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Canard explosions

Overview

These examples resolve the abrupt growth of an oscillation in a slow-fast system. FitzHugh--Nagumo parameter scans show the macro and micro views of a canard explosion, while reduced HH and adaptation examples show related multiple time-scale behavior.

Core ideas

Near a canard, a small parameter change can move a trajectory from a tiny oscillation to a large relaxation cycle. The trajectory follows an otherwise repelling slow-manifold branch for an anomalously long interval. Macro scans show the global branches; micro scans reveal the extremely narrow parameter window where cycle amplitude changes sharply.

Essential model

The FitzHugh--Nagumo system is

$$ \dot v=v-v^3/3-n+I,\qquad \dot n=(av-n)/\tau_n. $$

The slow recovery variable $n$ has time scale $\tau_n$, $v$ is fast, and $I$ is the scanned control parameter. In the adaptation example, an additional slowly decaying feedback current shifts the effective drive during the oscillation.

Code examples

All six examples now live in one notebook, chapter15.ipynb: simulate_canard finds currents for selected FitzHugh-Nagumo amplitudes and plots phase-plane and time-trace views; simulate_canard_2 fixes a current inside the narrow transition and plots its phase-plane orbit; simulate_fitzhugh_nagumo_macro scans a broad current range for equilibrium and cycle envelopes; simulate_fitzhugh_nagumo_micro resolves the narrow near-critical scan and cycle amplitudes; simulate_hh_reduced_bif_diag plots reduced HH fixed-point and stable/unstable cycle envelopes; simulate_mmos adds slow adaptation to FitzHugh-Nagumo and plots its mixed-mode voltage trace (with an ipywidgets slider). Several of these cells are computationally expensive (minutes, not seconds) -- the notebook notes this inline.

What to look for

Compare the scale of FITZHUGH_NAGUMO_MACRO and FITZHUGH_NAGUMO_MICRO: the latter is needed to see the amplitude jump. CANARD makes the same transition visible as selected trajectories. In HH_REDUCED_BIF_DIAG, distinguish the stable cycle envelope from the backward-traced unstable one.

Suggested order

  1. Run FITZHUGH_NAGUMO_MACRO, then FITZHUGH_NAGUMO_MICRO.
  2. Run CANARD and CANARD_2 to inspect individual trajectories.
  3. Continue with HH_REDUCED_BIF_DIAG and MMOS.

Prerequisites and related chapters

Chapter 10 introduces slow-fast phase planes and Chapter 13 introduces Hopf and unstable cycles. Chapter 14 supplies the reduced HH type-2 setting; Chapter 19 uses a still slower current to make bursts.

Running the examples

Open chapter15.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The canard-scan cells are noticeably slower than the single-trajectory ones.