Poincare Sections and Connections

This guide covers the creation and analysis of Poincare sections, manifold intersections, and heteroclinic connections in the Circular Restricted Three-Body Problem.

Poincare Sections

Poincare sections are cross-sections of phase space that reveal the underlying dynamics of the system.

Creating Poincare Sections

HITEN provides several types of Poincare sections:

Synodic Sections

Synodic sections are defined in the rotating frame:

from hiten.system import SynodicMap, System

# Create system and orbit
system = System.from_bodies("earth", "moon")
l1 = system.get_libration_point(1)

# Create halo orbit
halo = l1.create_orbit("halo", amplitude_z=0.2, zenith="southern")
halo.correct()
halo.propagate()

# Generate section from manifold
stable_manifold = halo.manifold(stable=True, direction="positive")
stable_manifold.compute()

# Create synodic map from manifold
synodic_map = SynodicMap(stable_manifold)

# Compute section with configuration
overrides = {
    "interp_kind": "cubic",
    "segment_refine": 30,
}
synodic_map.compute(
    section_axis="x",
    section_offset=0.8,
    plane_coords=("y", "z"),
    overrides=overrides
)

Center Manifold Sections

Center manifold sections are defined in the center manifold coordinates:

from hiten.algorithms.poincare import CenterManifoldMapConfig

# Get center manifold
center_manifold = l1.get_center_manifold(degree=6)
center_manifold.compute()

# Create Poincare map
poincare_map = center_manifold.poincare_map(
    energy=0.7,                  # Energy level
    section_coord="p3",          # Section coordinate
    n_seeds=50,                  # Number of seed points
    n_iter=100,                  # Number of iterations
    seed_strategy="axis_aligned" # Seed strategy
)

Section Configuration

Control section parameters for different applications:

Basic Configuration

# Simple x-section
synodic_map.compute(
    section_axis="x",
    section_offset=0.8,
    plane_coords=("y", "z")
)

# Simple y-section
synodic_map.compute(
    section_axis="y",
    section_offset=0.0,
    plane_coords=("x", "z")
)

Advanced Configuration

# High resolution section
overrides = {
    "interp_kind": "cubic",
    "segment_refine": 50,
}
synodic_map.compute(
    section_axis="x",
    section_offset=0.8,
    plane_coords=("y", "z"),
    overrides=overrides
)

Section Analysis

Analyze computed sections:

Accessing Section Data

# Get section points
points = synodic_map.points
labels = synodic_map.labels

print(f"Number of section points: {len(points)}")
print(f"Section labels: {labels}")

# Access individual points
for i, point in enumerate(points):
    print(f"Point {i}: {point}")

Section Properties

# Section properties
print(f"Section axis: {synodic_map.section_axis}")
print(f"Section offset: {synodic_map.section_offset}")
print(f"Plane coordinates: {synodic_map.plane_coords}")

# Bounds
bounds = synodic_map.bounds
print(f"Section bounds: {bounds}")

Heteroclinic Connections

A heteroclinic connection is a path in phase space which joins two different equilibrium points.

Creating Connections

from hiten.algorithms.connections import ConnectionPipeline
from hiten.algorithms.connections.config import _ConnectionConfig
from hiten.algorithms.poincare import SynodicMapConfig

# Create two orbits
l1 = system.get_libration_point(1)
l2 = system.get_libration_point(2)

# L1 halo orbit
halo_l1 = l1.create_orbit("halo", amplitude_z=0.5, zenith="southern")
halo_l1.correct()
halo_l1.propagate()

# L2 halo orbit
halo_l2 = l2.create_orbit("halo", amplitude_z=0.3663368, zenith="northern")
halo_l2.correct()
halo_l2.propagate()

# Create manifolds
manifold_l1 = halo_l1.manifold(stable=True, direction="positive")
manifold_l1.compute(integration_fraction=0.9, step=0.005)

manifold_l2 = halo_l2.manifold(stable=False, direction="negative")
manifold_l2.compute(integration_fraction=1.0, step=0.005)

Once you created the manifolds, you can create a connection between them by configuring the section and search parameters.

# Create unified configuration
section_cfg = SynodicMapConfig(
    section_axis="x",
    section_offset=1 - system.mu,
    plane_coords=("y", "z"),
    interp_kind="cubic",
    segment_refine=30,
    tol_on_surface=1e-9,
    dedup_time_tol=1e-9,
    dedup_point_tol=1e-9,
    max_hits_per_traj=None,
    n_workers=None,
)

config = _ConnectionConfig(
    section=section_cfg,
    direction=None,
    delta_v_tol=1.0,
    ballistic_tol=1e-8,
    eps2d=1e-3,
)

# Create connection pipeline
conn = ConnectionPipeline.with_default_engine(config=config)

Then you can solve for connections between the manifolds.

