Quadratic Map

A quadratic map is a quadratic recurrence equation of the form

(1)

While some quadratic maps are solvable in closed form (for example, the three solvable cases of the logistic map), most are not.

A simple example of a quadratic map with a closed-form solution is

(2)

with , which has solution , the first few terms of which for , 1, ... are 2, 4, 16, 256, 65536, 4294967296, ... (OEIS A001146).

Another example is the number of "strongly" binary trees of height , given by

(3)

with . The first few terms are 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, ... (OEIS A003095) This recurrence has the "analytic" solution

(4)

where

			(5)
			(6)

(OEIS A077496) and is the floor function (Aho and Sloane 1973).

A third example is the closest strict underapproximation of the number 1,

(7)

where are integers. The solution is given by the recurrence

(8)

with . The resulting sequence is known as Sylvester's sequence and has first few terms 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (OEIS A000058). This has a closed solution as

(9)

where

			(10)
			(11)

(OEIS A076393; Aho and Sloane 1973, Vardi 1991, Graham et al. 1994).

The well-known recurrence

(12)

that is often called "the" quadratic map is not in general solvable in closed form. This is the real version of the complex map defining the Mandelbrot set. Fixed points of this map occur when

(13)

(14)

(15)

Period two fixed points occur when

			(16)
			(17)
			(18)
			(19)

Dropping the subscripts and factoring gives

(20)

Solutions therefore occur when

(21)

Period three fixed points occur when

(22)

Another example of a quadratic map with a closed-form solution is the case with

(23)

This has solution

(24)

where

(25)

Similarly, the case with

(26)

has solution

(27)

where

(28)

(Little).

The most general second-order two-dimensional map with an elliptic fixed point at the origin has the form

			(29)
			(30)

The map must have a determinant of 1 in order to be area-preserving, reducing the number of independent parameters from seven to three. The map can then be put in a standard form by scaling and rotating to obtain

			(31)
			(32)

The inverse map is

			(33)
			(34)

The fixed points are given by

(35)

for , ..., .

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