Neimark-Sacker point

• Neimark-Sacker point
• • Which method to use?
  • Normal form based on Poincaré return map
  • Normal form based on Iooss method
  • See also
  • References

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 a pair of complex conjugate eigenvalues $e^{\pm i \theta}$ on the unit circle with $\theta\in(0, \pi)$, which satisfy the non-resonance conditions

\[e^{i q \theta}-1 \neq 0, \quad q=1,2,3,4 \text { (no strong resonances). }\]

At such a bifurcation point, an invariant torus of the system (E) branches off from the periodic orbit.

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 NS bifurcation using

ns = 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 NeimarkSackerPO from which you can access

• ns.p the parameter value at the bifurcation point,
• ns.T the period of the periodic orbit,
• ns.ω the frequency $\theta$,
• ns.ζ, ns.ζ★ the right / left eigenvectors,
• ns.nf.nf the coefficients of the normal form.

You can pass verbose = true to get more information during the computation. Note that, for the periodic orbits based on orthogonal collocation, you can skip the (costly) computation of the normal form and only get the bifurcation point characteristics by using detailed = Val(false).

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, 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 $\Sigma$. 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]

\[d=\frac{1}{2} e^{-i \theta}\left\langle \zeta^*, \mathcal{C}(\zeta, \zeta, \bar{\zeta})+2 \mathcal{B}\left(\zeta,\left(I_{n-1}-\mathcal{A}\right)^{-1} \mathcal{B}(\zeta, \bar{\zeta})\right)+\mathcal{B}\left(\bar{\zeta},\left(e^{2 i \theta} I_{n-1}-\mathcal{A}\right)^{-1} \mathcal{B}(\zeta, \zeta)\right)\right\rangle\]

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

\[\mathcal{A} \zeta=e^{i \theta} \zeta, \mathcal{A}^{\mathrm{T}} \zeta^*=e^{-i \theta} \zeta^*, \text { and }\left\langle \zeta^*, \zeta\right\rangle=1\]

The coefficients a and b are returned in ns.nf.nf. The bifurcation is supercritical if $\Re(b) < 0$ and subcritical if $\Re(b) > 0$.

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 $Q_0(\tau)$ built from the Floquet eigenvectors.

The coefficients a and d are returned in ns.nf.nf. The bifurcation is supercritical if $\Re(d) < 0$ and subcritical if $\Re(d) > 0$.

See also

• Detection of bifurcation points of periodic orbits
• Continuation of Neimark-Sacker (NS) bifurcations of periodic orbits
• Steinmetz-Larter model: continuation of a curve of NS points of periodic orbits and computation of the NS normal form
• Lorenz-84 model, take 2: computation of a curve of NS points of periodic orbits from a Hopf-Hopf bifurcation

References