A Generation Mechanism of Canards in a Piecewise Linear System

Noboru ARIMA  Hideaki OKAZAKI  Hideo NAKANO  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E80-A   No.3   pp.447-453
Publication Date: 1997/03/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section of Selected Papers from the 9th Karuizawa Workshop on Circuits and Systems)
Category: 
Keyword: 
nonlinear oscillation,  slow-fast system,  piecewise linear vector field,  lost solutions,  ducks,  

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Summary: 
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.