Global Dynamic Behaviour of a Parallel Blower System

Hideaki OKAZAKI  Hideo NAKANO  Takehiko KAWASE  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E78-A   No.6   pp.715-726
Publication Date: 1995/06/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Nonlinear Problems
piecewise linear vector fields,  saddle-repellor-spiral connection,  boundary maps,  formation of horse shoe map,  one dimensional approximation analysis,  

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A parallel blower system is quite familiar in hydraulic machine systems and quite often employed in many process industries. It is dynamically dual to the fundamental functional element of digital computer, that is, the flip-flop circuit, which was extensively studied by Moser. Although the global dynamic behaviour of such systems has significant bearing upon system operation, no substantial study reports have hitherto been presented. Extensive research concern has primarily been concentrated upon the local stability of the equilibrium point. In the paper, a piecewise linear model is used to analytically and numerically investigate its manifold global dynamic behaviour. To do this, first the Poincar map is defined as a composition boundary map, each of which is defined as the point transformation from the entry point to the end point of any trajectory on some boundary planes. It was already shown that, in some parameter region, the system exhibits the so-called chaotic states. The chaos generating process is investigated using the above Poincar map and it is shown that the map contains the contracting, stretching and folding operations which, as we often see in many cases particularly in horse shoe map, produce the chaotic states. Considering the one dimensional motions of the orbits by such Poincar map, that is, the stretching and folding operations, a one dimensional approximation of the Poincar map is introduced to more closely and exactly study manifold bifurcation processes and some illustrative bifurcation diagrams in relation to system parameters are presented. Thus it is shown how many types of bifurcations like the Hopf, period doubling, saddle node, and homoclinic bifurcations come to exist in the system.