If (where denotes the closed interval from to on the real line) satisfies a Lipschitz condition with constant , i.e., iffor all , then the iteration schemewhere , converges to a fixed point of .
A fixed point for which the stability matrix has both eigenvalues of the same sign (i.e., both are positive or both are negative). If , then the node is called stable; if , then the node is called an unstable node.
If is a continuous function for all , then has a fixed point in . This can be proven by supposing that(1)(2)Since is continuous, the intermediate value theorem guarantees that there exists a such that(3)so there must exist a such that(4)so there must exist a fixed point .
A point which is mapped to itself under a map , so that . Such points are sometimes also called invariant points or fixed elements (Woods 1961). Stable fixed points are called elliptical. Unstable fixed points, corresponding to an intersection of a stable and unstable invariant manifold, are called hyperbolic (or saddle). Points may also be called asymptotically stable (a.k.a. superstable).
An -cycle is a finite sequence of points , ..., such that, under a map ,(1)(2)(3)(4)In other words, it is a periodic trajectory which comes back to the same point after iterations of the cycle. Every point of the cycle satisfies and is therefore a fixed point of the mapping . A fixed point of is simply a cycle of period 1.
A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function is a point such that(1)The fixed point of a function starting from an initial value can be computed in the Wolfram Language using FixedPoint[f, x]. Similarly, to get a list of the values obtained by iterating the function until a fixed point is reached, the command FixedPointList[f, x] can be used.The following table lists the smallest positive fixed points for several simple functions.functionfixed pointOEIScosecant1.1141571408A133866cosine0.7390851332A003957cotangent0.8603335890A069855hyperbolic cosecant0.9320200293A133867hyperbolic cosine----hyperbolic cotangent1.1996786402A085984hyperbolic secant0.7650099545A069814hyperbolic sine0--hyperbolic tangent0--inverse cosecant1.1141571408A133866inverse cosine0.7390851332A003957inverse cotangent0.8603335890A069855inverse..
An elliptic fixed point of a differential equation is a fixed point for which the stability matrix has purely imaginary eigenvalues (for ).An elliptic fixed point of a map is a fixed point of a linear transformation (map) for which the rescaled variables satisfy
The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian..
A hyperbolic fixed point of a differential equation is a fixed point for which the stability matrix has eigenvalues , also called a saddle point.A hyperbolic fixed point of a map is a fixed pointfor which the rescaled variables satisfy
Let be a closed convex subset of a Banach space and assume there exists a continuous map sending to a countably compact subset of . Then has fixed points.