A GENERATION MECHANISM OF CANARDS IN A PIECEWISE-LINEAR SYSTEM

Periodic solutions of slow-fast systems called ''canards,'' ''ducks,'' or ''lost solutions'' are examined in a second order autonomous syste m. A characteristic feature of the canard is that the solution very sl owly moves along the negative resistance of the slow curve. This featu re comes from that the solution moves on or very close to a curve whic h 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 cu bic function is replaced by piesewise linear functions with three or f our segments. Firstly, we pick out the rough characteristic feature of the vector field of the original nonlinear system with the cubic func tion. Then, using a piecewise linear model with three segments, it is shown that the slow manifold corresponding to the less eigenvalue of t wo positive real ones is the river in the segment which has the negati ve resistance. However, it is also shown that canards are never genera ted in the three segments piecewise linear model because of the existe nce of the ''nodal type'' invariant manifolds in the segment. In order to generate the canard, we propose a four segments piecewise linear m odel in which the property of the equilibrium point is an unstable foc us. Using the model, we consider that how the canards with no heads an d with a head are generated. Finally, the estimated mechanism is confi rmed to be correct using numerical calculations.