International Journal of Bifurcation and Chaos, Vol. 12, No. 12 (2002) 2847–2867 c  World Scientific Publishing Company BRANCHING PROCESSES AND COMPUTATIONAL COLLAPSE OF DISCRETIZED UNIMODAL MAPPINGS P. DIAMOND ∗ and I. VLADIMIROV † Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia ∗ pmd@maths.uq.edu.au † igv@maths.uq.edu.au Received December 5, 2000; Revised November 2, 2001 In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping. Keywords : Dynamical systems; computational collapse; discretization. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2848 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849 3. Qualitative Structure of Negative Semitrajectories . . . . . . . . . . . . . . . 2850 4. Definition of Quantization Errors . . . . . . . . . . . . . . . . . . . . . . . 2851 5. Representation of Preimages in Terms of Quantization Errors . . . . . . . . . . 2852 6. Quasiperiodic Representation of Quantization Errors . . . . . . . . . . . . . . 2856 7. Asymptotically Uniform Distribution of Toral Projections . . . . . . . . . . . . 2859 8. Asymptotic Independence and Uniform Distribution of Quantization Errors . . . . 2859 9. Associated Branching Process 1 . . . . . . . . . . . . . . . . . . . . . . . 2860 10. Associated Branching Process 2 . . . . . . . . . . . . . . . . . . . . . . . 2861 11. Asymptotic Distribution of Cardinalities of Preimages . . . . . . . . . . . . . . 2862 12. Frequency Functions for Sets of Reachable Points . . . . . . . . . . . . . . . . 2863 13. Application to Logistic Mapping . . . . . . . . . . . . . . . . . . . . . . . 2864 14. Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2865 † Permanent address: Institute for Information Transmission Problems, 19 Bolshoi Karetny Lane, GSP–4, 101447 Moscow, Russia. 2847