Period-doubling point

• Period-doubling point
• • Which method to use?
  • Predictor
  • Normal form based on Poincaré return map
  • Normal form based on Iooss method
  • See also
  • References

At a period-doubling (PD) bifurcation of a periodic orbit $\gamma$ (with period $T$) for parameter value $p_0$ for the Cauchy problem

\[\frac{du}{dt}=F(u,p),\tag{E}\]

the eigenvalues (Floquet coefficients) of the monodromy operator $\mathcal M=Y(T)$ solution to

\[\frac{dY}{dt}=A(t)Y(t), Y(0)=I_n\]

contain the simple eigenvalue $\mu=-1$.

There are two ways to compute the normal form of this bifurcation:

1. using the Poincaré return map ^[Kuznetsov],
2. using the method of ^[Iooss], see also ^[Kuz2].

You can obtain the normal form of a PD bifurcation using

pd = get_normal_form(br, ind; prm = Val(false))

where prm indicates whether you want to use the method based on Poincaré return map (Val(true)) or the one based on Iooss method (Val(false)). The call returns a point of type PeriodDoublingPO from which you can access

• pd.nf.p the parameter value at the bifurcation point,
• pd.T the period of the periodic orbit (the period of the bifurcated orbit is $2T$),
• pd.ζ, pd.ζ★ the right / left eigenvectors,
• pd.nf.nf the coefficients of the normal form.

You can pass verbose = true to get more information during the computation and, for the periodic orbits based on orthogonal collocation, detailed = Val(false) to skip the (costly) computation of the normal form.

Which method to use?

Depending on the method used for computing the periodic orbits, you have several possibilities:

• For shooting, you can only use the PRM method. Shooting is the preferred way for large scale systems. Note that the PRM method is not very precise numerically.
• For collocation, you can use PRM and Iooss methods. The Iooss method (prm = Val(false), the default) is the most precise.
• For Trapezoid method, PD normal form is not yet implemented.

Predictor

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

predictor(
    nf::BifurcationKit.PeriodDoublingPO{<:Shooting},
    δp,
    ampfactor;
    override
) -> NamedTuple{(:orbitguess, :pnew, :prob, :ampfactor, :δp, :po), <:NTuple{6, Any}}

If override = true, then the predictor is simply x0 .+ ampfactor .* e for the parameter p0 + δp and where e is a bifurcating eigenvector.

Normal form based on Poincaré return map

Given a transversal section $\Sigma$ to $x_0$ at $x_0(0)$, the Poincaré return map $\mathcal P$ associates to each point $x\in\Sigma$ close to $x_0(0)$ the first point $\mathcal P(x,p)\in\Sigma$ where the orbit of (E) with initial condition $x$ intersects again $\Sigma$. Hence, the discrete map $x_{n+1}=\mathcal P(x_n,p)$ has normal form

\[x_{n+1} = (a_{11}\cdot(p-p_0)-1)\cdot x_n + cx_n^3 + ...\]

where ^[Kuz2]

\[c =\frac{1}{6}\left\langle \zeta^*, \mathcal{C}(\zeta, \zeta, \zeta)+3 \mathcal{B}\left(\zeta,\left(I_{n-1}-\mathcal{A}\right)^{-1} \mathcal{B}(\zeta, \zeta)\right)\right\rangle\]

where $\mathcal C=d_1^3\mathcal P(x_0(0), p_0)$, $\mathcal B = d_1^2\mathcal P(x_0(0), p_0)$ and $\mathcal A = d_1\mathcal P(x_0(0), p_0)$. Also:

\[\mathcal{A} \zeta=-\zeta, \mathcal{A}^{\mathrm{T}} \zeta^*=-\zeta^*\]

The coefficients a and c are returned in pd.nf.nf. The bifurcation is supercritical if $c > 0$ and subcritical if $c < 0$.

Normal form based on Iooss method

This is based on ^[Iooss],^[Kuz2]. Suppose that the $T$ periodic orbit $x_0(\tau)$ has a Period-Doubling bifurcation for a parameter value $p_0$. Locally, the orbits can be represented by $p-p_0:=\mu$ and

\[x(\tau) = x_0(\tau)+\xi v(\tau)+H(\tau, \xi, \mu)\]

where

\[\left\{\begin{array}{l} \frac{d \tau}{d t}=1+a_{01}\cdot(p-p_0)+a_2 \xi^2+\cdots \\ \frac{d \xi}{d \tau}=c_{11}\cdot(p-p_0)\xi+c_3 \xi^3+\cdots \end{array}\right.\]

with center manifold correction $H(\tau, \xi, \mu)$ being $2T$ periodic in $\tau$ and $v(\tau)$ is a Floquet eigenvector for the eigenvalue -1.

The coefficients a₀₁, a₂, c₁₁ and c₃ are returned in pd.nf.nf. The bifurcation is supercritical if $c_3 < 0$ and subcritical if $c_3 > 0$.

See also

• Detection of bifurcation points of periodic orbits
• Continuation of Period-doubling (PD) bifurcations of periodic orbits
• Period doubling in Lur'e problem (PD aBS): computation of the PD normal form (with both methods) and branch switching for all the periodic orbit methods

References