Phase plane

Abstract: : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.

Abstract: We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.

Abstract: In a recent paper, Lai introduced a lattice-gas model. In this paper we generalize Lai's model, making application to various systems such as dilute Heisenberg magnets, higher-spin systems, and a lattice of SU(3) triplets. By a careful consideration of general thermodynamic stability, and by variational arguments, we demonstrate Lai's solution to be incorrect, and in turn produce the correct solution in this case and in other cases including higher-dimensional models. The remaining cases we treat in one dimension by Bethe's ansatz, reducing the problem to coupled integral equations. We locate the singularities of the ground-state energy in the phase plane; and we explicitly calculate the absolute-ground-state energy, excitations above the absolute ground state, and the first correction to the absolute ground state for small concentrations of impurities.