Periodicity and convergence for xn + 1 = | xn − xn − 1 |
نویسنده
چکیده
Each solution {xn} of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of {xn} is rational or irrational. Further, the sequence of ratios {xn/xn−1} satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p = 3 are explicitly determined in terms of the Fibonacci numbers. In spite of the non-existence of period 3, the unique positive fixed point of the first-order equation is shown to be a snap-back repeller so the irrational ratios behave chaotically. 2003 Elsevier Inc. All rights reserved.
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