6. Hopf Normal Form

Examples & Use-Cases·Bifurcation track

from IPython.display import Markdown
from tvbo import Dynamics, Continuation, SimulationExperiment
import matplotlib.pyplot as plt
import numpy as np

HOPF_YAML = r"""
name: HopfNF
description: |
    Supercritical Hopf normal form (Cartesian).
    For $a < 0$ the origin is a stable focus; at $a = 0$ a Hopf bifurcation
    gives birth to a stable limit cycle of radius $\sqrt a$ and angular
    frequency $w$. The legacy figures
    `HopfPhasePortrait_and_BifDiag3D_a={-2,-1,-0.15,0.5,1,1.5}.png` show, for
    each $a$, the 2D phase portrait alongside the 3D bifurcation diagram with
    the periodic-orbit tube.
parameters:
  a: { name: a, value: 0.5 }
  w: { name: w, value: 1.0 }
state_variables:
  x1:
    name: x1
    domain: { lo: -2.0, hi: 2.0 }
    equation:
      lhs: Derivative(x1, t)
      rhs: (a - x1**2 - x2**2)*x1 - w*x2
    initial_value: 0.0
  x2:
    name: x2
    domain: { lo: -2.0, hi: 2.0 }
    equation:
      lhs: Derivative(x2, t)
      rhs: (a - x1**2 - x2**2)*x2 + w*x1
    initial_value: 0.0
"""

dyn_a = Dynamics.from_string(HOPF_YAML)

Markdown(dyn_a.render('markdown'))

HopfNF

Supercritical Hopf normal form (Cartesian). For \(a < 0\) the origin is a stable focus; at \(a = 0\) a Hopf bifurcation gives birth to a stable limit cycle of radius \(\sqrt a\) and angular frequency \(w\). The legacy figures HopfPhasePortrait_and_BifDiag3D_a={-2,-1,-0.15,0.5,1,1.5}.png show, for each \(a\), the 2D phase portrait alongside the 3D bifurcation diagram with the periodic-orbit tube.

autonomous: True; modes: 1; state variables: 2; parameters: 2.

State Equations

\[ \dot{x_{1}} = x_{1}*\left(a - x_{1}^{2} - x_{2}^{2}\right) - w*x_{2} \] \[ \dot{x_{2}} = w*x_{1} + x_{2}*\left(a - x_{1}^{2} - x_{2}^{2}\right) \]

State Variables

Variable	Initial Value	Equation	Domain / Sampling	Flags
\(x_{1}\)	0	differential (order 1)	[-2, 2]	recorded
\(x_{2}\)	0	differential (order 1)	[-2, 2]	recorded

Parameters

Parameter	Value
\(a\)	0.5
\(w\)	1

Phase portraits across \(a\) — animation

The legacy figure series HopfPhasePortrait_and_BifDiag3D_a=*.png shows six static panels for fixed values of \(a\). Below we use BifurcationResult.animate(...) further down to generate a single animation that captures the whole transition from stable focus to limit cycle alongside the 3D bifurcation diagram. First, the static reference grid:

Phase portraits across \(a\) — static grid (legacy figures)

For reference, the same six values rendered as a static grid using kind='phaseplane':

a_values = [-2.0, -1.0, -0.15, 0.5, 1.0, 1.5]
fig, axes = plt.subplots(2, 3, figsize=(11, 7.5))
for ax, a in zip(axes.flat, a_values):
    dyn_a.parameters['a'].value = a
    dyn_a.plot("x1", "x2", kind="phaseplane", ax=ax,
               grid_n=22, n_trajectories=2, duration=15)
    ax.set_title(f"$a = {a}$")
    ax.set_xlim(-2, 2); ax.set_ylim(-2, 2)
plt.tight_layout(); plt.show()

Bifurcation diagram with limit-cycle continuation

A single 1-parameter sweep in \(a\) for the equilibrium branch, plus a nested branch that picks up periodic orbits from each detected Hopf point:

dyn = Dynamics.from_string(HOPF_YAML)

CONT_YAML = """
name: hopf_in_a
dynamics: HopfNF
free_parameters:
  - name: a
    domain: { lo: -2.0, hi: 2.0 }
max_steps: 400
ds: 0.01
bothside: true
branches:
  - name: po_from_hopf
    source_point: "hopf:all"
    bothside: true
"""

cont = Continuation.from_string(CONT_YAML)
exp = SimulationExperiment(dynamics=dyn, continuations=[cont])
result = exp.run("bifurcationkit.jl")
hopf = result.continuations["hopf_in_a"]
hopf.plot(VOI="x1")

Detected IPython. Loading juliacall extension. See https://juliapy.github.io/PythonCall.jl/stable/compat/#IPython
CT: 1.063861 seconds (2.60 M allocations: 129.708 MiB, 4.02% gc time, 99.93% compilation time)
CT: 0.000237 seconds (2.14 k allocations: 191.859 KiB)
CT: 5.140689 seconds (3.97 M allocations: 555.047 MiB, 7.41% gc time, 83.07% compilation time)
CT: 0.267288 seconds (226.88 k allocations: 390.487 MiB, 5.74% gc time)

