Neimark-Sacker Bifurcation 📂Dynamics

Neimark-Sacker Bifurcation

Definition

Simple Definition

Neimark-Sacker Bifurcation is a bifurcation in which an invariant closed curve arises or disappears at a fixed point as a parameter changes in a dynamical system¹.

Complex Definition

Let’s denote $n \ge 2$. $$ \dot{x} \mapsto f \left( x , \alpha \right) \qquad , x \in \mathbb{R}^{n} , \alpha \in \mathbb{R}^{1} $$ Given that the dynamical system $f$ is smooth under $x$ and $\alpha$. Suppose $\bar{x}$ is the hyperbolic fixed point of the system, and let $D f \left( \bar{x} \right)$ be the Jacobian matrix of this system. Among the eigenvalues of $D f \left( \bar{x} \right)$, let us denote two of them as $\lambda_{k_{1}}$ and $\lambda_{k_{2}}$. The bifurcation related to the existence of $\lambda_{k_{1}, k_{2}} = \exp \left( \pm i \theta \right)$ for $0 < \theta < \pi$ is called Neimark-Sacker Bifurcation².

Normal Form

$$ \begin{align*} \theta =& \theta (\alpha) \\ a =& a (\alpha) \\ b =& b (\alpha) \\ d =& d (\alpha) = a + i b \\ c =& c (\alpha) = e^{i \theta} d (\alpha) \\ \mu =& \mu (\alpha) = \left( 1 + \alpha \right) e^{i \theta} \\ R_{\theta} =& \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \end{align*} $$ From now on, we assume that the parameters mentioned below are functions dependent on the key parameter $\alpha$ and that $R_{\theta}$ represents the rotation matrix. Suppose the complex number $z$ is expressed as $z = x + iy$, or in polar coordinates as $z = r e^{i \phi}$. The Neimark-Sacker bifurcation can be classified into two types: supercritical and subcritical, denoted as supercritical when $a(0) < 0$ and subcritical when $a(0) > 0$. Both types share the following normal form:

In Cartesian coordinates: $$ \begin{bmatrix} x \\ y \end{bmatrix} \mapsto (1 + \alpha) R_{\theta} \begin{bmatrix} x \\ y \end{bmatrix} + \left( x^{2} + y^{2} \right) R_{\theta} \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$ In the complex plane: $$ z \mapsto e^{i \theta} z \left( 1 + \alpha + d \left| z \right|^{2} \right) = \mu z + c z \left| z \right|^{2} $$

Diagram

supercritical: Let $\alpha$ increase gradually. In $\alpha \le 0$, $z = 0$ was a stable node, but in $\alpha > 0$, $z = 0$ changes to an unstable node, and a stable invariant closed curve arises.
subcritical: Let $\alpha$ decrease gradually. In $\alpha \ge 0$, $z = 0$ was an unstable node, but in $\alpha < 0$, $z = 0$ changes to a stable node, and an unstable invariant closed curve arises.

Explanation

Neimark-Sacker bifurcation, also sometimes referred to as torus bifurcation, is not a phenomenon that is easily encountered, but it can be seen as an analog to the Hopf bifurcation in discrete systems. Thus, it is good to be familiar with its name.