Modeling of Dynamic Systems with Interval

A. Yu. Morozov ; D. L. Reviznikov

Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 479-490

Résumé

In solving applied and research problems, there often arise situations where certain parameters are not exactly known, but there is information about their ranges. For such problems, it is necessary to obtain an interval estimate of the solution based on interval values of parameters. In practice, the dynamic systems where bifurcations and chaos occur are of interest. But the existing interval methods are not always able to cope with such problems. The main idea of the adaptive interpolation algorithm is to build an adaptive hierarchical grid based on a kdtree where each cell of adaptive hierarchical grid contains an interpolation grid. The adaptive grid should be built above the set formed by interval initial conditions and interval parameters. An adaptive rebuilding of the partition is performed for each time instant, depending on the solution. The result of the algorithm at each step is a piecewise polynomial function that interpolates the dependence of the problem solution on the parameter values with a given precision. Constant grid compaction will occur at the corresponding points if there are unstable states or dynamic chaos in the system; therefore, the minimum cell size is set. The appearance of such cells during the operation of the algorithm is a sign of the presence of unstable states or chaos in a dynamic system. The effectiveness of the proposed approach is demonstrated in representative examples.

Détail

Keywords: instability, dynamic chaos, kd-tree.
Mots-clés : interval ODE, bifurcations, adaptive interpolation algorithm