3D bifurcation diagram

The full picture: equilibrium spine (with stability colouring), Hopf point, and the limit-cycle tube whose radius grows like \(\sqrt a\):

hopf.plot_3d(VOI="x1")

Combined sweep: phase portrait + 3D diagram

BifurcationResult.animate(...) ties the two views together. The left panel re-renders the phase portrait at the current value of \(a\); the right panel shows the 3D diagram once with a moving red marker that tracks the position on the equilibrium backbone.

anim = hopf.animate(dyn, "a", np.linspace(-2.0, 1.5, 24),
                    "x1", "x2", VOI="x1",
                    grid_n=18, n_trajectories=3, duration=15,
                    figsize=(11, 4.8),
                    title_fmt=r"$a = {value:+.2f}$")
anim.save("hopf_sweep.gif", writer="pillow", fps=8)

A limit cycle being born as the bifurcation parameter crosses the Hopf point: the fixed point loses stability and trajectories spiral out onto a growing closed orbit.

AUTO-07p (NumCont)

Mirrors the standalone PhasePortraitHopf.py example: AUTO-07p time-integrates from the YAML defaults to a stable focus (or limit cycle), sweeps \(a\) in both directions, and – when the equilibrium branch yields a Hopf label – continues the periodic orbit out via the nested branch declared above as source_point: "hopf:all".

result_auto = exp.run("auto-07p")
hopf_auto = result_auto.continuations["hopf_in_a"]
hopf_auto.plot(VOI="x1")

/opt/homebrew/bin/gfortran -arch arm64 -O -fopenmp -O -c /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model.f90 -o /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model.o
/opt/homebrew/bin/gfortran -arch arm64 -O -fopenmp -O /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model.o -o /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model.exe /Applications/auto-07p/lib/*.o
Starting /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ...

ld: warning: duplicate -rpath '/Library/Developer/CommandLineTools/SDKs/MacOSX26.4.sdk/usr/lib' ignored

  BR    PT  TY  LAB       a          L2-NORM          x1            x2      
   1     1  EP    1   5.00000E-01   0.00000E+00   0.00000E+00   0.00000E+00
   1    10        2   1.06000E+00   0.00000E+00   0.00000E+00   0.00000E+00
   1    20  EP    3   2.06000E+00   0.00000E+00   0.00000E+00   0.00000E+00

 Total Time    0.895E-02
/var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ... done
Starting /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ...

  BR    PT  TY  LAB       a          L2-NORM          x1            x2      
   1     1  EP    1   5.00000E-01   0.00000E+00   0.00000E+00   0.00000E+00
   1    10  HB    2   6.00002E-09   0.00000E+00   0.00000E+00   0.00000E+00
   1    20        3  -1.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00
   1    30  EP    4  -2.00000E+00   0.00000E+00   0.00000E+00   0.00000E+00

 Total Time    0.126E-02
/var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ... done
Merge done
Saving to hopf_in_a, hopf_in_a, and hopf_in_a ... done
Starting /var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ...

