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Hopf bifurcations

Overview

These normal-form examples contrast supercritical and subcritical Hopf bifurcations through radial equations, phase planes, and bifurcation diagrams.

Core ideas

A Hopf bifurcation changes the stability of an equilibrium as a complex pair of eigenvalues crosses the imaginary axis. A supercritical Hopf creates a small stable cycle, whereas a subcritical Hopf has an unstable small cycle and can coexist with a larger attracting cycle. Solid and dashed branches in the plots distinguish attracting and repelling structures.

Essential model

With polar radius $r$, the normal forms used here are

$$ \dot r=Ir-r^3 \quad\text{(supercritical)},\qquad \dot r=Ir+r^3 \quad\text{(subcritical)}. $$

Here $I$ is the bifurcation parameter and $r\ge0$ is distance from the equilibrium. The extended subcritical examples add a $-r^5$ term, which supports both an unstable inner and a stable outer cycle.

Code examples

All nine examples now live in one notebook, chapter13.ipynb: plot_hopf_sup plots the supercritical radial vector field; plot_hopf_sup_bif_diag draws its equilibrium and stable cycle branches; plot_hopf_sup_phase_plane integrates supercritical spirals and cycles; plot_hopf_sub plots the subcritical radial vector field; plot_hopf_sub_bif_diag draws the subcritical unstable cycle branch; plot_hopf_sub_phase_plane shows the corresponding repelling cycle and phase-plane trajectories; plot_hopf_sub_2 adds the quintic saturation term; plot_hopf_sub_bif_diag_2 plots its inner repelling and outer attracting cycles; plot_hopf_sub_phase_plane_2 traces the two-cycle phase-plane organization.

What to look for

Compare HOPF_SUP_BIF_DIAG with HOPF_SUB_BIF_DIAG: the stable cycle appears on opposite sides of onset. Then use the phase-plane scripts to connect a branch's line style to trajectories moving toward or away from its cycle.

Suggested order

  1. Run the three HOPF_SUP examples.
  2. Run HOPF_SUB, HOPF_SUB_BIF_DIAG, and HOPF_SUB_PHASE_PLANE.
  3. Use the three _2 examples to study coexistence of two cycles.

Prerequisites and related chapters

Chapter 11 introduces local bifurcations and Chapter 12 uses Jacobian eigenvalues. Chapters 14--15 apply Hopf and subcritical-cycle geometry to conductance-based neuron reductions.

Running the examples

Open chapter13.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.