Internals

ParameterizationType

A Union type for the different types of parameterizations that can be used when parameterizing a periodic orbit or its associated invairiant stable or intable manifold.

AbstractIndependantVariable

An abstract type to encompas all supported independant variables.

AbstractInterpolationWrapper

And abstract type that is used to define the interface for interpolation objects.

AbstractManifold

Abstract type for manifolds.

FastChebInterpolation(it::IT) <: AbstractInterpolationWrapper

A wrapper for FastChebInterp.ChebPoly that's functional for both the FastChebInterp.jl implementations of interpolation and least-squares regression (see FastChebInterp.jl for details).

Fields

• it::IT: The FastChebInterp.ChebPoly struct.

ParameterListType

A Union type of the acceptable types for a list of parameter values.

get_full_initial_state(orbit::AbstractPeriodicOrbit)

Get the full initial reference state of the periodic orbit.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.

Returns

• x0::SVector{6,Float64}: The full initial state of the periodic orbit.

get_jacobi_integral(orbit::AbstractPeriodicOrbit)

Get the Jacobi integral of the periodic orbit.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.

Returns

• C::Float64: The Jacobi integral of the periodic orbit.

get_orbit_length(orbit::TypeAPeriodicOrbit, type::Type{AbstractIndependantVariable})

Get the length of the periodic orbit in the desired variable.

Arguments

• orbit::TypeAPeriodicOrbit: The periodic orbit.
• type::Type{AbstractIndependantVariable}: The type of length to return.

Returns

• length::Float64: The length of the periodic orbit in the desired variable.

get_stable_manifold_initial_condition(
    man::InvariantManifold, left_pert::Bool, τ1::AbstractFloat, PT::ParameterizationType;
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
)

Returns the initial condition for a stable manifold trajectory as parameterized by τ1 ∈ [0,1].

Arguments

• man::InvariantManifold: The invariant manifold.
• left_pert::Bool: If true, the perturbation is to the left of the periodic orbit.
• τ1::AbstractFloat: The parameter for the initial condition on the periodic orbit.
• PT::ParameterizationType: The parameterization type for the τ1 parameter.

Keyword Arguments

• solver = Vern9(): The DifferentialEquations.jl solver to use.
• reltol = 1e-14: The relative tolerance for the solver.
• abstol = 1e-14: The absolute tolerance for the solver.

Returns

• x0m::SVector{6,Float64}: The initial condition for the stable manifold trajectory.

get_state_and_monodromy_matrix(orbit::AbstractPeriodicOrbit, τ1::AbstractFloat, PT::ParameterizationType)

Get the state and monodromy matrix of the periodic orbit at a give state parameterized by τ1 ∈ [0, 1] in terms of PT.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.
• τ1::AbstractFloat: The parameterization of the final state on the orbit.
• PT::ParameterizationType: The type of parameterization to use.

get_state_and_stm(z::SMatrix{6,7,T,42}) where T

Get the state and state transition matrix (STM) from the full integration matrix.

Arguments

• z::SMatrix{6,7,T,42}: Matrix with first column as the state and the remaining 6 x 6 as the STM.

Returns

• Tuple{SVector{6,T},SMatrix{6,6,T,36}: The state vector and the state transition matrix in a return tuple.

get_stm(z::SMatrix{6,7,T,42}) where T

Get the state transition matrix (STM) from the full integration matrix.

Arguments

• z::SMatrix{6,7,T,42}: Matrix with first column as the state and the remaining 6 x 6 as the STM.

Returns

• SMatrix{6,6,T,36}: The state transition matrix.

get_unstable_manifold_initial_condition(
    man::InvariantManifold, left_pert::Bool, τ1::AbstractFloat, PT::ParameterizationType;
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
)

Returns the initial condition for a unstable manifold trajectory as parameterized by τ1 ∈ [0,1].

Arguments

• man::InvariantManifold: The invariant manifold.
• left_pert::Bool: If true, the perturbation is to the left of the periodic orbit.
• τ1::AbstractFloat: The parameter for the initial condition on the periodic orbit.
• PT::ParameterizationType: The parameterization type for the τ1 parameter.

Keyword Arguments

• solver = Vern9(): The DifferentialEquations.jl solver to use.
• reltol = 1e-14: The relative tolerance for the solver.
• abstol = 1e-14: The absolute tolerance for the solver.

Returns

• x0m::SVector{6,Float64}: The initial condition for the stable manifold trajectory.

initial_state_with_stm(x0::SVector{6,T}) where T

Create the initial state with the state transition matrix (STM).

Arguments

• x0::SVector{6,T}: Initial state vector.

Returns

• SMatrix{6,7,T,42}: Matrix with first column as the state and the remaining 6 x 6 as the STM initial condition (i.e., [x0, I]).

jacobi_integral_gradient(x, μ)

Computes the gradient of the Jacobi integral for the CRTBP given the state x and mass parameter μ.

Arguments

• x::AbstractVector: State vector.
• μ::Real: Mass parameter.

natural_crtbp_and_time_eom_with_independant_arclen(x, p, t)

Compute the equations of motion for the CRTBP model with an extra state for the time with only natural motion. Here the independant variable has been transformed from time to arc-length.

Arguments

• x::SVector{7}: State vector, i.e., [position^T; velocity^T].
• p::Tuple{Float64,}: Tuple of parameters, (mu,)
• t::Float64: Time.

natural_crtbp_eom(x, p, t)

Compute the equations of motion for the CRTBP model with only natural motion.