  BR    PT  TY  LAB       a          L2-NORM        MAX x1        MAX x2        PERIOD    
   3     1        7   6.00002E-09   0.00000E+00   0.00000E+00   0.00000E+00   6.28319E+00
   3     2        8   1.00000E-04   1.00000E-02   1.00000E-02   1.00000E-02   6.28319E+00
   3     3        9   3.99900E-04   1.99975E-02   1.99951E-02   1.99951E-02   6.28319E+00
   3     4       10   8.99222E-04   2.99870E-02   2.99828E-02   2.99828E-02   6.28319E+00
   3     5       11   1.59717E-03   3.99646E-02   3.99590E-02   3.99590E-02   6.28319E+00
   3     6       12   2.49265E-03   4.99264E-02   4.99193E-02   4.99193E-02   6.28319E+00
   3     7       13   3.58425E-03   5.98686E-02   5.98601E-02   5.98601E-02   6.28319E+00
   3     8       14   4.87030E-03   6.97875E-02   6.97776E-02   6.97776E-02   6.28319E+00
   3     9       15   6.34883E-03   7.96795E-02   7.96682E-02   7.96682E-02   6.28319E+00
   3    10       16   8.01762E-03   8.95412E-02   8.95285E-02   8.95285E-02   6.28319E+00
   3    11       17   9.87423E-03   9.93692E-02   9.93551E-02   9.93551E-02   6.28319E+00
   3    12       18   1.19160E-02   1.09160E-01   1.09145E-01   1.09145E-01   6.28319E+00
   3    13       19   1.41400E-02   1.18912E-01   1.18895E-01   1.18895E-01   6.28319E+00
   3    14       20   1.65432E-02   1.28620E-01   1.28602E-01   1.28602E-01   6.28319E+00
   3    15       21   1.91224E-02   1.38284E-01   1.38264E-01   1.38264E-01   6.28319E+00
   3    16       22   2.18742E-02   1.47899E-01   1.47878E-01   1.47878E-01   6.28319E+00
   3    17       23   2.47952E-02   1.57465E-01   1.57442E-01   1.57442E-01   6.28319E+00
   3    18       24   2.78817E-02   1.66978E-01   1.66954E-01   1.66954E-01   6.28319E+00
   3    19       25   3.11301E-02   1.76437E-01   1.76412E-01   1.76412E-01   6.28319E+00
   3    20       26   3.45368E-02   1.85841E-01   1.85814E-01   1.85814E-01   6.28319E+00
   3    21       27   3.80978E-02   1.95187E-01   1.95159E-01   1.95159E-01   6.28319E+00
   3    22       28   4.18095E-02   2.04474E-01   2.04445E-01   2.04445E-01   6.28319E+00
   3    23       29   4.56680E-02   2.13701E-01   2.13670E-01   2.13670E-01   6.28319E+00
   3    24       30   4.96695E-02   2.22867E-01   2.22835E-01   2.22835E-01   6.28319E+00
   3    25       31   5.38102E-02   2.31970E-01   2.31937E-01   2.31937E-01   6.28319E+00
   3    26       32   5.80863E-02   2.41011E-01   2.40977E-01   2.40977E-01   6.28319E+00
   3    27       33   6.24942E-02   2.49988E-01   2.49953E-01   2.49953E-01   6.28319E+00
   3    28       34   6.70301E-02   2.58902E-01   2.58865E-01   2.58865E-01   6.28319E+00
   3    29       35   7.16903E-02   2.67750E-01   2.67713E-01   2.67713E-01   6.28319E+00
   3    30       36   7.64714E-02   2.76535E-01   2.76495E-01   2.76495E-01   6.28319E+00
   3    31       37   8.13697E-02   2.85254E-01   2.85213E-01   2.85213E-01   6.28319E+00
   3    32       38   8.63819E-02   2.93908E-01   2.93866E-01   2.93866E-01   6.28319E+00
   3    33       39   9.15046E-02   3.02497E-01   3.02454E-01   3.02454E-01   6.28319E+00
   3    34       40   9.67344E-02   3.11022E-01   3.10977E-01   3.10977E-01   6.28319E+00
   3    35       41   1.02068E-01   3.19481E-01   3.19436E-01   3.19436E-01   6.28319E+00
   3    36       42   1.07503E-01   3.27876E-01   3.27830E-01   3.27830E-01   6.28319E+00
   3    37       43   1.13035E-01   3.36207E-01   3.36160E-01   3.36160E-01   6.28319E+00
   3    38       44   1.18663E-01   3.44475E-01   3.44426E-01   3.44426E-01   6.28319E+00
   3    39       45   1.24382E-01   3.52678E-01   3.52628E-01   3.52628E-01   6.28319E+00
   3    40       46   1.30190E-01   3.60819E-01   3.60768E-01   3.60768E-01   6.28319E+00
   3    41       47   1.36085E-01   3.68898E-01   3.68845E-01   3.68845E-01   6.28319E+00
   3    42       48   1.42064E-01   3.76914E-01   3.76861E-01   3.76861E-01   6.28319E+00
   3    43       49   1.48124E-01   3.84869E-01   3.84815E-01   3.84815E-01   6.28319E+00
