Ultra fractal buddhabrot7/7/2023 For instance, beginning at 1 + √8 [ (approximately 3.82843) there is a range of parameters r that show oscillation among three values, and for slightly higher values of r oscillation among 6 values, then 12 etc. Most values of r beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of r that show non-chaotic behavior these are sometimes called islands of stability.Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos. From almost all initial conditions, we no longer see oscillations of finite period. At r ≈ 3.56995 (sequence A098587 in the OEIS) is the onset of chaos, at the end of the period-doubling cascade.This behavior is an example of a period-doubling cascade. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly the ratio between the lengths of two successive bifurcation intervals approaches the Feigenbaum constant δ ≈ 4.66920. With r increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc.The latter number is a root of a 12th degree polynomial (sequence A086181 in the OEIS).
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