Nonlinear Coupled Oscillator
The recurrence equations constitute a two-parameter canonical form of a nonlinear coupled oscillator.
More information is in the following two papers.
W. Metzler, W. Beau, W. Frees, and A. Ueberla, "Symmetry and Self-Similarity with Coupled Logistic Maps," Zeitschrift fur Naturforschung, 42(3), 1987 pp. 310–318.
W. Metzler, A. Brelle, and K. D. Schmidt, "Nonanalytic Dynamics for Generating the Mandelbrot Set: A Tutorial," International Journal of Bifurcation and Chaos, 2(2), 1992 pp. 241–250.