The Topological Analysis of Jump Phenomena in Nonlinear Systems

Multi-Vibrator

Abstract

The purpose of this thesis is to deal with jump phenomena in nonlinear systems from a topological point of view. As a typical example of such a nonlinear system we take up an electric circuit system called multi vibrator.
In the beginning, we explain ordinary phase plane analysis of the above system in order to compare with topological analysis and then describe jump phenomena on a phase plane. Secondarily, we represent multi vibrator system as an oriented graph with 6 nodes and 9 branches. The currents and the voltages in the branches of the system can be specified on 9-dimensional vector space respectively. Consequently we consider the system as two manifolds in R¹⁸. So we construct two differentiable manifolds which are 9-dimensional Kirchhoff space K⁹ and 9-dimensional branch characteristics space ∧⁹ where K⁹ is a linear subspace determined by the Kirchhoff laws and ∧⁹ is a space specified by resistor branches and capacitor branches. Then the dynamical system is described on the configuration space Σ⁹=K⁹∩ ∧⁹ which inverse function theorem guarantees. Moreover we derive the equation of the state on which two manifolds K⁹ and ∧⁹ are nontransversal systematically. As a result of this derivation we can see constructively that the transversality condition does not hold on the jump points of multi vibrator. It is also shown, however, that there is a state on which jump phenomena do not occur though the transversality is destroyed. This fact gives us a conjecture that Morse Index will classify the jump points. The above approach will be applied to not only electric circuit systems but also other types of nonlinear systems.