“logistic map”的概念、定义、翻译、参考文献-科学参考

logistic map

Mathematics

A discrete version of the logistic map is given by the recurrence relation

$\begin{array}{l} x_{n + 1} = r x_{n} (1 - x_{n}) \\ where 0 < r < 4 and 0 < x_{0} < 1 . \end{array}$

(Rewrite the continuous logistic equation in terms of N/K to see how this discrete version is arrived at.) By restricting r as above, the sequence remains positive.

This simple iteration leads to surprisingly complex behaviour. If r < 1, then the sequence x_n tends to 0, signifying extinction. If 1 < r < 2, then the sequence monotonically converges to 1−1/r. If 2 < r < 3, then the sequence converges to 1−1/r in an oscillatory fashion. But at r = 3 this equilibrium becomes unstable and bifurcates; for $3 < r < 1 + \sqrt{6}$ the stable behaviour is an oscillation between two values of r. Then at $r = 1 + \sqrt{6}$ we get a further bifurcation, and the stable behaviour is oscillations between four values of r. Further period-doubling bifurcations occur until the behaviour becomes chaotic at around r = 3.57, but then oscillations still occur at some greater values of r.

By cobwebbing with the graphs of y = rx(1−x) and y = x for different values of r, a qualitive appreciation of how these behaviours arise is possible.

单词	logistic map
释义	logistic map Mathematics A discrete version of the logistic map is given by the recurrence relation $\begin{array}{l} x_{n + 1} = r x_{n} (1 - x_{n}) \\ where 0 < r < 4 and 0 < x_{0} < 1 . \end{array}$ (Rewrite the continuous logistic equation in terms of N/K to see how this discrete version is arrived at.) By restricting r as above, the sequence remains positive. This simple iteration leads to surprisingly complex behaviour. If r < 1, then the sequence x_n tends to 0, signifying extinction. If 1 < r < 2, then the sequence monotonically converges to 1−1/r. If 2 < r < 3, then the sequence converges to 1−1/r in an oscillatory fashion. But at r = 3 this equilibrium becomes unstable and bifurcates; for $3 < r < 1 + \sqrt{6}$ the stable behaviour is an oscillation between two values of r. Then at $r = 1 + \sqrt{6}$ we get a further bifurcation, and the stable behaviour is oscillations between four values of r. Further period-doubling bifurcations occur until the behaviour becomes chaotic at around r = 3.57, but then oscillations still occur at some greater values of r. By cobwebbing with the graphs of y = rx(1−x) and y = x for different values of r, a qualitive appreciation of how these behaviours arise is possible.
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