Neimark-Sacker point
At a Neimark-Sacker (NS) 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 eigenvalues $e^{\pm i \theta}$ with $\theta$ and
\[e^{i q \theta}-1 \neq 0, \quad q=1,2,3,4 \text { (no strong resonances). }\]
There are two ways to compute the normal form of this bifurcation
- using the Poincaré return map [Kuznetsov]
- using the method of [Iooss], see also [Kuz2]
You can obtain the normal form of a NS 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, NS normal form is not yet implemented.
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
\[z_{n+1} = z_ne^{i\theta}(1+d|z_n|^2)\]
where[Kuz2]
\[d=\frac{1}{2} e^{-i \theta}\left\langle v^*, \mathcal{C}(v, v, \bar{v})+2 \mathcal{B}\left(v,\left(I_{n-1}-\mathcal{A}\right)^{-1} \mathcal{B}(v, \bar{v})\right)+\mathcal{B}\left(\bar{v},\left(e^{2 i \theta} I_{n-1}-\mathcal{A}\right)^{-1} \mathcal{B}(v, v)\right)\right\rangle\]
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:
\[\mathcal{A} v=e^{i \theta} v, \mathcal{A}^{\mathrm{T}} v^*=e^{-i \theta} v^*, \text { and }\left\langle v^*, v\right\rangle=1\]
Normal form based on Iooss method
This is based on [Iooss],[Kuz2]. Suppose that the $T$ periodic orbit $\gamma(\tau)$ has a Neimark-Sacker bifurcation for a parameter value $p_0$. We also assume that there are no strong resonances. Locally, the orbits can be represented by
\[x(\tau) = \gamma(\tau)+Q_0(\tau)\xi+\Phi(\tau, \xi)\]
where
\[\left\{\begin{aligned} \frac{d \tau}{d t} & =1+a|\xi|^2+\cdots \\ \frac{d \xi}{d t} & =\frac{i \theta}{T} \xi+d \xi|\xi|^2+\cdots \end{aligned}\right.\]
with center manifold correction $\Phi(\tau, \xi)$ being $T$ periodic in $\tau$ and $Q_0(\tau)$ is the Floquet operator.