Topological methods in analysis of periodic and chaotic canard-type trajectories

This paper investigates the role of topological methods in the analysis of canardtype periodic and chaotic trajectories. In Sections 1 – 5 we apply topological degree [1, 2] to the analysis of multi-dimensional canards. This part of the paper was written mainly by the first and the last authors. Sections 6 – 7 are devoted to an application of a special corollary of the Poincaré–Bendixson theorem to the existence of periodic two-dimensional canards. This fragment was written mainly by the first and the second authors. If W : R → R is a continuous mapping, Ω ⊂ R is a bounded open set, and y ∈ R does not belong to the image W (∂Ω) of the boundary ∂Ω of Ω, then the symbol deg(W,Ω, y) denotes the topological degree [1] of W at y with respect to Ω. If 0 6∈ W (∂Ω), then the integer number γ(W,Ω) = deg(W,Ω, 0), called the rotation of the vector field W at ∂Ω, is well defined. A detailed description of properties of the number γ(W,Ω) can be found, for example, in [2]. In particular, if I denotes the identity mapping, I(x) ≡ x, then the number γ(I −W,Ω) measures the algebraic number of fixed points of the mapping W in Ω. Consider the slow-fast system ẋ = X(x, y, ε) + X̂(x, y, z, ε), εẏ = Y (x, y, ε) + Ŷ (x, y, z, ε), ż = Z(x, y, z, ε). (1)