# Download PDF by Carl Friedrich Gauss: [Article] General Propositions relating to Attractive and

By Carl Friedrich Gauss

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Additional info for [Article] General Propositions relating to Attractive and Repulsive Forces acting in the inverse ratio of the square of the distance

Example text

Singu­ larity) to occur. We call such conditions the bifurcation conditions. 48 3. 2.

This frequency-domain technique has been considered as an efficient alternative to the classical time-domain method in solving many practical problems. 1, we first give some historical remarks on the theory. 2, the Hopf bifurcation theorem is presented in the classical time13 14 2. com by NANYANG TECHNOLOGICAL UNIVERSITY on 09/30/15. For personal use only. 3 is devoted to the frequency-domain version of the theorem. 4, a circuit example is given to show that these two different versions of the bifurcation theorem are indeed equiv­ alent.

On the other hand, it is possible that for certain values of /i, the system only has a unique (local) steady-state solution, while for other values of fi a steady-state solution and a periodic solution co-exist. In the study of nonlinear dynamics, we usually need to determine for what values of fi a branch of periodic solutions exists, under the assumption that all the other eigenvalues of the Jacobian are located on the open left-half plane. This is, in fact, the main contribution of Hopf in his fundamental bifurcation theorem.