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To get started with Poscidyn, only a few core concepts need to be understood. Poscidyn provides built-in classes for oscillator models (e.g. the Duffing oscillator and the Van der Pol oscillator) as well as excitation models (e.g. single-tone and multi-tone forcing).

However, Poscidyn introduces a fundamentally different workflow compared to traditional continuation-based tools. To use the package effectively—and correctly—it is important to understand how it operates conceptually, and how it compares to established continuation and bifurcation analysis software such as AUTO, MATCONT, and COCO.

Numerical frequency sweeping

In experiments, a frequency sweep is typically performed by slowly varying the excitation frequency while allowing the system to reach steady state at each step. Once steady state is reached, a response metric (e.g. amplitude) is recorded.

This process must be quasi-static: if the frequency step is too large, the system may jump between coexisting solution branches.

In computational dynamics, this procedure is known as continuation. Software packages such as AUTO, MATCONT, and COCO are specifically designed for this purpose. They are widely used for analyzing:

  • bifurcations
  • limit cycles
  • frequency response curves

A key limitation of continuation methods is that they are inherently sequential. Each solution depends on the previous one, which:

  • limits parallelization
  • increases computation time for large problems

Additionally, many of these tools are built in environments such as MATLAB or FORTRAN, which may limit accessibility and flexibility.

Poscidyn approach: multistart + artificial sweeps

Poscidyn takes a fundamentally different approach.

Instead of following a single solution branch, it computes many steady-state responses in parallel using a multistart strategy. For each excitation condition, the system is simulated from a large set of initial conditions. This typically reveals multiple coexisting attractors at the same frequency.

While this approach is computationally intensive, it is highly parallelizable and often faster in practice on modern hardware.

However, experiments typically observe only a single branch. To bridge this gap, Poscidyn introduces artificial sweep methods, which:

  • select one solution per frequency
  • enforce continuity in amplitude–phase space
  • reconstruct continuation-like curves

In addition, Poscidyn provides several response measures (e.g. demodulation, extrema, \(L^2\) metrics) to extract physically meaningful quantities from the simulated signals.

Future development aims to include hybrid approaches combining continuation and parallel methods (see Future work).

Multistarting

For each combination of drive frequency and amplitude, Poscidyn defines a search space of initial conditions. From this space, \(\texttt{n\_init\_cond}\) initial conditions are sampled.

This increases the probability of capturing all relevant stable attractors.

The search space is defined as:

\[ \begin{aligned} x_{0,i} &\in [-x_{\max,i},\, x_{\max,i}], \\ v_{0,i} &\in [-v_{\max,i},\, v_{\max,i}], \end{aligned} \]

Increasing \(\texttt{n\_init\_cond}\) improves coverage of the state space and is therefore a key simulation hyperparameter.

Currently, Poscidyn supports one method to determine these bounds:

Artificial sweeps

At a given frequency, multiple steady-state responses may exist due to different basins of attraction. In contrast, experiments typically observe only a single response.

Poscidyn reconstructs a synthetic sweep by selecting one solution per frequency such that the resulting curve remains continuous in amplitude–phase space. This is analogous to continuation.

Sweeps

Currently implemented:

  • Nearest neighbour method:
    selects solutions by minimizing local amplitude–phase mismatch between consecutive frequency steps, while penalizing unnecessary switching between initial condition seeds.

Response measures

Poscidyn provides multiple ways to extract response characteristics from steady-state signals:

  • Demodulation:
    extracts phasors at the drive frequency, superharmonics, or subharmonics

  • Minimum and maximum:
    extracts lower and upper bounds directly in the time domain

  • L2:
    computes the root-mean-square (RMS) magnitude of the signal

Solvers

Currently, Poscidyn computes steady-state responses using time integration.

The system is defined as:

\[ \dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), t), \qquad \mathbf{x}(t_0) = \mathbf{x}_0 \]

with solution:

\[ \mathbf{x}(t) = \boldsymbol{\Phi}(t; t_0, \mathbf{x}_0) \]

Implemented method:

  • Time integration:
    integrates each initial condition, retains the steady-state window, and evaluates the chosen response measure there

Limitations

Synthetic sweep methods approximate continuation behaviour but are not equivalent to true continuation algorithms. Important limitations and caveats are discussed on the Limitations page.