   3    44       50   1.54263E-01   3.92764E-01   3.92708E-01   3.92708E-01   6.28319E+00
   3    45       51   1.60479E-01   4.00598E-01   4.00541E-01   4.00541E-01   6.28319E+00
   3    46       52   1.66769E-01   4.08373E-01   4.08315E-01   4.08315E-01   6.28319E+00
   3    47       53   1.73130E-01   4.16089E-01   4.16030E-01   4.16030E-01   6.28319E+00
   3    48       54   1.79562E-01   4.23747E-01   4.23687E-01   4.23687E-01   6.28319E+00
   3    49       55   1.86061E-01   4.31348E-01   4.31286E-01   4.31286E-01   6.28319E+00
   3    50       56   1.92626E-01   4.38891E-01   4.38829E-01   4.38829E-01   6.28319E+00
   3    51       57   1.99254E-01   4.46379E-01   4.46316E-01   4.46316E-01   6.28319E+00
   3    52       58   2.05945E-01   4.53812E-01   4.53747E-01   4.53747E-01   6.28319E+00
   3    53       59   2.12696E-01   4.61190E-01   4.61124E-01   4.61124E-01   6.28319E+00
   3    54       60   2.19505E-01   4.68514E-01   4.68447E-01   4.68447E-01   6.28319E+00
   3    55       61   2.26371E-01   4.75785E-01   4.75717E-01   4.75717E-01   6.28319E+00
   3    56       62   2.33292E-01   4.83003E-01   4.82935E-01   4.82935E-01   6.28319E+00
   3    57       63   2.40266E-01   4.90170E-01   4.90100E-01   4.90100E-01   6.28319E+00
   3    58       64   2.47293E-01   4.97286E-01   4.97215E-01   4.97215E-01   6.28319E+00
   3    59       65   2.54370E-01   5.04351E-01   5.04279E-01   5.04279E-01   6.28319E+00
   3    60       66   2.61496E-01   5.11367E-01   5.11294E-01   5.11294E-01   6.28319E+00
   3    61       67   2.68670E-01   5.18334E-01   5.18260E-01   5.18260E-01   6.28319E+00
   3    62       68   2.75890E-01   5.25253E-01   5.25178E-01   5.25178E-01   6.28319E+00
   3    63       69   2.83156E-01   5.32124E-01   5.32049E-01   5.32049E-01   6.28319E+00
   3    64       70   2.90466E-01   5.38949E-01   5.38872E-01   5.38872E-01   6.28319E+00
   3    65       71   2.97818E-01   5.45727E-01   5.45650E-01   5.45650E-01   6.28319E+00
   3    66       72   3.05212E-01   5.52460E-01   5.52382E-01   5.52382E-01   6.28319E+00
   3    67       73   3.12647E-01   5.59148E-01   5.59069E-01   5.59069E-01   6.28319E+00
   3    68       74   3.20121E-01   5.65792E-01   5.65712E-01   5.65712E-01   6.28319E+00
   3    69       75   3.27633E-01   5.72392E-01   5.72311E-01   5.72311E-01   6.28319E+00
   3    70       76   3.35183E-01   5.78950E-01   5.78868E-01   5.78868E-01   6.28319E+00
   3    71       77   3.42770E-01   5.85465E-01   5.85382E-01   5.85382E-01   6.28319E+00
   3    72       78   3.50392E-01   5.91939E-01   5.91855E-01   5.91855E-01   6.28319E+00
   3    73       79   3.58048E-01   5.98371E-01   5.98287E-01   5.98287E-01   6.28319E+00
   3    74       80   3.65739E-01   6.04763E-01   6.04678E-01   6.04678E-01   6.28319E+00
   3    75       81   3.73462E-01   6.11116E-01   6.11029E-01   6.11029E-01   6.28319E+00
   3    76       82   3.81218E-01   6.17429E-01   6.17341E-01   6.17341E-01   6.28319E+00
   3    77       83   3.89005E-01   6.23703E-01   6.23614E-01   6.23614E-01   6.28319E+00
   3    78       84   3.96823E-01   6.29939E-01   6.29849E-01   6.29849E-01   6.28319E+00
   3    79       85   4.04670E-01   6.36137E-01   6.36047E-01   6.36047E-01   6.28319E+00
   3    80       86   4.12547E-01   6.42298E-01   6.42207E-01   6.42207E-01   6.28319E+00
   3    81       87   4.20452E-01   6.48423E-01   6.48331E-01   6.48331E-01   6.28319E+00
   3    82       88   4.28385E-01   6.54511E-01   6.54418E-01   6.54418E-01   6.28319E+00
   3    83       89   4.36345E-01   6.60564E-01   6.60471E-01   6.60471E-01   6.28319E+00
   3    84       90   4.44332E-01   6.66582E-01   6.66488E-01   6.66488E-01   6.28319E+00
   3    85       91   4.52344E-01   6.72565E-01   6.72470E-01   6.72470E-01   6.28319E+00
   3    86       92   4.60382E-01   6.78515E-01   6.78418E-01   6.78418E-01   6.28319E+00
   3    87       93   4.68445E-01   6.84430E-01   6.84333E-01   6.84333E-01   6.28319E+00
