Canards and chaos in nonlinear systems

Canards are a new phenomenon in slow-fast systems. The canard phenomenon in three types of nonlinear systems is studied. The authors first study the behavior of the Hopf bifurcation for the following two-dimensional systems: (a) a slow-fast system with a cubic nonlinearity, (b) a system with a constrained curve, and (c) a slow-fast system with a piecewise linear nonlinearity. It is shown that systems (a) and (b) have canard cycles, but the other forgets them. The Hopf bifurcation scheme of the system (a) is continuous, but (b) and (c) are discontinuous. The same questions are considered for three-dimensional systems. The canard with a pseudosingular saddle point is studied, and its role in the system dynamics is explained. It is shown that the slow-fast system with a piecewise linear nonlinearity drops this kind of canard. By using this result, the existence of a chaotic attractor is shown