Dynamic behavior of a distributed parameter system described by the one-dimensional wave equation with a nonlinear boundary condition is examined in detail using a graphical method proposed by Witt on a digital computer. The bifurcation diagram, homoclinic orbit and one-dimensional map are obtained and examined. Results using an analog simulator are introduced and compared with that of the graphical method. The discrepancy between these results is considered, and from the comparison among the bifurcation diagrams obtained by the graphical method, it is denoted that the energy dissipation in the system considerably restrains the chaotic state in the bifurcation process.

Periodic solutions of slow-fast systems called "canards," "ducks," or "lost solutions" are examined in a second order autonomous system. A characteristic feature of the canard is that the solution very slowly moves along the negative resistance of the slow curve. This feature comes from that the solution moves on or very close to a curve which is called slow manifolds or "rivers." To say reversely, solutions which move very close to the river are canards, this gives a heuristic definition of the canard. In this paper, the generation mechanism of the canard is examined using a piecewise linear system in which the cubic function is replaced by piecewise linear functions with three or four segments. Firstly, we pick out the rough characteristic feature of the vector field of the original nonlinear system with the cubic function. Then, using a piecewise linear model with three segments, it is shown that the slow manifold corresponding to the less eigenvalue of two positive real ones is the river in the segment which has the negative resistance. However, it is also shown that canards are never generated in the three segments piecewise linear model because of the existence of the "nodal type" invariant manifolds in the segment. In order to generate the canard, we propose a four segments piecewise linear model in which the property of the equilibrium point is an unstable focus.

This paper provides algorithms in order to solve an interval implicit function of the Poincare map generated by a continuous piece-wise linear (CPWL) vector field, with the use of interval arithmetic. The algorithms are implemented with the use of MATLAB and INTLAB. We present an application to verification of canards in two-dimensional CPWL vector field appearing in nonlinear piecewise linear circuits frequently, and confirm that the algorithms are effective.

Global dynamic behavior particularly the bifurcation of periodic orbits of a parallel blower system is studied using a piecewise linear model and the one-dimensional map defined by the Poincare map. First several analytical tools are presented to numerically study the bifurcation process particularly the bifurcation point of the fixed point of the Poincare map. Using two bifurcation diagrams and a bifurcation set, it is shown how periodic orbits bifurcate and leads to chaotic state. It is also shown that the homoclinic bifurcations occur in some parameter regions and that the Li & Yorke conditions of the chaotic state hold in the parameter region which is included in the one where the homoclinic bifurcation occurs. Together with the above, the stable and unstable manifolds of a saddle closed orbit is illustrated and the existence of the homoclinic points is shown.

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.

This paper introduces an adaptive distributed routing algorithm for the faulty star graph. The algorithm is based on that the n-star graph has uniform node degree n-1 and is n-1-connected. By giving two routing rules based on the properties of nodes, an optimal routing function for the fault-free star graph is presented. For a given destination in the n-star graph, n-1 node-disjoint and edge-disjoint subgraphs, which are derived from n-1 adjacent edges of the destination, can be constructed by this routing function and the concept of Breadth First Search. When faults are encountered, according to that there are n-1 node-disjoint paths between two arbitrary nodes, the algorithm can route messages to the destination by finding a fault-free subgraphs based on the local failure information (the status of all its incident edges). As long as the number f of faults (node faults and/or edge faults) is less than the degree n-1 of the n-star graph, the algorithm can adaptively find a path of length at most d+4f to route messages successfully from a source to a destination, where d is the distance between source and destination.

Roundness is one of the most important geometric measures for circular objects in the process of mechanical assembly. It is the amount of variation in a circular size which can be permitted. To compute roundness, the authors have already proposed an exact polynomial-time algorithm whose time complexity is O(n2). In this paper, we show that this roundness algorithm can be improved more efficiently, by introducing the deletion of the unnecessary points, in practical applications. In addition, the computational experience of this revised algorithm is also presented.