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

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

Bifurcation initially intended "splitting into elements. " specifically, a approach lower than is going a bifurcation while 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 structures are concerned, has been the topic of study for lots of scientists and engineers for the previous hundred years just because bifurcations are fascinating. a great approach of realizing bifurcations will be to work out them first and examine theories moment. in a different way will be to first understand the elemental recommendations and theories after which see what they seem like. In any occasion, you should either notice experiments and comprehend the theories of bifurcations. This booklet makes an attempt to supply a basic viewers with either avenues towards realizing bifurcations. in particular, (1) a number of concrete experimental effects acquired from digital circuits are given in bankruptcy 1. all of the circuits are extremely simple, that's the most important in any test. The circuits, although, shouldn't be too easy, another way not anything attention-grabbing can take place. Albert Einstein as soon as stated "as easy 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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3 Structure of the Double Scroll (e) (f) (g) Fig. 4. 25. (f) Cross section at VOl = 1:(0. 20. All of them have the same scaling. @1985 IEEE. space x > 0 and another in the half space x < O. Note that the origin belongs to the Lorenz attractor and that the same is true in the Double Scroll. At the origin in the Lorenz attractor, however, all eigenvalues are real, whereas in our case, the origin has one positive real eigenvalue and a pair of complex-conjugate eigenvalues which is responsible for the curled up structure.

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. It is a lot of fun. A circuit is an interconnection of components.

Fig. 4(a) shows the location of the three equilibrium points. Note the equilibrium points are independent of Cl. 8 p+ and P- are stable. 5)2 + 280) a pair of eigenvalues crosses the imaginary axis and a (piecewise-linear) Hopf bifurcation occurs. 0. 2(a) and note that any asymmetric periodic orbit must occur in a pair because (DS 1) is symmetric with respect to the origin. 0, a period-doubling is initiated. Fig. 2, while Fig. 44. Further period-doubling was difficult to observe. 5, the attractor in Fig.