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| We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delay self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. The presence of delays significantly enriches the dynamics, and varying the delay induces canard explosion, mixed mode oscillations as well as transitions to complex bursting periodic orbits. We show that as the delay
is increased a family of `classical' canard explosions ends as Bogdanov-Takens bifurcation occur at the folds points of the S-shaped critical manifold. Canard explosion and mixed-mode oscillations are investigated by means of geometric perturbation analysis,
and bursting by means of slow-fast periodic averaging. |
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