Boundary Crisis

In the 1D quadratic map, the single-band chaotic attractor (CA) disappears when A passes through 2.

1D quadratic map : 2

1 1)( nnn Axxfx

Eui-Sun Lee

Department of Physics

Kangwon National University

Bifurcation diagram

The initial points inside the basin are attracted to a given attractor,while the initial points outside of the basin would be expelled , and never return to the attractor.

Boundary crisis

A

Axu 2

411*

• Basin-boundary

• Basin : Region between

and .: unstable fixed point

• Boundary crisis occurs through the collision between the CA and the boundary of its basin .

The unstable fixed point exists on the boundary of the CA’s basin boundary.

*ux

*ux

*ux *

ux*ux

The Chaotic Transient

After the boundary crisis , a trajectory starting from the initial point in the interval (1-A,1)

exhibits the chaotic behavior before it diverges away.→ Chaotic Transient

When the parameter increases through 2, the boundary crisis occurs, and then the CA

transforms into the chaotic transient .

Lifetime of The Chaotic Transient

.2

1

)2(

A

• As the parameter increases, the lifetime of the chaotic transient becomes shorter.

• Average lifetime of the trajectories, starting from 1,000 randomly chosen initial point with uniform probability in the interval(1-A,1) for a given parameter, may be regarded as iteration time which when a trajectories (|x|) becomes larger than 10.0 .