Manifold and Tori Creation

This guide covers the creation and analysis of invariant manifolds and tori in the Circular Restricted Three-Body Problem, including stable/unstable manifolds and invariant tori.

Invariant Manifolds

Invariant manifolds are geometric structures that organize the dynamics around periodic orbits. They provide natural transport channels in the CR3BP.

Creating Manifolds

Manifolds are created from periodic orbits:

from hiten import System

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

# Create and correct halo orbit
halo = l1.create_orbit("halo", amplitude_z=0.2, zenith="southern")
halo.correct(max_attempts=25)
halo.propagate(steps=1000)

# Create stable manifold
stable_manifold = halo.manifold(stable=True, direction="positive")

# Create unstable manifold
unstable_manifold = halo.manifold(stable=False, direction="negative")

Computing Manifolds

Once created, manifolds must be computed to generate trajectories:

Basic Computation

# Compute stable manifold
stable_manifold.compute()

# Access computed trajectories
trajectories = stable_manifold.trajectories
print(f"Number of trajectories: {len(trajectories)}")

Advanced Computation

Control computation parameters:

# High accuracy computation
stable_manifold.compute(
    step=0.01,                    # Smaller step for higher resolution
    integration_fraction=0.9,     # Integrate for 90% of period
    displacement=1e-6,            # Small displacement along eigenvector
    dt=1e-3,                      # Integration time step
    method="adaptive",            # Integration method
    order=8                       # Integration order
)

Manifold Results

Computed manifolds provide access to trajectory data:

Accessing Trajectories

# Get trajectory data after computing
trajectories = stable_manifold.trajectories

print(f"Number of trajectories: {len(trajectories)}")
print(f"Trajectory shapes: {[traj.states.shape for traj in trajectories]}")

# Access individual trajectory
traj = trajectories[0]
print(f"First trajectory time range: {traj.t0} to {traj.tf}")
print(f"First trajectory states shape: {traj.states.shape}")

Invariant Tori

Invariant tori can be computed for certain orbits:

Creating Tori

from hiten import InvariantTori

# Create invariant torus
torus = InvariantTori(halo)

# Compute the torus
torus.compute(
    epsilon=1e-2,         # Perturbation parameter
    n_theta1=512,         # Grid resolution in theta1
    n_theta2=512          # Grid resolution in theta2
)

Torus Parameters

Control torus computation:

# Different schemes
torus.compute(epsilon=1e-2)

# Grid resolution
torus.compute(n_theta1=256, n_theta2=256)  # Lower resolution
torus.compute(n_theta1=1024, n_theta2=1024)  # Higher resolution

# Perturbation parameter
torus.compute(epsilon=1e-1)   # Large perturbation
torus.compute(epsilon=1e-4)   # Small perturbation

Examples

Earth-Moon L1 Halo Manifolds

from hiten import 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=0.3, zenith="southern")
halo.correct(max_attempts=25)
halo.propagate(steps=1000)

# Create and compute manifolds
stable_manifold = halo.manifold(stable=True, direction="positive")
unstable_manifold = halo.manifold(stable=False, direction="negative")

# Compute manifolds
stable_manifold.compute(
    step=0.02,
    integration_fraction=0.8,
    displacement=1e-6
)

unstable_manifold.compute(
    step=0.02,
    integration_fraction=0.8,
    displacement=1e-6
)

# Plot both manifolds
stable_manifold.plot()
unstable_manifold.plot()

Sun-Earth L2 Halo Tori

# Sun-Earth system
system = System.from_bodies("sun", "earth")
l2 = system.get_libration_point(2)

# Create L2 halo
halo_l2 = l2.create_orbit("halo", amplitude_z=0.1, zenith="northern")
halo_l2.correct()
halo_l2.propagate()

# Create and compute torus
torus = InvariantTori(halo_l2)
torus.compute(
    epsilon=1e-2,
    n_theta1=512,
    n_theta2=512
)

# Plot torus
torus.plot()

Multiple Manifold Types

# Create all manifold types
manifolds = {
    'stable_pos': halo.manifold(stable=True, direction="positive"),
    'stable_neg': halo.manifold(stable=True, direction="negative"),
    'unstable_pos': halo.manifold(stable=False, direction="positive"),
    'unstable_neg': halo.manifold(stable=False, direction="negative")
}

# Compute all manifolds
for name, manifold in manifolds.items():
    manifold.compute(
        step=0.05,
        integration_fraction=0.75
    )
    trajectories = manifold.trajectories
    print(f"{name}: {len(trajectories)} trajectories computed")

Next Steps

Once you understand manifolds and tori, you can:

Analyze Poincare sections (see Poincare Sections and Connections)
Find heteroclinic connections (see Poincare Sections and Connections)
Use center manifold methods (see Center Manifold Analysis)

For advanced manifold analysis, see Connection Analysis and Custom Detection.