logistic map

(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 xn 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 the stable behaviour is an oscillation between two values of r. Then at 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.

• lcm
• leading coefficient
• leading diagonal
• least action, principle of
• least common denominator
• least common multiple
• least squares
• least squares theorem
• least upper bound
• least value
• leaving variable
• Lebesgue, Henri
• Lebesgue integration
• left and right derivative
• left inverse
• left‐handed system
• Legendre, Adrien‐Marie
• Legendre polynomials
• Legendre's conjecture
• Legendre's differential equation
• Legendre symbol
• Leibniz, Gottfried Wilhelm
• Leibniz's integral rule
• Leibniz's Theorem
• lemma