Arguments

• x::SVector{6}: State vector, i.e., [position^T; velocity^T].
• p::Tuple{Float64,Float64,Float64}: Tuple of parameters, (mu,)
• t::Float64: Time.

natural_crtbp_eom_with_arclen(x, p, t)

Compute the equations of motion for the CRTBP model with only natural motion, employing the arclength along the trajectory as an additional state.

Arguments

• x::SVector{7}: State vector, i.e., [position^T; velocity^T; arc-len].
• p::Tuple{Float64,}: Tuple of parameters, (mu,)
• t::Float64: Time.

natural_crtbp_eom_with_independant_arclen(x, p, t)

Compute the equations of motion for the CRTBP model with only natural motion where the independant variable has been transformed from time to arc-length.

Arguments

• x::SVector{7}: State vector, i.e., [position^T; velocity^T].
• p::Tuple{Float64,}: Tuple of parameters, (mu,)
• t::Float64: Time.

natural_crtbp_eom_with_stm(x, p, t)

Compute the equations of motion for the CRTBP model with the state transition matrix.

Arguments

• x::SMatrix{6,7,Float64}: Matrix with first column ast the state and the remaining 6 x 6 as the STM.
• p::Tuple{Float64,}: Tuple of parameters, (mu,)
• t::Float64: Time.

propagate_from_initial_conditions(
    orbit::AbstractPeriodicOrbit, τ1::AbstractFloat, PT::ParameterizationType,
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
)

Propagate the periodic orbit from the initial conditions to another state on the orbit, as parameterized by τ1 ∈ [0, 1] in terms of PT::ParameterizationType.

Arguments

• orbit::AbstractPeriodicOrbit: The periodic orbit.
• τ1::AbstractFloat: The parameterization of the final state on the orbit.
• PT::ParameterizationType: The type of parameterization to use.

Keyword Arguments

• solver::OrdinaryDiffEq.AbstractODESolver: The ODE solver to use for the integration.
• reltol::Real: The relative tolerance for the ODE solver.
• abstol::Real: The absolute tolerance for the ODE solver.

Returns

• x1::SVector{6,Float64}: The final state on the periodic orbit.

propagate_return_final_state_and_stm(
    x0::SVector{6,TX}, tspan::Tuple{TT,TT}, mu;
    solver = Vern9(),
    reltol = 1e-14,
    abstol = 1e-14,
) where {TX,TT}

Propagate the state x0 over the time span tspan and return the final state and the state transition matrix (STM).

arguments

• x0::SVector{6,TX}: The initial state with x0 = [r0; v0].
• tspan::Tuple{TT,TT}: The span of the independant variable forwhich to solve the ode.
• mu::Real: The mass parameter of the three-body system (mu = m2 / (m1 + m2)).

keywords

• solver::OrdinaryDiffEq.AbstractODESolver: The solver to use for the integration.
• reltol: The relative tolerance for the solver.
• abstol: The absolute tolerance for the solver.

returns

• SMatrix{6,6,TX,36}: The stm for the full trajectory

typeA_shooting_function(u, p; constraint = (:x_start_coordinate, 0.0))

Shooting function for computing Type A periodic orbits in the CRTBP using the methodology described by Restrepo and Russell in A database of planar axisymmetric periodic orbits for the solar system doi: 10.1007/s10569-018-9844-6.

Arguments

• u::SVector{4,T}: Vector of decision variables, [rx, rz, vy, P / 2], where P is half of the orbital period.
• p::Tuple{Float64,}: Tuple of parameters, (mu,), where mu is the mass parameter.

Keyword Arguments

• constraint::Tuple{Symbol,Real}: Tuple of the constraint type and value corresponding to the final constraint considered to provide a fully defined system of nonlinear eqautions. The constraint type can be either :x_start_coordinate or :jacobi_integral. The constraint value is the value that the constraint should be equal to for the desired periodic orbit. Note that new constraints can be added with relative ease through modification of this function and its corresponding jacobian function typeA_shooting_jacobian.
• solver::OrdinaryDiffEq.AbstractODESolver: The solver to use for the integration.
• reltol: The relative tolerance for the solver.
• abstol: The absolute tolerance for the solver.

Returns

• SVector{4,T}: Vector of residuals for the nonlinear system of equations.

typeA_shooting_jacobian(u, p; constraint = (:x_start_coordinate, 0.0))

The Jacobian of the shooting function for computing Type A periodic orbits in the CRTBP using the methodology described by Restrepo and Russell in A database of planar axisymmetric periodic orbits for the solar system doi: 10.1007/s10569-018-9844-6.

Arguments

• u::SVector{4,T}: Vector of decision variables, [rx, rz, vy, P / 2], where P is half of the orbital period.
• p::Tuple{Float64,}: Tuple of parameters, (mu,), where mu is the mass parameter.

Keyword Arguments

• constraint::Tuple{Symbol,Real}: Tuple of the constraint type and value corresponding to the final constraint considered to provide a fully defined system of nonlinear eqautions. The constraint type can be either :x_start_coordinate or :jacobi_integral.
• solver::OrdinaryDiffEq.AbstractODESolver: The solver to use for the integration.
• reltol: The relative tolerance for the solver.
• abstol: The absolute tolerance for the solver.

Returns

• SMatrix{4,4,T,16}: Jacobian of the nonlinear system of equations.