   3    88       94   4.76532E-01   6.90313E-01   6.90215E-01   6.90215E-01   6.28319E+00
   3    89       95   4.84642E-01   6.96163E-01   6.96064E-01   6.96064E-01   6.28319E+00
   3    90       96   4.92776E-01   7.01980E-01   7.01881E-01   7.01881E-01   6.28319E+00
   3    91       97   5.00933E-01   7.07766E-01   7.07666E-01   7.07666E-01   6.28319E+00
   3    92       98   5.09111E-01   7.13520E-01   7.13419E-01   7.13419E-01   6.28319E+00
   3    93       99   5.17311E-01   7.19244E-01   7.19142E-01   7.19142E-01   6.28319E+00
   3    94      100   5.25533E-01   7.24936E-01   7.24834E-01   7.24834E-01   6.28319E+00
   3    95      101   5.33775E-01   7.30599E-01   7.30496E-01   7.30496E-01   6.28319E+00
   3    96      102   5.42038E-01   7.36232E-01   7.36128E-01   7.36128E-01   6.28319E+00
   3    97      103   5.50320E-01   7.41836E-01   7.41731E-01   7.41731E-01   6.28319E+00
   3    98      104   5.58622E-01   7.47410E-01   7.47304E-01   7.47304E-01   6.28319E+00
   3    99      105   5.66944E-01   7.52957E-01   7.52850E-01   7.52850E-01   6.28319E+00
   3   100      106   5.75283E-01   7.58474E-01   7.58367E-01   7.58367E-01   6.28319E+00
   3   101      107   5.83642E-01   7.63964E-01   7.63856E-01   7.63856E-01   6.28319E+00
   3   102      108   5.92018E-01   7.69427E-01   7.69318E-01   7.69318E-01   6.28319E+00
   3   103      109   6.00412E-01   7.74862E-01   7.74753E-01   7.74753E-01   6.28319E+00
   3   104      110   6.08823E-01   7.80271E-01   7.80160E-01   7.80160E-01   6.28319E+00
   3   105      111   6.17251E-01   7.85653E-01   7.85542E-01   7.85542E-01   6.28319E+00
   3   106      112   6.25696E-01   7.91009E-01   7.90897E-01   7.90897E-01   6.28319E+00
   3   107      113   6.34157E-01   7.96340E-01   7.96227E-01   7.96227E-01   6.28319E+00
   3   108      114   6.42634E-01   8.01645E-01   8.01531E-01   8.01531E-01   6.28319E+00
   3   109      115   6.51127E-01   8.06924E-01   8.06810E-01   8.06810E-01   6.28319E+00
   3   110      116   6.59635E-01   8.12179E-01   8.12064E-01   8.12064E-01   6.28319E+00
   3   111      117   6.68158E-01   8.17409E-01   8.17293E-01   8.17293E-01   6.28319E+00
   3   112      118   6.76696E-01   8.22615E-01   8.22499E-01   8.22499E-01   6.28319E+00
   3   113      119   6.85249E-01   8.27797E-01   8.27680E-01   8.27680E-01   6.28319E+00
   3   114      120   6.93815E-01   8.32956E-01   8.32838E-01   8.32838E-01   6.28319E+00
   3   115      121   7.02396E-01   8.38091E-01   8.37972E-01   8.37972E-01   6.28319E+00
   3   116      122   7.10991E-01   8.43203E-01   8.43083E-01   8.43083E-01   6.28319E+00
   3   117      123   7.19599E-01   8.48292E-01   8.48172E-01   8.48172E-01   6.28319E+00
   3   118      124   7.28221E-01   8.53359E-01   8.53237E-01   8.53237E-01   6.28319E+00
   3   119      125   7.36855E-01   8.58403E-01   8.58281E-01   8.58281E-01   6.28319E+00
   3   120      126   7.45503E-01   8.63425E-01   8.63303E-01   8.63303E-01   6.28319E+00
   3   121      127   7.54163E-01   8.68425E-01   8.68302E-01   8.68302E-01   6.28319E+00
   3   122      128   7.62835E-01   8.73404E-01   8.73281E-01   8.73281E-01   6.28319E+00
   3   123      129   7.71520E-01   8.78362E-01   8.78237E-01   8.78237E-01   6.28319E+00
   3   124      130   7.80216E-01   8.83299E-01   8.83173E-01   8.83173E-01   6.28319E+00
   3   125      131   7.88925E-01   8.88214E-01   8.88088E-01   8.88088E-01   6.28319E+00
   3   126      132   7.97645E-01   8.93110E-01   8.92983E-01   8.92983E-01   6.28319E+00
   3   127      133   8.06376E-01   8.97984E-01   8.97857E-01   8.97857E-01   6.28319E+00
   3   128      134   8.15119E-01   9.02839E-01   9.02711E-01   9.02711E-01   6.28319E+00
   3   129      135   8.23872E-01   9.07674E-01   9.07545E-01   9.07545E-01   6.28319E+00
   3   130      136   8.32637E-01   9.12489E-01   9.12360E-01   9.12360E-01   6.28319E+00
   3   131      137   8.41412E-01   9.17285E-01   9.17155E-01   9.17155E-01   6.28319E+00
   3   132      138   8.50197E-01   9.22061E-01   9.21931E-01   9.21931E-01   6.28319E+00
