費根鮑姆(Feigenbaum)常數是倍週期分叉(Period-doubling bifurcation)中，相鄰分叉點間隔的極限比率(Feigenbaum constant is the ratio of convergence) 用δ表示：
This constant—named after the mathematician Mitchell Feigenbaum—refers to a certain property of chaotic systems. A universal constant for functions approaching Chaos via period doubling.
Chaotic systems: A Dynamical System is chaotic if it
- 1. Has a Dense collection of points with periodic orbits,
- 2. Is sensitive to the initial condition of the system (so that initially nearby points can evolve quickly into very different states), and
- 3. Is Topologically Transitive.
One of the features of many chaotic systems, Jeff Goldblum goes on about in Jurassic Park(youtube), is that these systems exhibit certain types of behaviors as they transition to chaos. Specifically, if you take a chaotic system and tune its parameters, on the way to chaos you get period doubling—a doubling of the number of points in the system's period, or the places that the system oscillates between—before the chaos (you can see this in the above picture with the logistic map as *r *is increased).
And the ratio between the values where the period doubles ends up approaching the Feigenbaum constant, approximately 4.669.【ref】Happy Feigenbaum Constant Day!
An example is the bifurcation diagram of the logistic map: 【Wiki】
The bifurcation parameter r is shown on the horizontal axis of the plot, and the vertical axis shows the set of values of the logistic function visited asymptotically(漸進趨近) from almost all initial conditions. 橫軸是 parameter r ，縱軸是asymptotic values(收斂值趨近值) of the logistic function。
【舉例】r 是種植高麗菜的面積增加率，x 是種植高麗菜的收益。(把橫軸上一一些，有沒有點像？)
The bifurcation diagram shows the forking of the periods of stable orbits(穩定軌道，收斂值曲線) from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.
The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth.(HOW?)
A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
1975年，費根鮑姆計算得出週期倍增分岔（period-doubling bifurcations）發生時，參數 r 之間的差率是一個常數。他為此提供了數學證明。也進一步揭示了同樣的現象、同樣的常數適用於廣泛的數學函數領域。這個普適的結論使數學家們能夠在對表像不可捉摸的混沌系統的解密道路上邁出了第一步。這個（ratio of convergence）通稱為費根鮑姆常數。
Reference:
1978年費根鮑姆發表了關於映射的研究的重要論文Quantitative Universality for a Class of Nonlinear Transformations 《非線性轉換類型的定量普適性》，其中談到了對混沌理論具特別意義的Logistic函數。[from Wiki]
The bifurcation parameter r is shown on the horizontal axis of the plot, and the vertical axis shows the set of values of the logistic function visited asymptotically(漸進趨近) from almost all initial conditions. 橫軸是 parameter r ，縱軸是asymptotic values(收斂值趨近值) of the logistic function。
【舉例】r 是種植高麗菜的面積增加率，x 是種植高麗菜的收益。(把bifurcation diagram 橫軸上移一些，有沒有點像？)
【啟示】在一定範圍增加生產(r，努力、嘗試或風險)可以提高收斂的穩定值；超過一定範圍的r(嘗試、涉險)，將導致多狀態的收斂。
【結論】持續超投入，達混沌態。
【Apocalypse】不設限地過日子，方知人生極限；或者走入滅亡！(essentially the same, no difference)。
【註】Apocalypse: Total destruction and end of the world.
