Normal form of the Zero-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:

\[\mathbf L q_0=0, \quad \mathbf L q_1=i \omega_{0} q_1.\]

Under some conditions, $x(t)\approx x_0+2\Re w_1(t)q_1+w_0(t)q_0$ where $w_i$ satisfy the normal form:

\[\left\{\begin{aligned} \dot{w}_0= & \frac{1}{2} G_{200} w_0^2+G_{011}\left|w_1\right|^2+\frac{1}{6} G_{300} w_0^3 +G_{111} w_0\left|w_1\right|^2+O\left(\left\|\left(w_0, w_1, \bar{w}_1\right)\right\|^4\right) \\ \dot{w}_1= & i \omega_0 w_1+G_{110} w_0 w_1+\frac{1}{2} G_{210} w_0^2 w_1+\frac{1}{2} G_{021} w_1\left|w_1\right|^2 +O\left(\left\|\left(w_0, w_1, \bar{w}_1\right)\right\|^4\right) . \end{aligned}\right.\tag{E}\]

This normal form is usually computed in order to branch from a Zero-Hopf bifurcation point to curves of Neimark-Sacker bifurcations of periodic orbits (see ^[Kuznetsov2]). The flag hasNS (see below) tells whether such a curve emanates from the point. Passing detailed = Val(false) returns only the data needed for this branching procedure while detailed = Val(true) (the default through get_normal_form) returns all the coefficients of (E).

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 ZeroHopf.

Returned object

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

• x0, params, lens: the bifurcation point, the full parameter set and the two parameter axes,
• ζ = (; q0, q1) (resp. ζ★ = (; p0, p1)): the real null vector $q_0$ and the complex Hopf vector $q_1$ (resp. the left vectors), normalized so that $\langle q_0, p_0\rangle = \langle q_1, p_1\rangle = 1$,
• nf: a named tuple holding
  • λ0: the (real, null) eigenvalue $\mathbf L q_0 = 0$, ideally $\approx 0$,
  • ω: the imaginary frequency of the Hopf pair ($\omega = \operatorname{imag}(\lambda_1)$ where $\mathbf L q_1 = \lambda_1 q_1$, $\lambda_1 \approx i\omega_0$),
  • the coefficients of (E): G200, G110, G011, G111, G021 together with the derived quantities g110 = G110, f011 = G011 and the flag hasNS indicating whether a curve of Neimark-Sacker points emanates from the point,
  • the remaining data used by the predictors: the homological-equation terms h200, h110, h020, h011, h00010, h00001 and v10, v01, β1, β2, τ1, τ2, x.

Returned object (non-detailed)

Passing detailed = Val(false) returns only nf = (; ω, λ0, dFp) where ω is the (complex) eigenvalue $\lambda_1$ of the Hopf pair, together with ζ and ζ★. This is the minimal information required to start the continuation of the curves of equilibria (fold or Hopf curves) emanating from the point.

Predictors

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

predictor(
    zh::BifurcationKit.ZeroHopf,
    ::Val{:HopfCurve},
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:hopf, :ω, :EigenVec, :EigenVecAd, :x0), <:Tuple{BifurcationKit.var"#832#838"{BifurcationKit.var"#HopfCurve#835"{_A, _B}} where {_A, _B}, BifurcationKit.var"#833#839"{BifurcationKit.var"#HopfCurve#835"{_A, _B}} where {_A, _B}, BifurcationKit.var"#EigenVec#836"{BifurcationKit.ZeroHopf{Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}} where {Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}, BifurcationKit.var"#EigenVecAd#837"{BifurcationKit.ZeroHopf{Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}} where {Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}, BifurcationKit.var"#834#840"}}

predictor(
    zh::BifurcationKit.ZeroHopf,
    ::Val{:FoldCurve},
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:fold, :λ0, :EigenVec, :EigenVecAd, :x0), <:Tuple{BifurcationKit.var"#842#848"{BifurcationKit.var"#FoldCurve#845"{_A, _B}} where {_A, _B}, BifurcationKit.var"#843#849"{BifurcationKit.var"#FoldCurve#845"{_A, _B}} where {_A, _B}, BifurcationKit.var"#EigenVec#846"{BifurcationKit.ZeroHopf{Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}} where {Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}, BifurcationKit.var"#EigenVecAd#847"{BifurcationKit.ZeroHopf{Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}} where {Tv, Tτ, T, Tpar, Tlens, Tevl, Tevr, Tnf}, BifurcationKit.var"#844#850"}}

predictor(
    zh::BifurcationKit.ZeroHopf,
    ::Val{:NS},
    ϵ;
    verbose,
    ampfactor
) -> NamedTuple{(:orbit, :hasNS, :params, :T, :k), <:Tuple{BifurcationKit.var"#852#854"{BifurcationKit.var"#NS#853"{_A, _B, _C}} where {_A, _B, _C}, Vararg{Any, 4}}}

References