Editing Logistic map (section)

=== When there is a time delay ===
[[File:遅延ロジスティック写像.png|class=skin-invert-image|thumb|The trajectory of the delayed logistic map. The initial values <math>(x_0 , y_0)</math> are the same in both figures, but at the bifurcation point r = 2, the trajectory is attracted to a closed curve (left) and a point (right).]]
If we interpret the logistic map as a model of the population of each generation of organisms, it is possible that the population of the next generation will affect not only the population of the current generation, but also the population of the generation before that. <!--[ 341 ]--> An example of such a case is

{{NumBlk|:|<math>{\displaystyle x_{n+1}=ax_{n}(1-x_{n-1})}</math>|{{EquationRef|6-4}}}}

where the number of individuals in the previous generation, <math>x_{n-1}</math>, is included in the equation as a negative density effect <!--[ 341 ]-->. If <math>x_{n+1} = y_n</math>, then equation ( 6-4 ) can be replaced by the following two-variable difference equation <!--[ 342 ]-->.

{{NumBlk|:|<math>{\displaystyle {\begin{cases}x_{n+1}=y_{n}\\y_{n+1}=ay_{n}(1-x_{n})\end{cases}}}</math>|{{EquationRef|6-5}}}}

This dynamical system is used to study bifurcation of quasi-periodic attractors and is called the delayed logistic map <!--[ 342 ]--> <!--[ 343 ]-->. The delayed logistic map exhibits a Neimark–Sakher bifurcation at r = 2, where the asymptotically stable fixed point becomes unstable and an asymptotically stable invariant curve forms around the unstable fixed point <!--[ 344 ]-->.