   3   133      139   8.58993E-01   9.26819E-01   9.26687E-01   9.26687E-01   6.28319E+00
   3   134      140   8.67799E-01   9.31557E-01   9.31425E-01   9.31425E-01   6.28319E+00
   3   135      141   8.76615E-01   9.36277E-01   9.36144E-01   9.36144E-01   6.28319E+00
   3   136      142   8.85441E-01   9.40979E-01   9.40845E-01   9.40845E-01   6.28319E+00
   3   137      143   8.94277E-01   9.45662E-01   9.45528E-01   9.45528E-01   6.28319E+00
   3   138      144   9.03122E-01   9.50327E-01   9.50192E-01   9.50192E-01   6.28319E+00
   3   139      145   9.11976E-01   9.54975E-01   9.54839E-01   9.54839E-01   6.28319E+00
   3   140      146   9.20840E-01   9.59604E-01   9.59468E-01   9.59468E-01   6.28319E+00
   3   141      147   9.29713E-01   9.64216E-01   9.64080E-01   9.64080E-01   6.28319E+00
   3   142      148   9.38595E-01   9.68811E-01   9.68674E-01   9.68674E-01   6.28319E+00
   3   143      149   9.47486E-01   9.73389E-01   9.73251E-01   9.73251E-01   6.28319E+00
   3   144      150   9.56385E-01   9.77949E-01   9.77811E-01   9.77811E-01   6.28319E+00
   3   145      151   9.65293E-01   9.82493E-01   9.82354E-01   9.82354E-01   6.28319E+00
   3   146      152   9.74210E-01   9.87021E-01   9.86881E-01   9.86881E-01   6.28319E+00
   3   147      153   9.83135E-01   9.91531E-01   9.91391E-01   9.91391E-01   6.28319E+00
   3   148      154   9.92068E-01   9.96026E-01   9.95885E-01   9.95885E-01   6.28319E+00
   3   149      155   1.00101E+00   1.00050E+00   1.00036E+00   1.00036E+00   6.28319E+00
   3   150      156   1.00996E+00   1.00497E+00   1.00482E+00   1.00482E+00   6.28319E+00
   3   151      157   1.01892E+00   1.00941E+00   1.00927E+00   1.00927E+00   6.28319E+00
   3   152      158   1.02788E+00   1.01384E+00   1.01370E+00   1.01370E+00   6.28319E+00
   3   153      159   1.03685E+00   1.01826E+00   1.01812E+00   1.01812E+00   6.28319E+00
   3   154      160   1.04583E+00   1.02266E+00   1.02251E+00   1.02251E+00   6.28319E+00
   3   155      161   1.05482E+00   1.02704E+00   1.02690E+00   1.02690E+00   6.28319E+00
   3   156      162   1.06381E+00   1.03141E+00   1.03127E+00   1.03127E+00   6.28319E+00
   3   157      163   1.07282E+00   1.03577E+00   1.03562E+00   1.03562E+00   6.28319E+00
   3   158      164   1.08183E+00   1.04011E+00   1.03996E+00   1.03996E+00   6.28319E+00
   3   159      165   1.09084E+00   1.04443E+00   1.04429E+00   1.04429E+00   6.28319E+00
   3   160      166   1.09987E+00   1.04874E+00   1.04860E+00   1.04860E+00   6.28319E+00
   3   161      167   1.10890E+00   1.05304E+00   1.05289E+00   1.05289E+00   6.28319E+00
   3   162      168   1.11793E+00   1.05732E+00   1.05717E+00   1.05717E+00   6.28319E+00
   3   163      169   1.12698E+00   1.06159E+00   1.06144E+00   1.06144E+00   6.28319E+00
   3   164      170   1.13603E+00   1.06584E+00   1.06569E+00   1.06569E+00   6.28319E+00
   3   165      171   1.14508E+00   1.07009E+00   1.06993E+00   1.06993E+00   6.28319E+00
   3   166      172   1.15415E+00   1.07431E+00   1.07416E+00   1.07416E+00   6.28319E+00
   3   167      173   1.16321E+00   1.07852E+00   1.07837E+00   1.07837E+00   6.28319E+00
   3   168      174   1.17229E+00   1.08272E+00   1.08257E+00   1.08257E+00   6.28319E+00
   3   169      175   1.18137E+00   1.08691E+00   1.08676E+00   1.08676E+00   6.28319E+00
   3   170      176   1.19046E+00   1.09108E+00   1.09093E+00   1.09093E+00   6.28319E+00
   3   171      177   1.19955E+00   1.09524E+00   1.09509E+00   1.09509E+00   6.28319E+00
   3   172      178   1.20865E+00   1.09939E+00   1.09923E+00   1.09923E+00   6.28319E+00
   3   173      179   1.21776E+00   1.10352E+00   1.10336E+00   1.10336E+00   6.28319E+00
   3   174      180   1.22687E+00   1.10764E+00   1.10748E+00   1.10748E+00   6.28319E+00
   3   175      181   1.23599E+00   1.11175E+00   1.11159E+00   1.11159E+00   6.28319E+00
   3   176      182   1.24511E+00   1.11585E+00   1.11569E+00   1.11569E+00   6.28319E+00
