Library

Computation of homoclinic orbit to an hyperbolic saddle based on the projection boundary condition (PBC) method.

Internal fields

• disc::Any: Sructure encoding the discretization of the boundary value problem. For example, you can pass a Trapeze, a Collocation or an AbstractShootingProblem.

• lens::Any: Two lenses which are used to define 2 free parameters.

• T::Any: Return time T

• ϵ0::Any: Precision of how far the section is from the homoclinic point.

• ϵ1::Any: Precision of how far the section is from the homoclinic point.

• freelens::Any: Free parameters

• Qu0::Any: Orthonormal Projector on the unstable subspace orthogonal.

• Qs0::Any: Orthonormal Projector on the stable subspace orthogonal.

• N::Int64: Dimension of phase space

• updateEveryStep::Int64: updates the section every update_section_every_step step during continuation.

• jacobian::Symbol: How the jacobian of the problem is computed.

• test::Any

• testOrbitFlip::Bool

• testInclinationFlip::Bool

• nUnstable::Int64

• nStable::Int64

continuation(
    𝐇𝐨𝐦,
    homguess,
    lens,
    alg,
    _contParams;
    plot_solution,
    kwargs...
)

This is the continuation method for computing an homoclinic solution to a hyperbolic saddle. The parameter lens is the one from prob_vf::BifurcationProblem.

Arguments

Similar to continuation except that the problem is a HomoclinicHyperbolicProblemPBC.

continuation(
    prob_vf,
    bt,
    bvp,
    alg,
    _contParams;
    ϵ0,
    amplitude,
    freeparams,
    maxT,
    update_every_step,
    test_orbit_flip,
    test_inclination_flip,
    kwargs...
)

Perform automatic branch switching to homoclinic curve from a Bogdanov-Takens bifurcation point. It uses the homoclinic orbit predictor from the Bogdanov-Takens normal form.

Arguments

• prob::BifurcationProblem contains the vector field
• bt::BK.BogdanovTakens a Bogdanov-takens point. For example, you can get this from a call to bt = get_normal_form(br, ind_bt)
• bvp::BK.AbstractBoundaryValueDiscretization, for example Collocation(50, 4)
• alg continuation algorithm
• _contParams::ContinuationPar

Optional arguments

• ϵ0 = 1e-5 distance of the homolinic orbit from the saddle point
• amplitude = 1e-3 amplitude of the homoclinic orbit
• maxT = Inf limit on the "period" of the homoclinic cycle.

You can also pass the same arguments to the constructor of ::HomoclinicHyperbolicProblemPBC and those to continuation from BifurcationKit.

• kwargs arguments passed to continuation

generate_hom_problem(
    coll::Collocation,
    x::AbstractArray,
    pars,
    lensHom::Union{typeof(identity), IndexLens, PropertyLens, ComposedFunction};
    verbose,
    ϵ0,
    ϵ1,
    t0,
    t1,
    maxT,
    freeparams,
    kw...
) -> Tuple{Any, RecursiveArrayTools.ArrayPartition{_A, S} where {_A, S<:NTuple{5, Any}}, Any, RecursiveArrayTools.ArrayPartition{_A, S} where {_A, S<:NTuple{5, Any}}}

Generate a homoclinic to hyperbolic saddle problem from a periodic solution obtained with problem pb.

Arguments

• coll a Collocation which provide basic information, like the number of time slices M
• x::AbstractArray initial guess
• pars parameters
• lensHom parameter axis for continuation
• ϵ0, ϵ1: specify the distance to the saddle point of x₀, x₁
• t0, t1: specify the time corresponding to x₀, x₁. Overwrite the part with ϵ0, ϵ1 if set.

Optional arguments

You can pass the same arguments to the constructor of ::HomoclinicHyperbolicProblemPBC.

Output

• returns a HomoclinicHyperbolicProblemPBC and an initial guess.

generate_hom_problem(
    sh::Shooting,
    x::AbstractArray,
    pars,
    lensHom::Union{typeof(identity), IndexLens, PropertyLens, ComposedFunction};
    verbose,
    time,
    ϵ0,
    ϵ1,
    t0,
    t1,
    maxT,
    freeparams,
    kw...
) -> Tuple{Any, RecursiveArrayTools.ArrayPartition{_A, S} where {_A, S<:NTuple{5, Any}}, Any, RecursiveArrayTools.ArrayPartition{_A, S} where {_A, S<:NTuple{5, Any}}}

Generate a homoclinic to hyperbolic saddle problem from a periodic solution obtained with problem pb.

Arguments

• sh a Shooting which provide basic information, like the number of time slices M
• x::AbstractArray initial guess
• pars parameters
• lensHom::BK.AllOpticTypes parameter axis for continuation
• ϵ0, ϵ1: specify the distance to the saddle point of x₀, x₁
• t0, t1: specify the time corresponding to x₀, x₁. Overwrite the part with ϵ0, ϵ1 if set.

Optional arguments

You can pass the same arguments to the constructor of ::HomoclinicHyperbolicProblemPBC.

Output

• returns a HomoclinicHyperbolicProblemPBC and an initial guess.