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By Murray H. Protter

Greatest ideas are imperative to the speculation and functions of second-order partial differential equations and platforms. This self-contained textual content establishes the elemental ideas and gives a number of functions.

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These include cases like multiple limit cycles, multiple bifurcation points, emergence of new periodic solutions (the so-called isolas) that are disconnected from the steady-state curves, etc. The most remarkable work in this direction include the contributions of Chafee [1968, 1978], Takens [1973], Hassard k Wan [1978], Kielhofer [1979], Gobber k Willamowsky [1979], Golubitsky k Langford [1981], and Golubitsky k Schaeffer [1985]. Generalizations of the Hopf bifurcation theorem usually take into account, at least implicitly, a multi-parameter variation in the original ODE setting.

Since the locus of A(w; /i) passes through the point — 1 + iO when (i = 0, H = 0 is a bifurcation point. It then follows from the Nyquist criterion (and the time-domain analysis as well) that in this case the diode system has a stable equilibrium solution of zero amplitude, which corresponds to (*i,*2) = (0,0). 30 2. com by NANYANG TECHNOLOGICAL UNIVERSITY on 09/30/15. For personal use only. ),. 9. The frequency graph (eigenlocus) of the tunnel diode oscillator. The loci of the eigenvalue A(w;/I) corresponding to different values of Ji are shown in Fig.

Now, we are ready to state the following results about singularities. 1. A necessary condition for (wo. Mo) to be a singular point is Fi(u>0,fio) = F2(uj0,fio) = 0. 6) Proof. This follows immediately from a direct application of the generalized Nyquist stability criterion, because the closed-loop characteristic polynomial of the feedback system shown in Fig. 5 has at least one pole located on the imaginary axis. 0 We will show a few more necessary conditions in the following. At this point, we would like to mention that sufficient conditions, particularly suf­ ficient conditions for distinguishing the different types of bifurcations, are available in the literature [Hale & Kogak, 1991].

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