Editing Feigenbaum constants (section)

====Non-linear maps====
To see how this number arises, consider the [[real number|real]] one-parameter map
:<math>f(x) = a-x^2.</math>
Here {{mvar|a}} is the bifurcation parameter, {{mvar|x}} is the variable. The values of {{mvar|a}} for which the period doubles (e.g. the largest value for {{mvar|a}} with no {{nowrap|period-2}} orbit, or the largest {{mvar|a}} with no {{nowrap|period-4}} orbit), are {{math|''a''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}} etc. These are tabulated below:<ref>Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA503 p. 503].</ref>

:{| class="wikitable"
|-
! {{mvar|n}}
! Period
! Bifurcation parameter ({{mvar|a<sub>n</sub>}})
! Ratio {{math|{{sfrac|''a''{{sub|''n''−1}} − ''a''{{sub|''n''−2}}|''a''{{sub|''n''}} − ''a''{{sub|''n''−1}}}}}}
|-
| 1
|| 2
|| 0.75
|| —
|-
| 2
|| 4
|| 1.25
|| —
|-
| 3
|| 8
|| {{val|1.3680989}}
|| 4.2337
|-
| 4
|| 16
|| {{val|1.3940462}}
|| 4.5515
|-
| 5
|| 32
|| {{val|1.3996312}}
|| 4.6458
|-
| 6
|| 64
|| {{val|1.4008286}}
|| 4.6639
|-
| 7
|| 128
|| {{val|1.4010853}}
|| 4.6682
|-
| 8
|| 256
|| {{val|1.4011402}}
|| 4.6689
|-
|}

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the [[logistic map]]
:<math>f(x) = ax(1-x)</math>
with real parameter {{mvar|a}} and variable {{mvar|x}}. Tabulating the bifurcation values again:<ref>Alligood, [https://books.google.com/books?id=i633SeDqq-oC&pg=PA504 p. 504].</ref>

:{| class="wikitable"
|-
! {{mvar|n}}
! Period
! Bifurcation parameter ({{mvar|a<sub>n</sub>}})
! Ratio {{math|{{sfrac|''a''{{sub|''n''−1}} − ''a''{{sub|''n''−2}}|''a''{{sub|''n''}} − ''a''{{sub|''n''−1}}}}}}
|-
| 1
|| 2
|| 3
|| —
|-
| 2
|| 4
|| {{val|3.4494897}}
|| —
|-
| 3
|| 8
|| {{val|3.5440903}}
|| 4.7514
|-
| 4
|| 16
|| {{val|3.5644073}}
|| 4.6562
|-
| 5
|| 32
|| {{val|3.5687594}}
|| 4.6683
|-
| 6
|| 64
|| {{val|3.5696916}}
|| 4.6686
|-
| 7
|| 128
|| {{val|3.5698913}}
|| 4.6680
|-
| 8
|| 256
|| {{val|3.5699340}}
|| 4.6768
|-
|}