Period-doubling point

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 = false)

where prm indicates whether you want to use the method based on Poincaré return map (PRM) or the one based on Iooss method.

Which method to use?

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

  • For shooting, you can only 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. Note that the Iooss method 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

BifurcationKit.predictorMethod
predictor(nf, δp, ampfactor; override)

Compute the predictor for the period bifurcation of periodic orbit.

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 $\Sigma$ at $\mathcal P(x,p)$. Hence, the discrete map $x_{n+1}=\mathcal P(x_n,p)$ has normal form

\[x_{n+1} = -x_n+cx_n^3+...\]

where [Kuz2]

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

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

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

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.

References

  • Kuznetsov

    Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory", 2nd ed., 1998.

  • Kuz2

    Kuznetsov et al., “Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles.”

  • Iooss

    Iooss, "Global Characterization of the Normal Form for a Vector Field near a Closed Orbit.", 1988