Mathematics – Dynamical Systems
Scientific paper
2012-01-31
International Journal of Bifurcation and Chaos, Vol. 20, No. 5, pp. 1467-1475, 2010
Mathematics
Dynamical Systems
preprint version - for final version see journal reference
Scientific paper
10.1142/S0218127410026617
Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. A first-order asymptotic expansion of this location was found by Krupa and Szmolyan in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficient in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding use of this normal form. Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur.
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