In this paper we review several contributions made in the field of discrete dynamical systems, inspired by harmonic analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein (MSS). The MSS harmonics of a periodic orbit calculate the symbolic sequences of the period doubling cascade of the orbit. Based on MSS harmonics, Pastor, Romera and Montoya (PRM) introduced the PRM harmonics, which allow to calculate the structure of a 1D quadratic map. Likewise, we review the applications in 2D quadratic maps. In this case both MSS harmonics and PRM harmonics deal with external arguments instead of with symbolic sequences. Finally, we review pseudoharmonics and pseudoantiharmonics, which enable new interesting applications.
P.-F. Verhulst, “Recherches Mathématiques sur la Loi d’Accroissement de la Population,” Mémoires de l’Aca- démie Royale des Sciences et Belles-Lettres de Bruxelles, Bruxelles, 1844.
S. Ulam and J. von Neumann, “On Combination of Stochastic and Deterministic Processes,” Bulletin of the American Mathematical Society, Vol. 53, No. 11, 1947, p. 1120.
R. M. May, “Simple Mathematical Models with Very Complicated Dynamics,” Nature, Vol. 261, 1976, pp. 459- 467. doi:10.1038/261459a0
B. B. Mandelbrot, “Fractal Aspects of the Iteration of for Complex ? and z,” Annals of the New York Academy of Sciences, Vol. 357, 1980, pp. 249-259. doi:10.1111/j.1749-6632.1980.tb29690.x
B. B. Mandelbrot, “On the Quadratic Mapping of for Complex ? and z: The Fractal Structure of Its μ Set, and Scaling,” Physica D: Nonlinear Phenomena, Vol. 7, 1983, pp. 224-239. doi:10.1016/0167-2789(83)90128-8
B. Branner, “The Mandelbrot Set,” Proceedings of Symposia in Applied Mathematics, Vol. 39, 1989, pp. 75-105.
J. Milnor and W. Thurston, “On Iterated Maps of the Interval,” Dynamical Systems, Vol. 1342, 1988, pp. 465- 563. doi:10.1007/BFb0082847
H.-O. Peitgen, H. Jürgens and D. Saupe, “Chaos and Fractals,” Springer, New York, 1992, pp. 569-574.
R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” Addison-Wesley, Boston, 1989, pp. 44-48.
G. Pastor, M. Romera and F. Montoya, “An Approach to the Ordering of One-Dimensional Quadratic Maps,” Chaos, Solitons & Fractals, Vol. 7, No. 4, 1996, pp. 565-584. doi:10.1016/0960-0779(95)00071-2
M. Romera, G. Pastor, J. C. Sanz-Martín and F. Montoya, “Symbolic Sequences of One-Dimensional Quadratic Maps,” Physica A: Statistical Mechanics and Its Applications, Vol. 256, No. 3-4, 1998, pp. 369-382. doi:10.1016/S0378-4371(98)00083-1
G. Pastor, M. Romera and F. Montoya, “Harmonic Structure of One-Dimensional Quadratic Maps,” Physical Review E, Vol. 56, 1997, pp. 1476-1483. doi:10.1103/PhysRevE.56.1476
J. Hale and H. Ko?ak, “Dynamics and Bifurcations,” App- lied Mathematics, Vol. 3, 1991, p. 314. doi:10.1007/978-1-4612-4426-4
M. Misiurewicz and Z. Nitecki, “Combinatorial Patterns for Maps of the Interval,” Memoirs of the American Ma- thematical Society, Vol. 94, No. 456, 1991, pp. 109-110.
A. Douady and J. H. Hubbard, “Etude Dynamique des Polyn?mes Complexes,” Publications Mathematiques d’Orsay, 84-02, 1984 (Première Partie) and 85-04, 1985 (Deuxième Partie). http://portail.mathdoc.fr/PMO/PDF/D_DOUADY_84_02.pdf http://mathdoc.emath.fr/PMO/PDF/D_DOUADY_85_04.pdf
M. Romera, G. Pastor and F. Montoya, “Misiurewicz Points in One-Dimensional Quadratic Maps,” Physica A: Statistical Mechanics and Its Applications, Vol. 232, No. 1-2, 1996, pp. 517-535. doi:10.1016/0378-4371(96)00127-6
G. Pastor, M. Romera and F. Montoya, “On the Calculation of Misiurewicz Patterns in One-Dimensional Quadratic Maps,” Physica A: Statistical Mechanics and Its Applications, Vol. 232, No. 1-2, 1996, pp. 536-553. doi:10.1016/0378-4371(96)00128-8
G. Pastor, M. Romera, G. Alvarez and F. Montoya, “Misiurewicz Point Patterns Generation in One-Dimensional Quadratic Maps,” Physica A: Statistical Mechanics and Its Applications, Vol. 292, No. 1-4, 2001, pp. 207- 230. doi:10.1016/S0378-4371(00)00586-0
G. Pastor, M. Romera, J. C. Sanz-Martín and F. Montoya, “Symbolic Sequences of One-Dimensional Quadratic Maps Points,” Physica A: Statistical Mechanics and Its Applications, Vol. 256, No. 3-4, 1998, pp. 369-382. doi:10.1016/S0378-4371(98)00083-1
A. Douady and J. H. Hubbard, “Itération des Polyn?mes Quadratiques Complexes,” C. R. Academic Society: Série I, Paris, 1982.
