Jump behavior of circuits and systems

Some circuits exhibit jump behavior. for example, this occurs when the velocity field specified by the \dot{i}_L and \dot{\upsilon}_c of the inductor and capacitor characteristics cannot be "lifted" on to the resistive constraint manifold. The (jump) behavior is viewed as the limit as \epsilon \rightarrow 0 of the solutions of a regularized system of equations obtained by introducing suitably located \epsilon -parasitic L 's and C 's: this leads to a consistent way of defining discontinuous solutions. In particular, the behavior near a fold and cusp is examined. The concept of physically measurable operating point is defined and is related to that of strict local dissipativeness (which generalizes that of strict local passivity). Two examples are included.

[1]  J. J. Levin The asymptotic behavior of the stable initial manifolds of a system of nonlinear differential equations , 1957 .

[2]  S. Smale On Gradient Dynamical Systems , 1961 .

[3]  Jacob Katzenelson,et al.  Nonlinear RLC networks , 1965 .

[4]  A. Kelley Stability of the center-stable manifold , 1967 .

[5]  S. Smale Differentiable dynamical systems , 1967 .

[6]  I. W. Sandberg,et al.  Some network-theoretic properties of nonlinear DC transistor networks , 1969 .

[7]  I. W. Sandberg,et al.  Theorems on the analysis of nonlinear transistor networks , 1970, Bell Syst. Tech. J..

[8]  Leon O. Chua,et al.  The effects of parasitic reactances on nonlinear networks , 1971 .

[9]  F. Hoppensteadt Properties of solutions of ordinary differential equations with small parameters , 1971 .

[10]  S. Smale On the mathematical foundations of electrical circuit theory , 1972 .

[11]  C. Desoer,et al.  Trajectories of nonlinear RLC networks: A geometric approach , 1972 .

[12]  Felix F. Wu,et al.  NONLINEAR MONOTONE NETWORKS , 1974 .

[13]  L. Cesari ALTERNATIVE METHODS IN NONLINEAR ANALYSIS , 1975 .

[14]  Shui-Nee Chow,et al.  Applications of generic bifurcation. II , 1975 .

[15]  Floris Takens,et al.  Implicit differential equations: Some open problems , 1976 .

[16]  Leon O. Chua,et al.  Graph-theoretic properties of dynamic nonlinear networks , 1976 .

[17]  Jerrold E. Marsden,et al.  Qualitative methods in bifurcation theory , 1978 .

[18]  Alberto Sangiovanni-Vincentelli,et al.  On equivalent dynamic networks: Elimination of capacitor loops and inductor cutsets , 1978 .

[19]  Almost discontinuous oscillations: The generalized multivibrator , 1978 .

[20]  V. M. Popov,et al.  Monotonicity and mutability , 1979 .

[21]  Neil Fenichel Geometric singular perturbation theory for ordinary differential equations , 1979 .

[22]  Pravin Varaiya,et al.  Hierarchical stability and alert state steering control of interconnected power systems , 1980 .

[23]  L. Chua,et al.  Geometric properties of resistive nonlinear n-ports - Transversality, structural stability, reciprocity, and anti-reciprocity , 1980 .