A new class of chaotic systems is discovered that are generated in a practical, nonlinear, mutually coupled phase-locked loop (PLL) circuit. Presented theoretical results make it possible to understand experimental results of mutually coupled PLL's on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple geometry and complex dynamics is expected to have applications in other areas, e.g., power systems or interacting bar magnets. Motivated by the numerical study of this system, the topological horseshoe is proven to be generated in the codimension 3 unfolding of a degenerated orbit-flip homoclinic point for this system. Qualitatively different type of bifurcation phenomena are also observed to appear depending on the phase detector (PD) characteristics.

A globally convergent algorithm that is also quadratically convergent for solving bipolar transistor networks is proposed. The algorithm is based on the homotopy method using a rectangular subdivision. Since the algorithm uses rectangles, it is much more efficient than the conventional simplicial-type algorithms. It is shown that the algorithm is globally convergent for a general class of nonlinear resistive networks. Here, the term globally convergent means that a starting point which leads to the solution can be obtained easily. An efficient acceleration technique which improves the local convergence speed of the rectangular algorithm is proposed. By this technique, the sequence of the approximate solutions generated by the algorithm converges to the exact solution quadratically. Also, in this case the computational work involved in each iteration is almost identical to that of Newton's method. Therefore, the algorithm becomes as efficient as Newton's method when it arrives sufficiently close to the solution. It is also shown that sparse-matrix techniques can be introduced to the rectangular algorithm, and the partial linearity of the system of equations can be exploited to improve the computational efficiency. Some numerical examples are also given in order to demonstrate the effectiveness of the proposed algorithm.>