Boundary Crisis

1. Boundary Crisis Eui-Sun Lee Department of Physics Kangwon National University 1D quadratic map : Bifurcation diagram In the 1D quadratic map, the single-band chaotic attractor (CA) disappears when A passes through 2.

2. Boundary crisis • Basin-boundary 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. The unstable fixed point exists on the boundary of the CA’s basin boundary. • Basin : Region between and . : unstable fixed point • Boundary crisis occurs through the collision between the CA and the boundary of its basin .

3. The Chaotic Transient When the parameter increases through 2, the boundary crisis occurs, and then the CA transforms into 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

4. Lifetime of The Chaotic Transient • 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 .