J. Korean Math. Soc. 2011; 48(6): 1101-1123
Printed November 1, 2011
https://doi.org/10.4134/JKMS.2011.48.6.1101
Copyright © The Korean Mathematical Society.
Andrew Foster and Melusi Khumalo
Memorial University of Newfoundland, University of Johannesburg
Numerical schemes are routinely used to predict the behavior of continuous dynamical systems. All such schemes transform flows into maps, which can possess dynamical behavior deviating from their continuous counterparts. Here the common bifurcations of scalar dynamical systems are transformed under a class of algorithms known as \lopc\ methods. Through the use of normal forms, we prove that each such bifurcation in an originating flow gives rise to an exactly corresponding one in its discretization. The conditions for spurious period doubling behavior under this class of algorithm are derived. We discuss the global behavioral consequences of a singular set induced by the discretizing methods, including loss of monotonicity of solutions, intermittency, and distortion of attractor basins.
Keywords: collocation methods, spurious behavior, bifurcation
MSC numbers: 65L20, 65L06, 37N30
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