A Generation Mechanism of Canards in a Piecewise Linear System (Special Section of Selected Papers from the 9th Karuizawa Workshop on Circuits and Systems)

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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 piesewise 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. Using the model, we consider that how the canards with no heads and with a head are generated. Finally, the estimated mechanism is confirmed to be correct using numerical calculations.
社団法人電子情報通信学会の論文
1997-03-25

A Generation Mechanism of Canards in a Piecewise Linear System (Special Section of Selected Papers from the 9th Karuizawa Workshop on Circuits and Systems)

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