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.

Normal form based on Poincaré return map

Given a transversal section $\Sigma$ to $\gamma$ at $\gamma(0)$, the Poincaré return map $\mathcal P$ associates to each point $x\in\Sigma$ close to $\gamma(0)$ 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 $\mathcal C=d^3\mathcal P(\gamma(0))$, $\mathcal B = d^2\mathcal P(\gamma(0))$ and $\mathcal A = d\mathcal P(\gamma(0))$. 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 $\gamma(\tau)$ has a Period-Doubling bifurcation for a parameter value $p_0$. Locally, the orbits can be represented by

$$$x(\tau) = \gamma(\tau)+Q_0(\tau)\xi+\Phi(\tau, \xi)$$$

where

$$$\left\{\begin{array}{l} \frac{d \tau}{d t}=1+a_0\cdot(p-p_0)+a \xi^2+\cdots \\ \frac{d \xi}{d \tau}=c_0\cdot(p-p_0)\xi+c \xi^3+\cdots \end{array}\right.$$$

with center manifold correction $\Phi(\tau, \xi)$ being $2T$ periodic in $\tau$ and $Q_0(\tau)$ is the Floquet operator.

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