Center Manifold Analysis

This guide covers the creation and analysis of center manifolds in the Circular Restricted Three-Body Problem, including normal form theory, center manifold reduction, and Poincare maps.

Center Manifold Theory

Center manifolds are invariant manifolds associated with the center directions of libration points. They provide a powerful framework for understanding the local dynamics and finding periodic orbits.

Creating Center Manifolds

Center manifolds are created from libration points:

from hiten import System

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

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

# Compute the center manifold
center_manifold.compute()

print(f"Center manifold degree: {center_manifold.degree}")
print(f"Center manifold computed: {center_manifold.is_computed}")

Center Manifold Degrees

Control the degree of the center manifold expansion:

# Low degree (faster, less accurate)
cm_low = l1.get_center_manifold(degree=3)
cm_low.compute()

# Medium degree (balanced)
cm_med = l1.get_center_manifold(degree=6)
cm_med.compute()

# High degree (slower, more accurate)
cm_high = l1.get_center_manifold(degree=10)
cm_high.compute()

Center Manifold Properties

Access center manifold properties and data:

Basic Properties

# Basic properties
print(f"Degree: {center_manifold.degree}")
print(f"Libration point: {center_manifold.libration_point}")
print(f"Mass parameter: {center_manifold.mu}")
print(f"Position: {center_manifold.position}")

Coefficients

Access the computed coefficients:

# Get coefficients
coefficients = center_manifold.coefficients()
print(f"Coefficients shape: {coefficients.shape}")

# Access specific coefficients
print(f"First few coefficients: {coefficients[:10]}")

Normal Form Data

Access normal form transformation data:

# Get normal form data
normal_form = center_manifold.normal_form
print(f"Normal form type: {type(normal_form)}")

# Access transformation matrices
if hasattr(normal_form, 'C'):
    print(f"Transformation matrix C shape: {normal_form.C.shape}")
if hasattr(normal_form, 'Cinv'):
    print(f"Inverse transformation matrix shape: {normal_form.Cinv.shape}")

Center Manifold Coordinates

Work with center manifold coordinates:

Coordinate Transformations

# Transform from center manifold to physical coordinates
cm_coords = np.array([0.1, 0.05, 0.0, 0.0])  # [q1, q2, p1, p2]
physical_coords = center_manifold.to_synodic(cm_coords)
print(f"Physical coordinates: {physical_coords}")

# Transform from physical to center manifold coordinates
physical_state = np.array([0.8, 0.0, 0.1, 0.0, 0.15, 0.0])  # [x, y, z, vx, vy, vz]
cm_state = center_manifold.cm(physical_state)
print(f"Center manifold coordinates: {cm_state}")

Initial Conditions

Generate initial conditions for periodic orbits:

# Generate initial conditions
ic = center_manifold.to_synodic([0.0, 0.0], 0.6, "q3")
print(f"Initial conditions: {ic}")

# Different coordinate choices
ic_q1 = center_manifold.to_synodic([0.1, 0.0], 0.6, "q1")
ic_q2 = center_manifold.to_synodic([0.0, 0.1], 0.6, "q2")
ic_p1 = center_manifold.to_synodic([0.0, 0.0], 0.6, "p1")
ic_p2 = center_manifold.to_synodic([0.0, 0.0], 0.6, "p2")

Poincare Maps

Create Poincare maps from center manifolds:

Basic Poincare Maps

# Create Poincare map
poincare_map = center_manifold.poincare_map(energy=0.7)

# Compute with configuration
overrides = {
    "n_seeds": 50,
    "n_iter": 100,
    "seed_strategy": "axis_aligned",
}
poincare_map.compute(section_coord="p3", overrides=overrides)

print(f"Poincare map computed")

Poincare Map Configuration

Control Poincare map parameters:

# High resolution map
poincare_map = center_manifold.poincare_map(energy=0.7)
overrides = {
    "n_seeds": 100,
    "n_iter": 200,
    "seed_strategy": "axis_aligned",
}
poincare_map.compute(section_coord="p3", overrides=overrides)

