# 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]

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

## 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