By Takashi Matsumoto, Motomasa Komuro, Hiroshi Kokubu, Ryuji Tokunaga

ISBN-10: 4431682430

ISBN-13: 9784431682431

ISBN-10: 4431682457

ISBN-13: 9784431682455

Bifurcation initially intended "splitting into components. " particularly, a method below­ is going a bifurcation whilst there's a qualitative switch within the habit of the sys­ tem. Bifurcation within the context of dynamical platforms, the place the time evolution of platforms are concerned, has been the topic of analysis for plenty of scientists and engineers for the prior hundred years just because bifurcations are attention-grabbing. an exceptional manner of figuring out bifurcations will be to work out them first and research theories moment. otherwise will be to first understand the elemental ideas and theories after which see what they seem like. In any occasion, it's best to either realize experiments and comprehend the theories of bifurcations. This booklet makes an attempt to supply a common viewers with either avenues towards figuring out bifurcations. in particular, (1) quite a few concrete experimental effects got from digital circuits are given in bankruptcy 1. the entire circuits are extremely simple, that's the most important in any test. The circuits, even if, shouldn't be too easy, in a different way not anything fascinating can take place. Albert Einstein as soon as stated "as basic as pos­ sible, yet not more" . one of many significant purposes for the circuits mentioned being easy is because of their piecewise-linear features. specifically, the voltage­ present relationships are composed of numerous line segments that are effortless to construct. Piecewise-linearity additionally simplifies rigorous research in a drastic guy­ ner. (2) The piecewise-linearity of the circuits has a ways attaining consequences.

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0), (v) The saddle-type periodic orbit survives even after the death of the attractor (1/e 1 = 13,0). ©1986 Elsevier Science Publishers. 575 . 575, a periodic window of Fig. 2(f)). Further increase of 1/C1 results in a period-doubling cascade of this period-3 orbit and then 16 1. Bifurcations Observed from Electronic Circuits again the chaotic attractor given in Fig. 2(g)). E Interior Crisis (The Double Scroll) As one increases 1/01 further, the attractor abruptly enlarges itself and creates two holes located symmetrically with respect to the origin.

A) n = 1. (b) n = 2. 4 The set k=-n l(R) with n = 4. 1. 1 Introduction The purpose of this chapter is to give concrete experimental as well as numerical results of bifurcation phenomena in electronic circuits together with reasonable theoretical justifications. All circuits described are so simple that high school students can build them. All of them except for one, in addition, behave within audible frequencies. We strongly recommend that the reader build at least one of them, and take a look at and listen to the bifurcations.

0. With this value, a saddle-type periodic orbit has been found outside the periodic attractors. It is shown by the broken closed curve in Fig. 4(n). Since this orbit is not attracting, one cannot observe it on an oscilloscope. Nor, since it is saddle-type, can one observe it with a digital computer by integrating backward in time. Newton iteration was used to find it. As we increase C 1 , the saddle-type periodic orbit shrinks gradually as shown in Fig. 4(o-s). 5, the attractor is located very close to the saddle-type periodic orbit (Fig.

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Bifurcations: Sights, Sounds, and Mathematics by Takashi Matsumoto, Motomasa Komuro, Hiroshi Kokubu, Ryuji Tokunaga

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