Author | Matsumoto, Takashi. author |
---|---|

Title | Bifurcations [electronic resource] : Sights, Sounds, and Mathematics / by Takashi Matsumoto, Motomasa Komuro, Hiroshi Kokubu, Ryuji Tokunaga |

Imprint | Tokyo : Springer Japan, 1993 |

Connect to | http://dx.doi.org/10.1007/978-4-431-68243-1 |

Descript | XVI, 468 p. online resource |

SUMMARY

Bifurcation originally meant "splitting into two parts. " Namely, a system underยญ goes a bifurcation when there is a qualitative change in the behavior of the sysยญ tem. Bifurcation in the context of dynamical systems, where the time evolution of systems are involved, has been the subject of research for many scientists and engineers for the past hundred years simply because bifurcations are interesting. A very good way of understanding bifurcations would be to see them first and study theories second. Another way would be to first comprehend the basic concepts and theories and then see what they look like. In any event, it is best to both observe experiments and understand the theories of bifurcations. This book attempts to provide a general audience with both avenues toward understanding bifurcations. Specifically, (1) A variety of concrete experimental results obtained from electronic circuits are given in Chapter 1. All the circuits are very simple, which is crucial in any experiment. The circuits, however, should not be too simple, otherwise nothing interesting can happen. Albert Einstein once said "as simple as posยญ sible, but no more" . One of the major reasons for the circuits discussed being simple is due to their piecewise-linear characteristics. Namely, the voltageยญ current relationships are composed of several line segments which are easy to build. Piecewise-linearity also simplifies rigorous analysis in a drastic manยญ ner. (2) The piecewise-linearity of the circuits has far reaching consequences

CONTENT

1 Bifurcations Observed from Electronic Circuits -- 1.1 Introduction -- 1.2 The Double Scroll Circuit -- 1.3 Structure of the Double Scroll -- 1.4 The Double Scroll Circuit is Chaotic in the Sense of Shilโ{128}{153}nikov -- 1.5 Homoclinic Linkage -- 1.6 The Torus Breakdown Circuit -- 1.7 The Hyperchaotic Circuit -- 1.8 The Neon Bulb Circuit -- 1.9 The R-L-Diode Circuit -- 2 Bifurcations of Continuous Piecewise-Linear Vector Fields -- 2.1 Introduction -- 2.2 Definition and Standard Forms of Continuous Piecewise-Linear Maps -- 2.3 Normal Forms of Piecewise-Linear Vector Fields -- 2.4 Multiregion Systems and Chaotic Attractors -- 2.5 Bifurcation Equations of Piecewise-Linear Vector Fields -- 2.6 Bifurcation Sets -- 3 Fundamental Concepts in Bifurcations -- 3.1 Introduction -- 3.2 Fundamental Notions for Dynamical Systems -- 3.3 Local Bifurcations around Equilibrium Points in Vector Fields -- 3.4 Dynamics and Bifurcations for Discrete Dynamical Systems -- 3.5 Bifurcations of Homoclinic and Heteroclinic Orbits in Vector Fields -- References -- Credits

Mathematics
Dynamics
Ergodic theory
System theory
Electrical engineering
Mathematics
Dynamical Systems and Ergodic Theory
Systems Theory Control
Electrical Engineering