Normal form of the Hopf-Hopf bifurcation

We follow the paper^[Kuznetsov],^[Kuznetsov2] and consider a Cauchy problem

\[\dot x=\mathbf F(x,p).\]

We denote by $\mathbf L$ the jacobian of $\mathbf F$ at the bifurcation point $(x_0,p_0)$. We choose a basis such that:

\[\mathbf L q_1=i \omega_{1} q_1, \quad \mathbf L q_2=i \omega_{2} q_2.\]

Under some conditions, $x(t)\approx x_0+2\Re w_1(t)q_1+2\Re w_2(t)q_2$ where $w_i$ satisfy the normal form:

\[\left\{\begin{aligned} \dot{w}_1= & i \omega_1 w_1+\frac{1}{2} G_{2100} w_1\left|w_1\right|^2+G_{1011} w_1\left|w_2\right|^2 +\frac{1}{12} G_{3200} w_1\left|w_1\right|^4+\frac{1}{2} G_{2111} w_1\left|w_1\right|^2\left|w_2\right|^2+\frac{1}{4} G_{1022} w_1\left|w_2\right|^4 \\ & +O\left(\left\|\left(w_1, \bar{w}_1, w_2, \bar{w}_2\right)\right\|^6\right) \\ \dot{w}_2= & i \omega_2 w_2+G_{1110} w_2\left|w_1\right|^2+\frac{1}{2} G_{0021} w_2\left|w_2\right|^2 +\frac{1}{4} G_{2210} w_2\left|w_1\right|^4+\frac{1}{2} G_{1121} w_2\left|w_1\right|^2\left|w_2\right|^2+\frac{1}{12} G_{0032} w_2\left|w_2\right|^4 \\ & +O\left(\left\|\left(w_1, \bar{w}_1, w_2, \bar{w}_2\right)\right\|^6\right) \end{aligned}\right.\tag{E}\]

This normal form is usually computed in order to branch from a Hopf-Hopf bifurcation point to curves of Neimark-Sacker bifurcations of periodic orbits (see ^[Kuznetsov2]). Passing detailed = Val(false) returns only the data needed for this branching procedure while detailed = Val(true) (the default through get_normal_form) returns the cubic coefficients $G_{2100}, G_{0021}, G_{1011}, G_{1110}$ of (E) together with the coefficients required by the predictors.

Normal form computation

The normal form (E) can be automatically computed as follows

get_normal_form(br, ind_bif;    verbose = false, lens = getlens(br),    detailed = Val(true),            # full normal form    autodiff = true,                 # use ForwardDiff for the differentiations    start_with_eigen = Val(true),    # Val(false): kernel basis via bordered systems    bls = MatrixBLS(), bls_adjoint = bls)

br is a branch computed after a call to continuation with detection of bifurcation points enabled and ind_bif is the index of the bifurcation point on the branch br. The above call returns a point with information needed to compute the bifurcated branch. For more information about the optional parameters (nev, ζs, scaleζ, ...), we refer to get_normal_form. The result returns an object of type HopfHopf.

Returned object

The call get_normal_form(br, ind_bif) returns a HopfHopf point with the following fields

• x0, params, lens: the bifurcation point, the full parameter set and the two parameter axes,
• ζ = (; q1, q2) (resp. ζ★ = (; p1, p2)): the two complex eigenvectors of the Hopf pairs (resp. the left vectors), normalized by scaleζ and such that $\langle p_i, q_i\rangle = 1$. The pairs are ordered so that $\operatorname{imag}(\lambda_1) \geq \operatorname{imag}(\lambda_2) > 0$,
• nf: a named tuple holding
  • ω0: the frequency recorded by the continuation (i.e. that of the pair from which the Hopf-Hopf point was detected),
  • λ1, λ2: the eigenvalues of the two Hopf pairs (with the ordering above),
  • the cubic coefficients of (E): G2100, G0021 (self couplings) and G1011, G1110 (cross couplings),
  • the data required by the :NS predictor: γ₁₁₀, γ₁₀₁, γ₂₁₀, γ₂₀₁, the $2\times 2$ matrix Γ, the homological-equation terms h₁₁₀₀, h₀₀₁₁, h₂₀₀₀, h₀₀₂₀, the parameter terms h₀₀₀₀₁₀, h₀₀₀₀₀₁ and the two sets ns1 = (; dω1, dω2, α) and ns2 = (; dω1, dω2, α).

Predictors

The predictor for a non trivial guess at distance $\delta p$ from the bifurcation point is provided by the methods

predictor(
    hh::BifurcationKit.HopfHopf,
    ::Val{:HopfCurve},
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:hopf, :ω, :EigenVec, :EigenVecAd, :x0), <:Tuple{BifurcationKit.var"#871#877"{BifurcationKit.var"#HopfCurve#874"{_A, _B}} where {_A, _B}, BifurcationKit.var"#872#878"{BifurcationKit.var"#HopfCurve#874"{_A, _B}} where {_A, _B}, BifurcationKit.var"#EigenVec#875", BifurcationKit.var"#EigenVecAd#876", BifurcationKit.var"#873#879"}}

predictor(
    hh::BifurcationKit.HopfHopf,
    ::Val{:NS},
    ϵ;
    verbose,
    ampfactor
) -> NamedTuple{(:ns1, :ns2, :params1, :params2, :ω11, :ω12, :ω21, :ω22, :T1, :T2, :k1, :k2), <:Tuple{BifurcationKit.var"#881#885"{BifurcationKit.var"#NS1#883"{ϵ, q1, x1, h₂₀₀₀}} where {ϵ, q1, x1, h₂₀₀₀}, BifurcationKit.var"#882#886"{BifurcationKit.var"#NS2#884"{ϵ, q2, x2, h₀₀₂₀}} where {ϵ, q2, x2, h₀₀₂₀}, Vararg{Any, 10}}}

References