A method to construct irregular fractal curves

In this paper, we present a method of constructing irregular fractal curves. This method is based on a sequence (Φ[n])n∈N of piecewise differentiable maps defined succinctly in Narayaninsamy [T. Narayaninsamy, On basin boundaries, Appl. Math. Comput. 99 (1999) 261-274]. The experimental results concerning the basins of attraction of attractors which are given in Gardini, Mira et al. and Narayaninsamy [L. Gardini, Some global bifurcations of two-dimensional endomorphisms by use of critical lines, Nonlinear Anal. Theory Meth. Appl. 18(4) (1991) 361-399; C, Mira, D. Fournier-Prunaret, L. Gardini, H. Kawakami, J.-C. Cathala, Basin bifurcations of two-dimensional noninvertible maps: fractalization of basins, J. Bifurcat. Chaos 2 (1994) 343-381; T. Narayaninsamy, Contribution à l'Etude de l'Itération Fractionaire et à Celle des Endomorphismes Bi-Dimensionnels, Doctorat Mathématiques Appliquées de l'Université Paul Sabatier (Toulouse III)] constitute the basic tool of this sequence.