Continuous and Discrete Geometry

pyna uses separate object families for continuous-time and discrete-time dynamical systems.

Continuous-time side:

• Trajectory is sampled finite-time geometry.

• Cycle is a periodic orbit of a flow.

• Tube is a resonance zone around an elliptic cycle, possibly bounded by hyperbolic cycles.

• TubeChain groups tubes belonging to one resonance.

Discrete-time side:

• Orbit is sampled map iteration geometry.

• PeriodicOrbit is a closed orbit of a map.

• Island is one reduced resonance island on a section.

• IslandChain is the section-level chain of islands.

The bridge between the two sides is a section cut. Cutting a Cycle by a Poincare section produces a PeriodicOrbit of the return map. Cutting a Tube produces an IslandChain. Cutting a TubeChain merges the island chains from its tubes.

This separation is intentional. A numerical trajectory can be useful geometry without proving invariance. Builders and adapters therefore make promotion explicit: users can require closure checks before a sampled trajectory becomes a Cycle or before map samples become a PeriodicOrbit.

The same vocabulary is shared by generic finite-dimensional systems and by the toroidal magnetic-field-line specialization. Generic roots are available as pyna.topo.CoreTube and related names; toroidal defaults remain available as pyna.topo.Tube, pyna.topo.Cycle and pyna.topo.IslandChain.

See Also

• Mini Cases

• RMP Stellarator Resonance Analysis

• Tutorial: Monodromy Matrix and Orbit Shift on an Analytic X-type Saddle Cycle

• Island Jacobian Analysis: Periodic Orbits and Monodromy Matrices