Simple Hopf point

At a Hopf branch point $(x_0,p_0)$ for the problem $F(x,p)=0$, the spectrum of the linear operator $dF(x_0,p_0)$ contains two purely imaginary $\pm i\omega,\ \omega > 0$ which are simple. At such point, we can compute the normal form to transform the initial Cauchy problem

\[\mathbf{M}(x,p)\cdot\dot x = \mathbf{F}(x,p)\]

where $\mathbf{M}(x,p)$ is a mass matrix. For ODE, we have $\mathbf{M}(x,p)=Id$.

in large dimensions to a complex polynomial vector field:

\[\dot z = z\left(a \cdot (p-p_0) + i\omega + l_1|z|^2\right)\quad\text{(E)}\]

whose solutions give access to the solutions of the Cauchy problem in a neighborhood of $(x,p)$.

More precisely, if $\mathbf{J} \equiv d\mathbf{F}(x_0,p_0)$ and $M \equiv \mathbf{M}(x_0,p_0)$, then we have $\mathbf{J}\zeta = i\omega M\zeta$ and $\mathbf{J}^*\bar\zeta = -i\omega M^*\bar\zeta$ for some complex eigenvector $\zeta$. It can be shown that $x(t) \approx x_0 + 2\Re(z(t)\zeta)$ when $p=p_0+\delta p$.

Expression of the coefficients

The coefficients $a,l_1$ above are computed as follows^[Haragus]:

\[a=\left\langle\mathbf{F}_{11}(\zeta)+2 \mathbf{F}_{20}\left(\zeta, \Psi_{001}\right), \zeta^{*}\right\rangle,\]

\[l_1=\left\langle 2 \mathbf{F}_{20}\left(\zeta, \Psi_{110}\right)+2 \mathbf{F}_{20}\left(\bar{\zeta}, \Psi_{200}\right)+3 \mathbf{F}_{30}(\zeta, \zeta, \bar{\zeta}), \zeta^{*}\right\rangle.\]

where

\[\begin{aligned} -\mathbf{J} \Psi_{001} &=\mathbf{F}_{01} \\ (2 i \omega M-\mathbf{J}) \Psi_{200} &=\mathbf{F}_{20}(\zeta, \zeta) \\ -\mathbf{J} \Psi_{110} &=2 \mathbf{F}_{20}(\zeta, \bar{\zeta}). \end{aligned}\]

and where

\[\mathbf{F}(x,p)-\mathbf{J}x := \sum_{1\leq q+l\leq p}\mathbf{F}_{ql}(x^{(q)},p^{(l)})+o(\|u\|+\|p\|)^p.\]

with $\mathbf{F}_{ql}$ a $(q+l)$-linear map.

Normal form computation

The normal form (E) is automatically computed as follows

get_normal_form(br, ind_bif;	verbose = false, lens = getlens(br),	detailed = Val(true),            # full normal form (coefficients a, b)	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 above call returns an object of type Hopf.

Returned object

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

• x0, p, params, lens: the bifurcation point, its parameter value, the full parameter set and the parameter axis,
• ω: the imaginary frequency of the Hopf pair,
• ζ (resp. ζ★): the complex right (resp. left) eigenvector satisfying $\mathbf J \zeta = i\omega M\zeta$ and $\mathbf J^*\zeta^* = -i\omega M^*\zeta^*$, normalized by scaleζ and such that $\langle\zeta^*, \zeta\rangle = 1$,
• type: :SuperCritical, :SubCritical or :Singular, deduced from the sign of real(nf.b),
• nf: a HopfNormalForm holding
  • nf.a: the coefficient $a$ in (E), i.e. the linear dependence of the normal form on $p - p_0$,
  • nf.b: the Lyapunov coefficient $l_1$ of (E),
  • nf.Ψ001, nf.Ψ110, nf.Ψ200: the second order terms of the parametrization of the center manifold (see the equations defining $\Psi_{001}$, $\Psi_{200}$, $\Psi_{110}$ above).

Passing detailed = Val(false) skips the computation of the coefficients $a,b$ (they are set to missing) and returns only the data needed to start the continuation of the periodic orbit.

Predictor

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

predictor(
    hp::BifurcationKit.Hopf,
    ds;
    verbose,
    ampfactor
) -> NamedTuple{(:orbit, :Ψ001, :amp, :ω, :period, :p, :dsfactor), <:Tuple{BifurcationKit.var"#677#679"{BifurcationKit.Hopf{Tv, Tτ, T, Tω, Tpar, Tlens, Tevr, Tevl, Tnf}, _A, BifurcationKit.var"#A#678"{_A1}} where {Tv, Tτ, T, Tω, Tpar, Tlens<:Union{typeof(identity), IndexLens, PropertyLens, ComposedFunction}, Tevr, Tevl, Tnf, _A, _A1}, Any, Any, Any, Any, Any, Int64}}

References