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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 method lower than­ is going a bifurcation whilst there's a qualitative swap 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 plenty of scientists and engineers for the earlier hundred years just because bifurcations are attention-grabbing. an exceptional means of figuring out bifurcations will be to determine them first and learn theories moment. in a different way will be to first understand the fundamental innovations and theories after which see what they appear to be. In any occasion, it's best to either realize experiments and comprehend the theories of bifurcations. This publication makes an attempt to supply a basic viewers with either avenues towards realizing bifurcations. in particular, (1) numerous concrete experimental effects acquired from digital circuits are given in bankruptcy 1. the entire circuits are extremely simple, that's an important in any scan. The circuits, in spite of the fact that, shouldn't be too uncomplicated, differently 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 long way attaining consequences.

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Example text

2, the v-i characteristic naturally becomes passive for large voltages. 1(b) shows the v-i characteristic of the subcircuit N in Fig. 2(a) for a larger range. 4. ) of Fig. 1(b) does not have to be piecewiselinear to allow one to observe qualitatively the same attractor. ) of Fig. 1. Then with 1/C1 = 9, 1/C2 the chaotic attractor of Fig. 7 is observed. 65, 20 1. Bifurcations Observed from Electronic Circuits Fig. 7. 12). @1985 IEEE. 2(i) and Fig. 4(j). 3 shows the projections of the attractor which we will study.

If the graph is a single straight line, then nothing interesting can happen. The trajectory either converges to the origin or diverges to infinity. Even a sustained periodic orbit cannot occur! 1. Bifurcations Observed from Electronic Oircuits 4 (2) Note that the nonlinear resistor described by Fig. , VR iR = VR g(VR) ~ o. , VR iR ~ 0, then again, nothing interesting can happen. This is due to the simple fact that a passive resistor always dissipates power so that the trajectory has no choice except for converging to a stable equilibrium point.

Kirchhoff Current Law (KCL): The sum of the currents at any node is zero. It is easy to show that if there are m nodes, then only m - 1 of KVL can be independent. Similarly, only n - m + 1 of KCL can be independent. There are three classes of components: (1)resistors (2)capacitors and (3)inductors. Let nR, nc and nL be the number of resistors, capacitors, and inductors, respectively, so that n = nR + nc + nL. Let VR E IRnR ,Vc E IRna and VL E IRnL be the vectors of resistor voltages, capacitor voltages, and inductor voltages, respectively.

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