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This study explores the interior crisis and band-merging crisis in one-dimensional quadratic maps. As parameter A increases (A ≈ 1.543689), two-band chaotic attractors (CA) merge into a single-band CA. At this band-merging crisis, the boundaries of the two-band CA collapse simultaneously and yield an unstable period-1 orbit. Additionally, the paper delves into the behavior of four-band CAs, showcasing how they collapse to form two-band CAs through unstable orbits. The work further examines the attractor-widening crisis and the significant shifts in the Lyapunov exponent.
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Eui-Sun Lee Department of Physics Kangwon National University Interior Crisis • Band-merging crisis 1D quadratic map : • As a parameter increases, the 2-bands CA merge into the single band CA through the • band-merging crisis for A =1.543689… . • At the band-merging crisis, the boundaries of the 2-bands CA collapse with unstable period-1 • orbit simultaneously. Through the band-merging crisis, the unstable period-1orbit is embedded • in the single-band CA. In the 1D quadratic map, the transition to chaos occurs at the critical point (A∞), and then the chaotic attractor (CA) appears. Bifurcation diagram 2-bands CA single-band CA
Band-merging Crisis • A band-merging crisis of the 4-band CA . 4-band CA 2-band CA The boundaries of the 4-bands CA collapse with unstable period-2 orbit simultaneously. Through the band-merging crisis, the unstable period-2 orbit is embedded in the 2-band CA. • The 2n -bands CA collapse with the unstable period-2n-1 orbit , and then 2n-bands CA merge into the 2n-1–bands CA ,in which the unstable period-2n orbit is embedded. • → Band-merging crisis.
Attractor-widening Crisis • When the period-3window are close, the 3-bands CA suddenly widen by collapse with theunstable • period-3 orbit born via a saddle-node bifurcation for A≈1.7903… . • When the Attractor-widening Crisis occurs, the Lyapunov exponent of the CA exhibits the rough jump. Period-3 window Lyapunov exponent
