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Transformation of local bifurcations under collocation methods. (English) Zbl 1230.65137

Authors’ 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.

MSC:
65P30Bifurcation problems (numerical analysis)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
65L20Stability and convergence of numerical methods for ODE
65L06Multistep, Runge-Kutta, and extrapolation methods
37N30Dynamical systems in numerical analysis
37M20Computational methods for bifurcation problems
37G10Bifurcations of singular points
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