odeanalysis

odeanalysis is a SymPy-based package for exact symbolic, formal, structural, and certified analysis of scalar linear ODEs and first-order linear differential systems. It is designed for structural questions—singularities, Frobenius/Levelt data, Newton and Levelt--Turrittin structure, Stokes geometry, turning points, parameter transitions, exact continuation in selected families, and rigorously enclosed numerical transport—not as a generic nonlinear ODE solver.

The package distinguishes exact, certified-formal, certified-semialgebraic, certified-numerical, structural, and incomplete results. See the certification model before interpreting a complete, certified, or structural_only field.

Choose a workflow

The documentation index gives the complete map.

Capability overview

The theorem-level boundary is in capabilities and limitations.

Installation

python -m pip install odeanalysis

Python 3.11+ is required. Core dependencies are SymPy, funcprops, and semialg. Rigorous ball continuation uses:

python -m pip install "odeanalysis[certified]"

Scalar quick start

import sympy as sp
from odeanalysis import analyze_ode_singularities, frobenius_analysis

x = sp.symbols("x")
y = sp.Function("y")
ode = x**2*sp.diff(y(x), x, 2) + x*sp.diff(y(x), x) + (x**2-1)*y(x)

singularities = analyze_ode_singularities(ode, y, x)
frobenius = frobenius_analysis(ode, y, x, point=0, terms=6)

Continue with local analysis, Fuchsian analysis, Newton invariants, formal asymptotics, turning points, canonical equations, and analytic continuation.

System quick start

import sympy as sp
from odeanalysis import FirstOrderSystem, analyze_system_singularity, formal_system_analysis, system_stokes_geometry

x = sp.symbols("x")
system = FirstOrderSystem(x, sp.ImmutableMatrix.diag(x**-2, -x**-2))
local = analyze_system_singularity(system, 0)
formal = formal_system_analysis(system, 0, adaptive=True)
stokes = system_stokes_geometry(formal)
assert formal.certificate.verified

The system guides are singularities, formal reduction, Stokes geometry, and parameter stratification. The overview remains in systems and the deeper reducer reference is Levelt--Turrittin.

Parameter quick start

from odeanalysis import system_formal_type_stratification

a = sp.symbols("a", real=True)
family = FirstOrderSystem(x, sp.ImmutableMatrix.diag(a/x, -a/x))
strata = system_formal_type_stratification(family, (a,), max_resonance_order=2)

A formal-type stratum certifies a discrete bounded signature: singularity kind, leading rank, spectral multiplicity pattern, resonance orders within the configured bound, ramification, block dimensions, exponential-block data, and structural Stokes combinatorics. Continuously varying exponent values are not claimed to be constant merely because a cell is open.

Certified numerical continuation

from odeanalysis import certified_system_continuation

constant = FirstOrderSystem(x, sp.ImmutableMatrix([[0, 1], [0, 0]]))
transport = certified_system_continuation(constant, 0, 2, precision_bits=192)

The current certified backend is deliberately narrow: for constant homogeneous systems it encloses exp((b-a)A) with Arb complex balls. Variable-coefficient validated integration is refused rather than replaced by tolerance-only floating-point integration. See certified continuation.

Examples, verification, and boundaries

Start with the example gallery and worked failures. The older compact examples reference is retained for API-oriented snippets.

For evidence semantics and conventions see verification, mathematical conventions, API classification, public API index, architecture, and traceability. Dominance documents sector ordering, and the user guide remains a broad reference.

License

GPL-3.0-only. See LICENSE.