   3   177      183   1.25424E+00   1.11993E+00   1.11977E+00   1.11977E+00   6.28319E+00
   3   178      184   1.26337E+00   1.12400E+00   1.12384E+00   1.12384E+00   6.28319E+00
   3   179      185   1.27251E+00   1.12806E+00   1.12790E+00   1.12790E+00   6.28319E+00
   3   180      186   1.28166E+00   1.13210E+00   1.13194E+00   1.13194E+00   6.28319E+00
   3   181      187   1.29081E+00   1.13614E+00   1.13598E+00   1.13598E+00   6.28319E+00
   3   182      188   1.29996E+00   1.14016E+00   1.14000E+00   1.14000E+00   6.28319E+00
   3   183      189   1.30912E+00   1.14417E+00   1.14401E+00   1.14401E+00   6.28319E+00
   3   184      190   1.31829E+00   1.14817E+00   1.14801E+00   1.14801E+00   6.28319E+00
   3   185      191   1.32746E+00   1.15215E+00   1.15199E+00   1.15199E+00   6.28319E+00
   3   186      192   1.33664E+00   1.15613E+00   1.15597E+00   1.15597E+00   6.28319E+00
   3   187      193   1.34582E+00   1.16009E+00   1.15993E+00   1.15993E+00   6.28319E+00
   3   188      194   1.35500E+00   1.16405E+00   1.16388E+00   1.16388E+00   6.28319E+00
   3   189      195   1.36419E+00   1.16799E+00   1.16782E+00   1.16782E+00   6.28319E+00
   3   190      196   1.37339E+00   1.17192E+00   1.17175E+00   1.17175E+00   6.28319E+00
   3   191      197   1.38259E+00   1.17584E+00   1.17567E+00   1.17567E+00   6.28319E+00
   3   192      198   1.39179E+00   1.17974E+00   1.17958E+00   1.17958E+00   6.28319E+00
   3   193      199   1.40100E+00   1.18364E+00   1.18347E+00   1.18347E+00   6.28319E+00
   3   194      200   1.41022E+00   1.18753E+00   1.18736E+00   1.18736E+00   6.28319E+00
   3   195      201   1.41944E+00   1.19140E+00   1.19123E+00   1.19123E+00   6.28319E+00
   3   196      202   1.42866E+00   1.19527E+00   1.19510E+00   1.19510E+00   6.28319E+00
   3   197      203   1.43789E+00   1.19912E+00   1.19895E+00   1.19895E+00   6.28319E+00
   3   198      204   1.44712E+00   1.20296E+00   1.20279E+00   1.20279E+00   6.28319E+00
   3   199      205   1.45636E+00   1.20680E+00   1.20662E+00   1.20662E+00   6.28319E+00
   3   200      206   1.46560E+00   1.21062E+00   1.21045E+00   1.21045E+00   6.28319E+00
   3   201      207   1.47484E+00   1.21443E+00   1.21426E+00   1.21426E+00   6.28319E+00
   3   202      208   1.48409E+00   1.21823E+00   1.21806E+00   1.21806E+00   6.28319E+00
   3   203      209   1.49334E+00   1.22202E+00   1.22185E+00   1.22185E+00   6.28319E+00
   3   204      210   1.50260E+00   1.22581E+00   1.22563E+00   1.22563E+00   6.28319E+00
   3   205      211   1.51186E+00   1.22958E+00   1.22940E+00   1.22940E+00   6.28319E+00
   3   206      212   1.52113E+00   1.23334E+00   1.23317E+00   1.23317E+00   6.28319E+00
   3   207      213   1.53040E+00   1.23709E+00   1.23692E+00   1.23692E+00   6.28319E+00
   3   208      214   1.53967E+00   1.24083E+00   1.24066E+00   1.24066E+00   6.28319E+00
   3   209      215   1.54895E+00   1.24457E+00   1.24439E+00   1.24439E+00   6.28319E+00
   3   210      216   1.55823E+00   1.24829E+00   1.24811E+00   1.24811E+00   6.28319E+00
   3   211      217   1.56751E+00   1.25200E+00   1.25183E+00   1.25183E+00   6.28319E+00
   3   212      218   1.57680E+00   1.25571E+00   1.25553E+00   1.25553E+00   6.28319E+00
   3   213      219   1.58610E+00   1.25940E+00   1.25922E+00   1.25922E+00   6.28319E+00
   3   214      220   1.59539E+00   1.26309E+00   1.26291E+00   1.26291E+00   6.28319E+00
   3   215      221   1.60469E+00   1.26676E+00   1.26658E+00   1.26658E+00   6.28319E+00
   3   216      222   1.61399E+00   1.27043E+00   1.27025E+00   1.27025E+00   6.28319E+00
   3   217      223   1.62330E+00   1.27409E+00   1.27391E+00   1.27391E+00   6.28319E+00
   3   218      224   1.63261E+00   1.27774E+00   1.27756E+00   1.27756E+00   6.28319E+00
   3   219      225   1.64193E+00   1.28138E+00   1.28120E+00   1.28120E+00   6.28319E+00
   3   220      226   1.65124E+00   1.28501E+00   1.28483E+00   1.28483E+00   6.28319E+00
