📂Dynamics

Infinite Period Bifurcation

Definition

Infinite-period bifurcation is a bifurcation where the limit cycle, including a saddle point and a stable node, appears or disappears as the parameter of a dynamical system changes. The period converging to this limit cycle must diverge to infinity with the changing parameter.

Description

An infinite-period bifurcation is a bifurcation in which the flow diverges to infinity for the period as it approaches the limit cycle. This is a global bifurcation because examining only the vicinity of a fixed point does not provide an understanding of the limit cycle¹.

Among the fixed points on the limit cycle, saddle points and stable points are mentioned since the following form must be enforced, simply put.

Imagining two fixed points on a limit cycle, other combinations would not geometrically constitute an infinite-period bifurcation.

(1) Unstable nodes cannot exist

If unstable nodes exist, they cannot form a limit cycle.

(2) Only stable nodes cannot exist

If there are two stable nodes, there must be an unstable node; this is impossible due to (1).

(3) The unstable manifold of a saddle point is a limit cycle

For the same reason as (1), if a saddle point exists, its unstable manifold must be a limit cycle for it to form a limit cycle.

(4) Only saddle points cannot exist

If only saddle points exist, a stable node must exist at some point on the limit cycle. Saddles that are not unstable manifolds of the limit cycle are already excluded by (3).

(5) At least one saddle point and one stable node must exist

To satisfy all conditions discussed so far, it is necessary to have a saddle point that draws in the inner or outer flow and converges to stable nodes within the limit cycle.

Example ²

$$ \begin{align*} \dot{r} =& r \left( 1 - r^{2} \right) \\ \dot{\theta} =& \mu - \sin \theta \end{align*} $$ Suppose a system like the one above is given in the polar coordinate system. This system has a limit cycle $r = 1$ and an unstable node $r = 0$, regardless of $\mu$, and it has two fixed points $\left\{ \sin^{-1} \mu \right\}$ on $r = 1$ when $0 < \mu < 1$. When $\mu > 1$, there are no fixed points because $\dot{\theta} \ne 0$, and exactly $\mu = 1$ is the bifurcation point.

$\mu < 1$

In the case of $\mu < 1$, the flow of this system is not difficult to grasp. No matter where it starts, it converges to one stable node.

$\mu > 1$

Regardless of what $\mu$ is, the speed at which it approaches the limit cycle $\dot{r}$ does not change, but the rotational speed $\dot{\theta}$ is minimized when $\theta = \pi / 2$. Imagining $\mu \to 1^{+}$ as $\mu > 1$ decreases to $1$, the rotational period diverges to infinity as $\mu$ approaches $1$ and the speed at $\theta = \pi / 2$ slows down.