Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures

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For metals in tensile tests, discontinuous plastic deformation is a state where the stress oscillates rather than obtaining a stationary level. Estrin & Kubin have proposed a simple nonlinear model, valid in the cryogenic range, which contains a Hopf bifurcation, indicating the onset of discontinuous plastic deformation. We analyse the model for post-bifurcation behaviour and identify three parameter ranges with different transitions to relaxation oscillations (associated with the singular perturbation nature of the model). In one range, the transition is of the explosive canard type, where amplitude and period change dramatically over a very short parameter range. We derive general asymptotic formulae for the canard point and obtain excellent agreement with numerical simulations.