Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TRANSFORMATION OF LOCAL BIFURCATIONS UNDER COLLOCATION METHODS

  • Foster, Andrew (Department of Mathematics and Statistics Memorial University of Newfoundland) ;
  • Khumalo, Melusi (Department of Mathematics University of Johannesburg)
  • Received : 2009.07.17
  • Published : 2011.11.01
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Abstract

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 linearized one-point collocation 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.

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