A. Douady, “Chaotic Dynamics and Fractals” Academic Press, New York, 1986.
G. Pastor, M. Romera, G. álvarez and F. Montoya, “Operating with External Arguments in the Mandelbrot Set Antenna,” Physica D: Nonlinear Phenomena, Vol. 117, No. 1-2, 2002, pp. 52-71. doi:10.1016/S0167-2789(02)00539-0
N. Metropolis, M. L. Stein and P. R. Stein, “On Finite Limit Sets for Transformations on the Unit Interval,” Journal of Combinatorial Theory: Series A, Vol. 15, No. 1, 1973, pp. 25-44. doi:10.1016/0097-3165(73)90033-2
M. Morse and G. A. Hedlung, “Symbolic Dynamics,” American Journal of Mathematics, Vol. 60, No. 4, 1938, pp. 815-866. doi:10.2307/2371264
B.-L. Hao and W.-M. Zheng, “Symbolic Dynamics of Uni- modal Maps Revisited,” International Journal of Modern Physics B, Vol. 3, No. 2, 1989, pp. 235-246. doi:10.1142/S0217979289000178
M. Schroeder, “Fractals, Chaos, Power Laws,” W. H. Freeman, New York, 1991.
A. N. Sharkovsky, “Coexistence of Cycles of Continuous Mapping of the Line into Itself,” Ukrainian Mathematical Journal, Vol. 16, 1964, pp. 61-71.
A. N. Sharkovsky, Yu. L. Maistrenko and E. Yu. Romanenko, “Difference Equations and Their Applications,” Kluver Academic Publishers, Dordrecht, 1993.
M. Romera, G. Pastor and F. Montoya, “Graphic Tools to Analyse One-Dimensional Quadratic Maps,” Computer & Graphics, Vol. 20, No. 2, 1996, pp. 333-339. doi:10.1016/0097-8493(95)00134-4
M. Romera, G. Pastor and F. Montoya, “On the Cusp and the Tip of a Midget in the Mandelbrot Set Antenna,” Physica Letters A, Vol. 221, No. 3, 1996, pp. 158-162.
M. J. Feigenbaum, “Quantitative University for a Class of Nonlinear Transformations,” Journal of Statistical Physics, Vol. 19, No. 1, 1978, pp. 25-52. doi:10.1007/BF01020332
R. L. Devaney, “Complex Dynamical Systems,” American Mathematical Society, Providence, 1994.
M. Romera, G. Pastor, G. Alvarez and F. Montoya, “Shrubs in the Mandelbrot Set Ordering,” International Journal of Bifurcation and Chaos, Vol. 13, No. 8, 2003, pp. 2279-2300. doi:10.1142/S0218127403007941
G. Pastor, M. Romera, G. Alvarez and F. Montoya, “External Arguments for the Chaotic Bands Calculation in the Mandelbrot Set,” Physica A: Statistical Mechanics and Its Applications, Vol. 353, No. 1, 2005, pp. 145-158. doi:10.1016/j.physa.2005.02.025
G. Pastor, M. Romera, G. Alvarez, D. Arroyo, A. B. Orue, V. Fernández and F. Montoya, “A General View of Pseudoharmonics and Pseudoantiharmonics to Calculate External Arguments of Douady and Hubbard,” Applied Mathematic and Computation, Vol. 213, No. 2, 2009, pp. 484-497. doi:10.1016/j.amc.2009.03.038
G. Pastor, M. Romera, G. Alvarez, D. Arroyo and F. Montoya, “Equivalence between Subshrubs and Chaotic Bands in the Mandelbrot Set,” Discrete Dynamics in Nature and Society, 2006, Article ID: 45920. doi:10.1155/DDNS/2006/70471