TRANSFORMATION OF LOCAL BIFURCATIONS UNDER COLLOCATION METHODS
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Numerical schemes are routinely used to predict the behav-ior 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 trans-formed under a class of algorithms known as linearized one-point colloca-tion methods. Through the use of normal forms, we prove that each such bifurcation in an originating flow gives rise to an exactly correspond-ing one in its discretization. The conditions for spurious period dou-bling behavior under this class of algorithm are derived. We discuss the global behavioral consequences of a singular set induced by the discretiz-ing methods, including loss of monotonicity of solutions, intermittency,and distortion of attractor basins.
Abstract
Numerical schemes are routinely used to predict the behav-ior 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 trans-formed under a class of algorithms known as linearized one-point colloca-tion methods. Through the use of normal forms, we prove that each such bifurcation in an originating flow gives rise to an exactly correspond-ing one in its discretization. The conditions for spurious period dou-bling behavior under this class of algorithm are derived. We discuss the global behavioral consequences of a singular set induced by the discretiz-ing methods, including loss of monotonicity of solutions, intermittency,and distortion of attractor basins.
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