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**Displacement in the parameter space versus spurious solution of discretization with large time step.**
*(English)*
Zbl 1036.37031

Summary: In order to investigate a possible correspondence between differential and difference equations, it is important to possess a discretization of ordinary differential equations. It is well known that, when differential equations are discretized, the solution thus obtained depends on the time step used. In the majority of cases, such a solution is considered spurious when it does not resemble the expected solution of the differential equation. This often happens when the time step taken into consideration is too large.

In this work, we show that, even for quite large time steps, some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered. To reduce such a displacement, a judicious choice of the discretization scheme should be made. To this end, a recent discretization scheme, based on the Lie expansion of the original differential equations, proposed by S. Monaco and D. Normand-Cyrot [Lect. Notes Control Inf. Sci. 144, 788–797 (1990; Zbl 0709.93048)] is analysed. Such a scheme is shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo-period of the underlying dynamics.

In this work, we show that, even for quite large time steps, some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered. To reduce such a displacement, a judicious choice of the discretization scheme should be made. To this end, a recent discretization scheme, based on the Lie expansion of the original differential equations, proposed by S. Monaco and D. Normand-Cyrot [Lect. Notes Control Inf. Sci. 144, 788–797 (1990; Zbl 0709.93048)] is analysed. Such a scheme is shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo-period of the underlying dynamics.

### MSC:

37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |

37C10 | Dynamics induced by flows and semiflows |

65L99 | Numerical methods for ordinary differential equations |