The transition from simple to fractal basin boundary structure displayed by unimodal maps |
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| Institution: | 1. Laboratory of Engineering, Systems and Applications, National School of Applied Sciences, Sidi Mohamed Ben Abdellah-Fez University, Fez, Morocco;2. Laboratory of Electronic Signals and Systems of Information, Faculty of Science Dhar El Mahrez, Sidi Mohamed Ben Abdellah-Fez University, Fez, Morocco;1. Department of Computer Sciences, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates;2. School of Computer Sciences, Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia;3. Department of Electrical Engineering and Computer Science, Khalifa University, Abu Dhabi, United Arab Emirates;1. Department of Polymer Science and Chemical Engineering, Pusan National University, Busan, 609-735, Republic of Korea;2. Department of Physics, Pukyong National University, Busan, 608-737, Republic of Korea;3. Pharmcadd, IntelliumCenter, Centum 6-ro, Haeundae-gu, Busan, 48059, Korea |
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| Abstract: | One-dimensional unimodal maps of the form Xn + 1 = F (r, xn) exhibit periodic orbits, or p-cycles, confined within specific values of the control parameter r. For any p-cycle almost all initial conditions xO will iterate to one of attracting fixed points of the map under iteration of the pth iterate of the map F(p). The basin boundaries which confine points attracted to the fixed points are simple or fractal depending upon whether the cycle is a member of the period doubling cascade or is a cycle in the aperiodic region beyond the period doubling accumulation point. The topology of these basin structures and the transition from simple to fractal behaviour is analyzed as a function of control parameter in terms of a scenario which parallels that of the progression from period doubling to the aperiodic regime. |
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