from extensive simulations are becoming available not only for maps but also for a number of flows, i.e. for systems governed either by finite sets of ordinary nonlinear differential equations2–8 and, more recently, for infinite dimensional systems controlled by delay-differential equations, such as semiconductor lasers with delayed feedback9. Experimentally, the determination of stability boundaries beyond the Hopf lines is a very difficult problem. This is the challenge that we wish to address here. More specifically, we report experimental phase diagrams of high-resolution displaying detailed information about the stability boundaries measured for an electronic circuit. Such stability diagrams summarize the dynamics over extended intervals of control parameters and provide detailed information about virtually all oscillations present in the circuit, including chaotic ones. Our diagrams reveal an unanticipated and quite surprising organization of parameters, that form a large discontinuous spiral of the stability phases underlying the set of periodic oscillations. Whirling along the discontinuous phases of the spiral implies having to jump repeatedly between an infinite alternation of chaotic and periodic phases. As explained below, such discontinuities are signatures of systematic bifurcation cascades unfolding according to a hitherto unseen and more complex scenario not included, e.g. in the famous classification of Arnold10. Since all oscillations underlying the discontinuous spiral are stable oscillations, the spiral is directly accessible to experimental work. From a theoretical point of view, the determination of the frontiers between chaotic and periodic phases of oscillators is a complex task that needs to be addressed in all branches of the natural sciences. Unlike the relatively SCIENTIFIC REPORTS | 3 : 3350 | DOI: 10.1038/srep03350 1 www.nature.com/scientificreports tame situation for fixed equilibria, whose boundaries are accessible by straightforward calculations, not even approximate methods exist for delimiting parameter windows likely to contain, e.g. chaotic bursting or pulsation phases of high-periodicity and their rich accumulation limits which are so typical in laser dynamics11, in biological neural networks12, and in many other applications13. Even worse, it seems likely that no analytical methods will ever exist to delimit such stability domains. For instance, think of the problem of computing analytically the self-similar boundary of the Mandelbrot set. Thus, the theoretical delimitation of stability phases can be dealt numerically only. Regularities of complex phases are universal invariants that help unify seemingly distinct systems into universality classes whose knowledge is a great asset for applications. Results Our experiments are conducted with the help of the electronic circuit shown schematically in Fig. [1]. The circuit is a slight variation of an autonomous Duffing-like proxy introduced recently14,15, having the distinctive technological advantage of bypassing noisy spectra which normally pollute driven (i.e. non-autonomous) oscillators. An additional characteristic is that our oscillator is among the most precise devices presently in existence, allowing its modes of oscillation to be measured with very high accuracy. In a variety of configurations, such oscillators are also popular nowadays as components in experiments conceived to study the variegated complex phenomena predicted for nonlinear oscillators. Our circuit is governed by the equations 3 €x{bx{xzx _ zkz~0, _ z_ ~ðx{z Þv , ð1Þ 14,15 where variables and parameters are all adimensional . Equivalently, these equations can be also conveniently written as _ x~y, y_ ~x{x3 zby{kz, z_ ~ðy{z Þ v: ð2Þ Here, b is used t (...truncated)