Solutions to nonlinear recurrence equations. (English) Zbl 1514.65198

Withers, Christopher S.; Nadarajah, Saralees

Rocky Mt. J. Math. 52, No. 6, 2153-2168 (2022).

Summary: Let \(F(z)\) be any function. Suppose that \(w\) is a fixed point of \(F(z)\), that is, \(F(w)=w\). Then the recurrence equation \[x_{n + 1} = F(x_n)\] for \(n=0, 1, 2, \dots\) has a solution of the form \[x_n (w) = w + \sum_{i = 1}^\infty a_1^i A_i F_{\centerdot 1} (w)^{in} ,\] where \(F_{\centerdot 1}(z)=dF(z)/d\). So, for each \(w\) there is a set of complex \(x_0\) such that \(x_0(w)=x_0\). We assume that \(F(z)\) is analytic at \(w\). This solution appears to be new, even for such famous examples like the logistic map and the Mandelbrot equation.

MSC:

65Q99

Numerical methods for difference and functional equations, recurrence relations

Keywords:

exact solutions; logistic map; Mandelbrot equation

Software:

LogMapBaseFunction; Logistic Map

References:

[1]	M. Brio, “Bell polynomials of the second kind in matlab”, Matlab code, 2022, available at https://www.mathworks.com/matlabcentral/fileexchange/14483-bell-polynomials-of-the-second-kind.
[2]	L. Comtet, Advanced combinatorics, Reidel, Dordrecht, 1974. · Zbl 0283.05001
[3]	R. Devaney, “Unveiling the Mandelbrot set”, 2006, available at https://plus.maths.org/content/unveiling-mandelbrot-set.
[4]	H. Krieger, “The Mandelbrot set”, video, Numberphile, 2014, available at https://youtu.be/NGMRB4O922I.
[5]	B. Mandelbrot, “Fractals and the art of roughness”, TED talk, 2010, available at https://youtu.be/ay8OMOsf6AQ.
[6]	M. F. Maritz, “A note on exact solutions of the logistic map”, Chaos 30 (2020), art. id. 033136. · Zbl 1446.39018 · doi:10.1063/1.5125097
[7]	R. M. May, “Simple mathematical models with very complicated dynamics”, Nature 261 (1976), 459-467. · Zbl 1369.37088 · doi:10.1038/261459a0
[8]	R. M. May and G. F. Oster, “Bifurcations and dynamic complexity in simple ecological models”, The American Naturalist 110:974 (1976), 573-599. · doi:10.1086/283092
[9]	Y. Morimoto, “Hopf bifurcation in the simple nonlinear recurrence equation \[X(t+1)=AX(t)[1-X(t-1)]\]”, Phys. Lett. A 134:3 (1988), 179-182. · doi:10.1016/0375-9601(88)90816-X
[10]	Y. Morimoto, “The dynamics of nonlinear recurrence equations having integer variable”, Phys. Lett. A 227:3-4 (1997), 187-191. · Zbl 0962.39500 · doi:10.1016/S0375-9601(97)00020-0
[11]	D. Muller, “This equation will change how you see the world (the logistic map)”, video, Veritasium, 2020, available at https://youtu.be/ovJcsL7vyrk.
[12]	B. Sparks, “The Feigenbaum constant and logistic map”, video, Numberphile, 2017, available at https://youtu.be/ETrYE4MdoLQ.
[13]	E. W. Weisstein, “Logistic map”, from MathWorld, available at https://mathworld.wolfram.com/LogisticMap.html.
[14]	C. S. Withers and S. Nadarajah, “Solutions of linear recurrence equations”, Appl. Math. Comput. 271 (2015), 768-776. · Zbl 1410.39033 · doi:10.1016/j.amc.2015.09.079
[15]	“A cycling journey - Mandelbrot fractal zoom”, video, Maths Town, 2020, available at https://youtu.be/a3XDry3EwiU.
[16]	“Bell polynomials”, Wikipedia entry, available at https://en.wikipedia.org/wiki/Bell_polynomials.
[17]	“Bell polynomials of the second kind in matlab”, Matlab code, available at http://freesourcecode.net/matlabprojects/70276/bell-polynomials-of-the-second-kind-in-matlab