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Pseudo Singularities and Canards

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Geometric Singular Perturbation Theory Beyond the Standard Form

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 6))

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Abstract

There are important questions in the context of relaxation oscillatory behaviour observed in singularly perturbed systems that we have not addressed so far:

  • where do these oscillations originate from, i.e., what is the underlying mechanism that leads to the genesis of rhythmic behaviour?

  • how does the singular nature of the perturbation problem manifest itself in the rhythm generation?

Partial answers to the above questions can be found in classic bifurcation theory [63] which focuses on understanding significant changes in dynamical systems outputs under system parameter variations. The time-scale splitting in our singular perturbation problems creates additional complexity and sometimes surprising, counter-intuitive behaviour. We start with a couple of examples to motivate the development of the corresponding theory.

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Notes

  1. 1.

    Variations in the dimensionless parameter μ could be interpreted as variations in the initial precursor concentration p 0.

  2. 2.

    The terminology ‘pseudo singularity’ was introduced by Jose Argemi [2].

  3. 3.

    The terminology ‘canard’ was introduced by Eric Benoit et al. [8]; see also Remark 6.5.

  4. 4.

    French: ‘faux’ = false as opposed to ‘vrai’ = true; the term ‘vrai’ is seldom used in the canard theory literature.

  5. 5.

    More generally, it is observed under variation along a path α(s), \(s\in I\subset {\mathbb R}\), in the corresponding system parameter space \(\alpha \in \mathbb {R}^p\), p ≥ 1.

  6. 6.

    Theorems 6.1 and 6.2 apply to the forced vdP oscillator (6.1) as well.

  7. 7.

    The original two-stroke oscillator model (2.28) is given by α = 1, β = 1, γ = 0 and δ = 1.

  8. 8.

    Displacement dependent normal forces arise frequently in engineering problems [1, 26, 42, 114].

  9. 9.

    For γ = 0, there is no equilibrium; compare with the original stick-slip oscillator model (2.28) analysis.

  10. 10.

    While the set L does not constitute a set of isolated singularities of N(s, x, y) as stated in Assumption 3.3, the theory presented is still valid since s is a ‘true’ slow variable.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia

    Martin Wechselberger

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Wechselberger, M. (2020). Pseudo Singularities and Canards. In: Geometric Singular Perturbation Theory Beyond the Standard Form. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-36399-4_6

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