# Different section coordinates
poincare_map_q1 = center_manifold.poincare_map(energy=0.7)
overrides_q1 = {
    "n_seeds": 50,
    "n_iter": 100,
}
poincare_map_q1.compute(section_coord="q1", overrides=overrides_q1)

poincare_map_q2 = center_manifold.poincare_map(energy=0.7)
overrides_q2 = {
    "n_seeds": 50,
    "n_iter": 100,
}
poincare_map_q2.compute(section_coord="q2", overrides=overrides_q2)

Poincare Map Analysis

Analyze computed Poincare maps:

Map Properties

# Plot the computed map
poincare_map.plot(axes=("p2", "q3"))

Map Data

# Plot the map with custom axes
poincare_map.plot(axes=("p2", "q3"))

Map Visualization

# Plot Poincare map
poincare_map.plot(axes=("p2", "q3"))

Finding Orbits in Center Manifold

Use center manifolds to find periodic orbits:

Analytical Initial Conditions

# Generate initial conditions for different orbit types

# Halo orbit
halo_ic = center_manifold.to_synodic([0.0, 0.0], 0.6, "q3")
print(f"Halo initial conditions: {halo_ic}")

# Lyapunov orbit
lyapunov_ic = center_manifold.to_synodic([0.1, 0.0], 0.6, "q1")
print(f"Lyapunov initial conditions: {lyapunov_ic}")

# Vertical orbit
vertical_ic = center_manifold.to_synodic([0.0, 0.0], 0.6, "p2")
print(f"Vertical initial conditions: {vertical_ic}")

Orbit Creation from Center Manifold

# Create orbits using center manifold initial conditions

# Halo orbit
halo = l1.create_orbit("halo", initial_state=halo_ic)
halo.correct()
halo.propagate()

# Lyapunov orbit
lyapunov = l1.create_orbit("lyapunov", initial_state=lyapunov_ic)
lyapunov.correct()
lyapunov.propagate()

# Vertical orbit
vertical = l1.create_orbit("vertical", initial_state=vertical_ic)
vertical.correct()
vertical.propagate()

Practical Examples

Earth-Moon L1 Center Manifold

from hiten import System

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

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

# Create and compute Poincare map
poincare_map = center_manifold.poincare_map(energy=0.7)
overrides = {
    "n_seeds": 50,
    "n_iter": 100,
    "seed_strategy": "axis_aligned",
}
poincare_map.compute(section_coord="p3", overrides=overrides)

# Plot map
poincare_map.plot(axes=("p2", "q3"))

# Generate orbits
halo_ic = center_manifold.to_synodic([0.0, 0.0], 0.6, "q3")
halo = l1.create_orbit("halo", initial_state=halo_ic)
halo.correct()
halo.propagate()
halo.plot()

Sun-Earth L2 Center Manifold

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

# Create center manifold
center_manifold = l2.get_center_manifold(degree=8)
center_manifold.compute()

# Create and compute Poincare map
poincare_map = center_manifold.poincare_map(energy=0.5)
overrides = {
    "n_seeds": 100,
    "n_iter": 200,
    "seed_strategy": "axis_aligned",
}
poincare_map.compute(section_coord="q1", overrides=overrides)

# Plot map
poincare_map.plot(axes=("q2", "p1"))

# Generate orbits
lyapunov_ic = center_manifold.to_synodic([0.05, 0.0], 0.5, "q1")
lyapunov = l2.create_orbit("lyapunov", initial_state=lyapunov_ic)
lyapunov.correct()
lyapunov.propagate()
lyapunov.plot()

Custom Center Manifold Analysis

# Custom mass parameter
system = System.from_mu(0.1)  # 10% mass ratio
l1 = system.get_libration_point(1)

# Create center manifold
center_manifold = l1.get_center_manifold(degree=5)
center_manifold.compute()

# Generate multiple orbits
energies = np.linspace(0.3, 0.7, 5)

for energy in energies:
    ic = center_manifold.to_synodic([0.0, 0.0], energy, "q3")
    orbit = l1.create_orbit("halo", initial_state=ic)
    orbit.correct()
    orbit.propagate()

    print(f"Energy {energy:.2f}: Period {orbit.period:.3f}")

Next Steps

Once you understand center manifolds, you can:

Perform advanced dynamical analysis (see Connection Analysis and Custom Detection)
Create custom systems (see Custom Dynamical Systems)
Use advanced continuation methods (see Continuation Methods and Custom Steppers)

For more advanced center manifold analysis, see Polynomial Methods and Algebra.