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.

In this paper, oscillation modes produced in a Josephson circuit and its application to digital systems are described. The analysis is performed using an analog simulator to model the Josephson junction, in addition to computer simulation. The experimental results concerning oscillation modes agree well with the simulation results. The main advantage of the mapping for the oscillation modes is that it allows understanding of the relationships among oscillation modes and circuit parameters at first sight. In addition, a novel application of nonlinear oscillation to digital systems is described.

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.