   3   221      227   1.66057E+00   1.28863E+00   1.28845E+00   1.28845E+00   6.28319E+00
   3   222      228   1.66989E+00   1.29224E+00   1.29206E+00   1.29206E+00   6.28319E+00
   3   223      229   1.67922E+00   1.29585E+00   1.29566E+00   1.29566E+00   6.28319E+00
   3   224      230   1.68855E+00   1.29944E+00   1.29926E+00   1.29926E+00   6.28319E+00
   3   225      231   1.69788E+00   1.30303E+00   1.30284E+00   1.30284E+00   6.28319E+00
   3   226      232   1.70722E+00   1.30661E+00   1.30642E+00   1.30642E+00   6.28319E+00
   3   227      233   1.71656E+00   1.31018E+00   1.30999E+00   1.30999E+00   6.28319E+00
   3   228      234   1.72591E+00   1.31374E+00   1.31355E+00   1.31355E+00   6.28319E+00
   3   229      235   1.73525E+00   1.31729E+00   1.31710E+00   1.31710E+00   6.28319E+00
   3   230      236   1.74461E+00   1.32084E+00   1.32065E+00   1.32065E+00   6.28319E+00
   3   231      237   1.75396E+00   1.32437E+00   1.32418E+00   1.32418E+00   6.28319E+00
   3   232      238   1.76332E+00   1.32790E+00   1.32771E+00   1.32771E+00   6.28319E+00
   3   233      239   1.77268E+00   1.33142E+00   1.33123E+00   1.33123E+00   6.28319E+00
   3   234      240   1.78204E+00   1.33493E+00   1.33474E+00   1.33474E+00   6.28319E+00
   3   235      241   1.79141E+00   1.33843E+00   1.33824E+00   1.33824E+00   6.28319E+00
   3   236      242   1.80077E+00   1.34193E+00   1.34174E+00   1.34174E+00   6.28319E+00
   3   237      243   1.81015E+00   1.34542E+00   1.34523E+00   1.34523E+00   6.28319E+00
   3   238      244   1.81952E+00   1.34890E+00   1.34871E+00   1.34871E+00   6.28319E+00
   3   239      245   1.82890E+00   1.35237E+00   1.35218E+00   1.35218E+00   6.28319E+00
   3   240      246   1.83828E+00   1.35583E+00   1.35564E+00   1.35564E+00   6.28319E+00
   3   241      247   1.84766E+00   1.35929E+00   1.35910E+00   1.35910E+00   6.28319E+00
   3   242      248   1.85705E+00   1.36274E+00   1.36254E+00   1.36254E+00   6.28319E+00
   3   243      249   1.86644E+00   1.36618E+00   1.36598E+00   1.36598E+00   6.28319E+00
   3   244      250   1.87583E+00   1.36961E+00   1.36942E+00   1.36942E+00   6.28319E+00
   3   245      251   1.88523E+00   1.37304E+00   1.37284E+00   1.37284E+00   6.28319E+00
   3   246      252   1.89463E+00   1.37645E+00   1.37626E+00   1.37626E+00   6.28319E+00
   3   247      253   1.90403E+00   1.37986E+00   1.37967E+00   1.37967E+00   6.28319E+00
   3   248      254   1.91343E+00   1.38327E+00   1.38307E+00   1.38307E+00   6.28319E+00
   3   249      255   1.92284E+00   1.38666E+00   1.38647E+00   1.38647E+00   6.28319E+00
   3   250      256   1.93224E+00   1.39005E+00   1.38985E+00   1.38985E+00   6.28319E+00
   3   251      257   1.94165E+00   1.39343E+00   1.39324E+00   1.39324E+00   6.28319E+00
   3   252      258   1.95107E+00   1.39681E+00   1.39661E+00   1.39661E+00   6.28319E+00
   3   253      259   1.96048E+00   1.40017E+00   1.39997E+00   1.39997E+00   6.28319E+00
   3   254      260   1.96990E+00   1.40353E+00   1.40333E+00   1.40333E+00   6.28319E+00
   3   255      261   1.97932E+00   1.40688E+00   1.40669E+00   1.40669E+00   6.28319E+00
   3   256      262   1.98875E+00   1.41023E+00   1.41003E+00   1.41003E+00   6.28319E+00
   3   257      263   1.99818E+00   1.41357E+00   1.41337E+00   1.41337E+00   6.28319E+00
   3   258  EP  264   2.00760E+00   1.41690E+00   1.41670E+00   1.41670E+00   6.28319E+00

 Total Time    0.119E+01
/var/folders/ym/9kw1g21j1nd7kwfn8c0z3st40000gn/T/tvbo_numcont_HopfNF_nxadhqn6/model ... done
Saving to b.hopf_in_a_HB0, s.hopf_in_a_HB0, and d.hopf_in_a_HB0 ... done

The 3D view uses the unified style registry: stable LC segments are rendered as a tube surface, unstable ones as a wireframe.

hopf_auto.plot_3d(VOI="x1")

This single TVBO experiment reproduces the entire figure series of the legacy tutorial – without writing a model.f90, without a global continuation registry, and using only the YAML strings declared above.