Normal form of the Bautin bifurcation

• Normal form of the Bautin bifurcation
• • Normal form computation
  • Predictor
  • References

We follow the paper^[Kuznetsov] 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:

\[\mathbf L q=i \omega_{0} q, \quad \mathbf L^{T} p=-i \omega_{0} p, \quad \langle p, q\rangle=1.\]

Under some conditions, $x(t)\approx x_0+2\Re w(t)q$ where $w$ satisfies the normal form:

\[\dot{w}=i \omega_{0} w+\frac{1}{2} G_{21} w|w|^{2}+\frac{1}{12} G_{32} w|w|^{4}+O\left(|w|^{6}\right).\tag{E}\]

The second Lyapunov coefficient is

\[l_2:=\frac{1}{12} \operatorname{Re} G_{32}.\]

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    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 Bautin.

Returned object

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

• x0, params, lens: the bifurcation point, the full parameter set and the two parameter axes,
• ζ (resp. ζ★): the complex right (resp. left) eigenvector of the Hopf pair, normalized by scaleζ and such that $\langle \zeta*, \zeta\rangle = 1$,
• nf: a named tuple holding
  • ω: the frequency of the Hopf pair,
  • G21, G32: the complex cubic and quintic coefficients of (E),
  • l1 = G21/2, l2 = real(G32)/12: the first and second Lyapunov coefficients (the second Lyapunov coefficient is that of (E)),
  • in detailed mode, the additional data required to branch to the curve of folds of periodic orbits: h₂₀₀₀, h₁₁₀₀, h₀₀₁₀, h₀₀₀₁ (homological equation terms), γ₁₁₀, γ₁₀₁, γ₂₁₀, γ₂₀₁ and α (unfolding coefficients, see ^[Kuznetsov]).

The predictor below relies on the detailed normal form: call get_normal_form(br, ind_bif; detailed = Val(true)) (the default).

Predictor

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

predictor(
    gh::BifurcationKit.Bautin,
    ::Val{:FoldPeriodicOrbitCont},
    ϵ;
    verbose,
    ampfactor
) -> NamedTuple{(:orbit, :ω, :params, :x0), <:Tuple{BifurcationKit.var"#790#793"{BifurcationKit.var"#FoldPO#792"{ϵ, q0, x0, h₂₀₀₀}} where {ϵ, q0, x0, h₂₀₀₀}, Any, Any, BifurcationKit.var"#791#794"}}

References