|Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1992: ¥2,000,000 (Direct Cost: ¥2,000,000)
The aim of this project is to investigate mathematical models of dynamical systems which exhibit chaotic behavior, with particular emphasis on phenomena which may be relevant to the operation of engineering systems. The methods used are a combination of mathematical analysis and numerical computation, especially computer graphics displays of phase space structures. Simple mathematical models are chosen which have the most important nonlinear properties, and which represent a wide range of applications. Work began with forced oscillators of Duffing type with a potential well, with emphasis on escape over a smooth potential barrier, applicable to problems of ship capsize in rough seas and breaking of moleculor bonds with lasers, for example. Further efforts involve a differential-delay equation describing a phase-locked loop, and swing equations which model the stability of electric generators in a power grid. Again emphasis is on bifurcations causing loss of stability and escape, and si
tuations in which chaotic motions appear just prior to escape.
In the study of the delay-differential equation, a schematic bifurcation diagram was completed which summarizes the results of numerois computer simulations, and demonstrates that familiar low-dimensionals, bifurcations explain results observed so far.
A manuscript was completed for publication in a referred Journal. Investigation of coupled swing equations continued : transient time portraits of the basin of stable operation revealed no chaotic structure in the basin interior, so that only the previously observed fractal basin boundaries are relevant. An intensive effort to find underlying homoclinic structures was pursued, and a new approach based on long transient staddle orbits was developed, but success is still elusive.
Review of known generic bifurcations led to a more extensive and refined classification scheme, including the important phenomena of indeterminate outcome, and the regular or chaotic structure of saddle-type structures which cause instability. A major publication on these fundamental results is nearlying complete.
Joint work has been carried out on a differential-difference equation, namely a differential equation with delay, relevant to engineering control systems. The particular equation studied was fist introduced by Minorsky in connection with the stabilization of ship rolling motions.
Such a problem has strictly an infinite-dimensional space, but this can hopefully be approximated by a large but finite-dimensional space. In our studies we have worked in a space of over a hundred dimensions. An important problem is the location of basins of attraction, and in particular the basin of safe, non-failing starts. The key to understanding the structure of the basins lies with the basic sets within the boundaries. The straddle-orbit method has been used most successfully to locate these, and a particularly significant result is that these non-attracting sets take a period-doubling route to chaos. Less