This repository contains Python scripts to simulate rings of Janus oscillators and arrays of coupled pendula, AUTO scripts to perform numerical continuation, and jupyter notebooks to plot results.
Create a conda environment with required packages:
conda create -n oscillator_env numpy=1.23.5 pip wheel matplotlib scipy jupyter ffmpeg pyqt. Activate this environment to run the scripts: conda activate oscillator_env.
Our fork of auto-07p contains modified AUTO package for detecting SBPs: https://github.com/znicolaou/auto-07p. Clone and install the package, as in
cd ~
git clone https://github.com/znicolaou/auto-07p.git
mkdir auto && mv auto-07p auto/07p
cd auto/07p
./configure
make
source cmds/auto.env.sh
Batches were run on a SLURM cluster; some script modifications may be necessary if no such cluster is available.
The respository contains two main directorys.
The janus directory contains a script a Python script janus.py contains to numerically integrate a ring of Janus oscillators. Running ./janus.py --help produces the following usage message:
usage: janus.py [-h] --filebase FILEBASE [--output OUTPUT] [--num NUM] [--time TIME] [--beta BETA] [--sigma SIGMA] [--gamma GAMMA] [--omega OMEGA] [--rtime RTIME] [--dt DT]
[--seed SEED] [--symmetrize SYM]
Numerical integration of networks of phase oscillators.
options:
-h, --help show this help message and exit
--filebase FILEBASE Base string for file output.
--output OUTPUT Output style, 0 for no stdout and sparse output, 1 for stdout and sparse output, 2 for stdout and dense output. Default 1.
--num NUM Number of Janus oscillators. Default 16.
--time TIME Total integration time. Detault 2000.
--beta BETA Internal coupling strength. Default 0.25.
--sigma SIGMA Coupling strength. Default 1.
--gamma GAMMA Coefficient for amplitude terms. Default 0.1.
--omega OMEGA Natural frequency gap. Default 1.0.
--rtime RTIME Time to start averaging order parameter. Default 1000.
--dt DT Time step for averaging and output. Default 0.1.
--seed SEED Initial condition random seed. Default 1.
--symmetrize SYM Symmetrize the coupling. 1 for symmetric, 0 for chiral. Default 0.
Batches of this script are used to generate the initial limit cycles. To do so, the SLURM batch script sbatch sweep.sh, which should take an hour or so to complete. After the jobs complete, run the script ./makecycles.py to generate the starting data for the AUTO continuation.
To generate the main numerical continuation results, submit the SLURM script sbatch job.sh, which should take over a day to complete. After the job completes, run ./makesteadys.sh and sbatch steadyjob.sh to find and numerically continue the steady states visited by the heteroclinic cycles (these jobs run in a few minutes).
Some solution branches generated in the continuation are neutrally stable invariant limit cycles, which possess symmetry-constrained unity Floquet multipliers. The direciton vector in such branches can spuriously change signs because of the singularlity of the Jacobian, and are thus difficult to continue. The script invariant.auto contains code for tracking the number of neutrally stable multipliers and ensuring that the direction of the change in the order parameter does not change. (This was designed specifically for the exemplary solution branch; additional modifications would be required for branches that do exhibit extrema in the order parameter.) Run sbatch invariantjob.sh to run the script on the exemplary branch in the manuscript.
Finally, the Jupyter notebook plotjanus.ipynb contains code to plot and animate solutions. A few example solution branches are stored in data/janus/examples, which are plotted in the notebook.
The pendula directory contains a script a Python script pendula.py contains to numerically integrate an array of parameterically driven pendula. Running ./pendula.py --help produces the following usage message:
usage: pendula.py [-h] --filebase FILEBASE [--num NUM] [--frequency FREQ] [--initfrequency FREQ0] [--initamplitude AMP0] [--amplitude AMP] [--delta DELTA] [--cycles CYCLES]
[--initcycle INITCYCLE] [--outcycle OUTCYCLE] [--dt DT] [--seed SEED] [--damp DAMP] [--spring SPRING] [--init INIT] [--rtol RTOL] [--atol ATOL] [--verbose VERBOSE]
Driven pendula.
options:
-h, --help show this help message and exit
--filebase FILEBASE Base string for file output
--num NUM Number of pendula
--frequency FREQ Driving frequency
--initfrequency FREQ0
Initial driving frequency
--initamplitude AMP0 Driving amplitude
--amplitude AMP Driving amplitude
--delta DELTA Alternating pendulum length scale
--cycles CYCLES Simulation time in driving cycles
--initcycle INITCYCLE
Simulation time in driving cycles
--outcycle OUTCYCLE Cycle to start outputting
--dt DT Time between outputs in driving cycles
--seed SEED Seed for random initial conditions
--damp DAMP Damping coefficient
--spring SPRING Spring coefficient
--init INIT Initial random scale
--rtol RTOL Relative error tolerance
--atol ATOL Absolute error tolerance
--verbose VERBOSE Verbose output
Batches of this script are used to generate the initial limit cycles. To do so, the SLURM batch script sbatch sweep.sh, which should take an hour or so to complete. After the jobs complete, run the script ./makecycles.py to generate the starting data for the AUTO continuation.
To generate the main numerical continuation results from the starting cycles, submit the SLURM script sbatch job.sh, which should take a couple of hours to complete. To find the initial period doubling instabilities of the homogeneous state and the branches emerging from them, submit the batch sbatch flatjob.sh, which should complete in an hour or so.
The Jupyter notebook plotpendula.ipynb contains code to plot and animate solutions. A few example solution branches are stored in data/pendula/examples, which are plotted in the notebook.