# Solve for connections
conn.solve(manifold_l1, manifold_l2)

# Check results
print(f"Connections found: {conn}")

# Plot results
conn.plot(dark_mode=True)

Connection Analysis

Analyze found connections:

Connection Properties

# Plot and analyze results
conn.plot(dark_mode=True)

Connection Classification

# Connection results are displayed via plot
conn.plot(dark_mode=True)

Connection Visualization

# Plot connections
conn.plot(dark_mode=True)

Practical Examples

Earth-Moon L1-L2 Connection

from hiten import System
from hiten.algorithms.connections import ConnectionPipeline
from hiten.algorithms.connections.config import _ConnectionConfig
from hiten.algorithms.poincare import SynodicMapConfig

# Create system
system = System.from_bodies("earth", "moon")
mu = system.mu

# Get libration points
l1 = system.get_libration_point(1)
l2 = system.get_libration_point(2)

# Create orbits
halo_l1 = l1.create_orbit("halo", amplitude_z=0.5, zenith="southern")
halo_l1.correct()
halo_l1.propagate()

halo_l2 = l2.create_orbit("halo", amplitude_z=0.3663368, zenith="northern")
halo_l2.correct()
halo_l2.propagate()

# Create manifolds
manifold_l1 = halo_l1.manifold(stable=True, direction="positive")
manifold_l1.compute(integration_fraction=0.9, step=0.005)

manifold_l2 = halo_l2.manifold(stable=False, direction="negative")
manifold_l2.compute(integration_fraction=1.0, step=0.005)

# Configure connection
section_cfg = SynodicMapConfig(
    section_axis="x",
    section_offset=1 - mu,
    plane_coords=("y", "z"),
    interp_kind="cubic",
    segment_refine=30,
    tol_on_surface=1e-9,
    dedup_time_tol=1e-9,
    dedup_point_tol=1e-9
)

config = _ConnectionConfig(
    section=section_cfg,
    direction=None,
    delta_v_tol=1.0,
    ballistic_tol=1e-8,
    eps2d=1e-3,
)

# Find connections
conn = ConnectionPipeline.with_default_engine(config=config)
conn.solve(manifold_l1, manifold_l2)

# Display results
print(conn)
conn.plot(dark_mode=True)

Sun-Earth L1-L2 Connection

# Sun-Earth system
system = System.from_bodies("sun", "earth")
mu = system.mu

# Get libration points
l1 = system.get_libration_point(1)
l2 = system.get_libration_point(2)

# Create orbits
halo_l1 = l1.create_orbit("halo", amplitude_z=0.1, zenith="southern")
halo_l1.correct()
halo_l1.propagate()

halo_l2 = l2.create_orbit("halo", amplitude_z=0.1, zenith="northern")
halo_l2.correct()
halo_l2.propagate()

# Create manifolds
manifold_l1 = halo_l1.manifold(stable=True, direction="positive")
manifold_l1.compute(integration_fraction=0.8, step=0.01)

manifold_l2 = halo_l2.manifold(stable=False, direction="negative")
manifold_l2.compute(integration_fraction=0.8, step=0.01)

# Configure connection
section_cfg = SynodicMapConfig(
    section_axis="x",
    section_offset=1 - mu,
    plane_coords=("y", "z"),
    interp_kind="cubic",
    segment_refine=20,
    tol_on_surface=1e-10,
    dedup_time_tol=1e-10,
    dedup_point_tol=1e-10
)

config = _ConnectionConfig(
    section=section_cfg,
    direction=None,
    delta_v_tol=0.1,
    ballistic_tol=1e-10,
    eps2d=1e-4,
)

# Find connections
conn = ConnectionPipeline.with_default_engine(config=config)
conn.solve(manifold_l1, manifold_l2)

# Display results
print(conn)
conn.plot()

Next Steps

Once you understand Poincare sections and connections, you can:

Use center manifold methods (see Center Manifold Analysis)
Perform advanced dynamical analysis (see Connection Analysis and Custom Detection)
Create custom systems (see Custom Dynamical Systems)

For more advanced connection analysis, see Connection Analysis and Custom Detection.