pyGDIS

pyGDIS is the reference Python implementation of the Generalized Dynamical Instability Score (GDIS), a descriptor-based framework for quantifying nonlinear instability and localized critical-transition behavior across parameter-ordered trajectory families.

The package implements the formulation used in the accompanying manuscript and separates reusable scientific software from paper-specific benchmarking and plotting code.

Why GDIS?

Conventional measures often emphasize one manifestation of nonlinear behavior. GDIS combines complementary channels within a bounded and interpretable architecture.

Measure	Primary emphasis	Bounded cross-system score	Localized transition term
Lyapunov exponent / FTLE	Perturbation divergence	No	No
Entropy measures	Informational complexity	Often	No
Recurrence measures	Recurrence geometry	Depends	Indirect
GDIS	Dynamical, geometric, informational, and temporal evidence	Yes	Yes

GDIS is intended to complement—not replace—established nonlinear-dynamics tools.

Mathematical formulation

The reference implementation computes five descriptors:

• J: local dynamical sensitivity
• S: trajectory stretching
• A: attractor expansion
• H: nonlinear complexity
• T: temporal persistence

The principal descriptors are aggregated through a weighted geometric mean,

$$ C = \left(\widetilde J^{\alpha_J}\widetilde S^{\alpha_S}\widetilde A^{\alpha_A}\right)^{1/(\alpha_J+\alpha_S+\alpha_A)}, $$

with reference exponents

$$ \alpha_J=0.42,\qquad \alpha_S=0.33,\qquad \alpha_A=0.25. $$

After complexity and persistence modulation and Hill saturation, the sustained-instability functional is combined with an unweighted transition-localization term through

$$ \Phi = -\ln(1-I_{\mathrm{sustained}}) + \lambda_t I_{\mathrm{transition,base}}, $$

followed by

$$ \mathrm{GDIS}=1-e^{-\Phi},\qquad 0\leq \mathrm{GDIS}<1. $$

The default reference value is lambda_t = 0.18.

Mathematical properties

The implementation preserves the properties established in the manuscript:

• Boundedness: 0 <= GDIS < 1.
• Monotonicity: increasing sustained or transition instability cannot decrease GDIS.
• Baseline preservation: when the transition term is zero, GDIS equals the sustained-instability functional.
• Robust scale handling: percentile normalization is invariant to positive affine transformations of its input descriptor.

Installation

python -m pip install pygdis

For development:

git clone https://github.com/hamiddi/pygdis.git
cd pygdis
python -m pip install -e ".[dev]"
pytest

Quick start

from gdis import GDIS
from gdis.benchmarks import LorenzSystem

system = LorenzSystem()
trajectories, parameters = system.generate_sweep()

result = GDIS().fit_transform(
    trajectories,
    parameters,
    jacobian_function=system.jacobian,
    critical_value=system.critical_value,
)

print(result.to_dataframe().head())
result.plot()

Data-only use

When an analytical Jacobian is unavailable, omit jacobian_function. pyGDIS uses a trajectory-based local-divergence proxy.

result = GDIS().fit_transform(trajectories, parameters)

If critical_value is omitted, the reference implementation centers the transition-localization window at the maximum transition-energy estimate and records this choice in result.metadata.

Analyze trajectory data from CSV

The repository includes a complete long-format example dataset at examples/synthetic_parameter_sweep.csv. Each row contains a control parameter, time, and two observed state variables:

parameter,time,x1,x2
0.000000,0.000000,...,...
0.000000,0.100334,...,...

Run the included end-to-end analysis:

cd examples
python csv_data_example.py

The script creates gdis_results/ beside the CSV file.

The script loads and groups the CSV into parameter-ordered trajectories, computes GDIS in data-only mode, and writes:

• gdis_results.csv with the final score, potential, functionals, and descriptors;
• gdis_profile.png;
• gdis_components.png; and
• analysis_summary.txt.

To analyze a user-supplied file:

python csv_data_example.py \
  --input my_data.csv \
  --parameter-column load \
  --time-column time \
  --state-columns voltage angle

See CSV analysis in the usage guide for the required format and interpretation of outputs.

Package organization

src/gdis/
├── components.py       # J, S, A, H, and T descriptors
├── scaling.py          # robust normalization and saturation
├── transition.py       # transition energy and localization
├── potential.py        # Phi and bounded GDIS mapping
├── score.py            # reference end-to-end GDIS estimator
├── sensitivity.py      # transition-weight rescoring
├── validation.py       # optional validation metrics
├── plotting.py         # reusable plotting helpers
├── benchmarks/         # canonical benchmark access
└── systems/            # Lorenz, Rössler, Chen, and logistic systems

Paper-specific workflows are intentionally isolated under examples/publication/ and are not imported by the core package.

Reproducing the manuscript analyses

python scripts/reproduce_paper.py

Or run the stages separately:

python examples/publication/reproduce_benchmarks_and_sensitivity.py
python examples/publication/reproduce_publication_figures.py

These scripts reproduce the canonical benchmark sweep, transition-weight sensitivity analysis, validation tables, attractor galleries, and publication figures. They are research-reproduction workflows rather than core API modules.

Transition-weight sensitivity

A completed result can be rescored without recomputing trajectories or descriptors:

from gdis import transition_weight_sensitivity

sensitivity = transition_weight_sensitivity(
    result,
    weights=(0.0, 0.18, 0.25, 0.50, 0.75, 1.0),
)

Documentation

• Installation
• Usage
• Mathematics
• API
• Applications

Companion manuscript

Generalized Dynamical Instability Score (GDIS): A Universal Framework for Quantifying Instability and Critical Transitions in Nonlinear Dynamical Systems

The manuscript is being submitted to IEEE Access. Publication metadata and DOI will be added after acceptance. Supplementary material accompanies the manuscript.

Citation

Until the journal article is published, cite the software repository using CITATION.cff. A temporary BibTeX entry is:

@software{ismail_pygdis_2026,
  author  = {Ismail, Hamid D. and Harb, Ahmad and Bikdash, Marwan},
  title   = {pyGDIS: Generalized Dynamical Instability Score},
  year    = {2026},
  version = {1.0.0},
  url     = {https://github.com/hamiddi/pygdis}
}

Scope and interpretation

This release implements the manuscript's reference formulation. The descriptor set, weighting strategy, and transition-localization method are modular and may be extended for application-specific studies. The current benchmark evidence supports broad cross-system applicability; it is not an exhaustive empirical proof over every nonlinear system.

Contributing and license

Contributions are welcome; see CONTRIBUTING.md. pyGDIS is distributed under the MIT License.