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Quadratic Map

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A quadratic map in one variable is a quadratic recurrence equation of the form

x_(n+1)=a_2x_n^2+a_1x_n+a_0,
(1)

where a_2!=0 and n is a nonnegative integer. 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

x_n=x_(n-1)^2
(2)

with x_0=2, which has solution x_n=2^(2^n), the first few terms of which for n=0, 1, ... are 2, 4, 16, 256, 65536, 4294967296, ... (OEIS A001146).

Another example is the number of "strongly" rooted binary trees of height <=n, given by

y_n=y_(n-1)^2+1
(3)

Here each internal vertex has exactly two ordered children, and height is measured in edges. With y_0=1, the first few terms for n=0, 1, ... are 1, 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, ... (OEIS A003095). This recurrence equation has the "analytic" solution

y_n=|_c^(2^n)_|,
(4)

where

c=exp[sum_(j=0)^(infty)2^(-j-1)ln(1+y_j^(-2))]
(5)
=1.502836801....
(6)

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

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

S_n=sum_(i=1)^n1/(z_i),
(7)

where n is a positive integer and 1<z_1<...<z_n are integers. The solution is given by the recurrence

z_n=z_(n-1)^2-z_(n-1)+1,
(8)

with z_1=2. 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

z_n=|_d^(2^n)+1/2_|,
(9)

where

d=1/2sqrt(6)exp{sum_(j=1)^(infty)2^(-j-1)ln[1+(2z_j-1)^(-2)]}
(10)
=1.2640847353....
(11)

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

The well-known recurrence

x_(n+1)=x_n^2+c
(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

x^((1))=[x^((1))]^2+c
(13)
(x^((1)))^2-x^((1))+c=0
(14)
x_+/-^((1))=1/2(1+/-sqrt(1-4c)).
(15)

Points of period dividing 2 satisfy

x_(n+2)=x_(n+1)^2+c
(16)
=(x_n^2+c)^2+c
(17)
=x_n^4+2cx_n^2+(c^2+c)
(18)
=x_n.
(19)

Dropping the subscripts and factoring gives

x^4+2cx^2-x+(c^2+c)=(x^2-x+c)(x^2+x+1+c)=0.
(20)

For c<-3/4, the real periodic points of least period 2 are

x_+/-^((2))=1/2(-1+/-sqrt(-3-4c)).
(21)

There are no real periodic points of least period 2 for c>=-3/4. The periodic points of least period 3 satisfy

x^6+x^5+(3c+1)x^4+(2c+1)x^3+(3c^2+3c+1)x^2 +(c+1)^2x+(c^3+2c^2+c+1)=0.
(22)

The roots of this equation that are also fixed points must be excluded.

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

a_0=((a_1-4)(a_1+2))/(4a_2).
(23)

This has solution

x_n=(r^(2^n)+r^(-2^n)-1/2a_1)/(a_2),
(24)

where

r=1/4a_1+1/2x_0a_2+1/4sqrt((a_1+4+2x_0a_2)(a_1-4+2x_0a_2)).
(25)

Similarly, the case with

a_0=(a_1(a_1-2))/(4a_2)
(26)

has solution

x_n=((2r)^(2^n)-1/2a_1)/(a_2),
(27)

where

r=1/4a_1+1/2x_0a_2
(28)

(Little).

After a linear transformation of coordinates, a real two-dimensional quadratic map with an elliptic fixed point at the origin has the form

x^'=xcosalpha-ysinalpha+a_(20)x^2+a_(11)xy+a_(02)y^2
(29)
y^'=xsinalpha+ycosalpha+b_(20)x^2+b_(11)xy+b_(02)y^2.
(30)

Here alpha not in piZ. The Jacobian must be identically 1 for the map to be area-preserving, reducing the number of independent parameters from seven to three. If the quadratic part is nonzero, scaling and rotating give the standard Hénon map form (Hénon 1969)

x^'=xcosalpha-ysinalpha+x^2sinalpha
(31)
y^'=xsinalpha+ycosalpha-x^2cosalpha.
(32)

The inverse map is

x=x^'cosalpha+y^'sinalpha
(33)
y=-x^'sinalpha+y^'cosalpha+(x^'cosalpha+y^'sinalpha)^2.
(34)

The periodic points with period dividing a positive integer n are characterized by

x_i^2sinalpha+2x_icosalpha-x_(i-1)-x_(i+1)=0
(35)

for i=0, ..., n-1, with indices taken modulo n.

See also

Bogdanov Map, Hénon Map, Julia Set, Linear Recurrence Equation, Logistic Map, Lozi Map, Mandelbrot Set, Quadratic, Quadratic Equation, Quadratic Formula, Quadratic Recurrence Equation, Recurrence Equation, Strange Attractor, Sylvester's Sequence

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References

Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart. 11, 429-437, 1973.Finch, S. R. "Quadratic Recurrence Constants." §6.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 443-448, 2003.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Research problem 4.65 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Hénon, M. "Numerical Study of Quadratic Area-Preserving Mappings." Quart. Appl. Math. 27, 291-312, 1969. https://doi.org/10.1090/qam/253513.Little, M. "Novel Exact Solutions to Special Cases of the Quadratic Map." http://homepage.ntlworld.com/little_mm/quadrams.html.Sloane, N. J. A. Sequences A000058/M0865, A001146/M1297, A003095/M1544, A076393, and A077496 in "The On-Line Encyclopedia of Integer Sequences."Sprott, J. C. "Automatic Generation of Strange Attractors." Comput. & Graphics 17, 325-332, 1993. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 53-60, 1998.Vardi, I. "Are All Euclid Numbers Squarefree?" and "PowerMod to the Rescue." §5.1 and 5.2 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89, 1991.

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Quadratic Map

Cite this as:

Weisstein, Eric W. "Quadratic Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuadraticMap.html

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