Dmitry Reviznikov
Publications:
Morozov A. Y., Reviznikov D. L.
Modeling of Dynamic Systems with Interval Parameters in the Presence of Singularities
2020, Vol. 16, no. 3, pp. 479490
Abstract
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.
