Application of the adaptive filtering
for the construction of
two-parameter
bifurcation diagrams of the flow systems
A.A. Kipchatov and S.V. Podin
Saratov State University,
Saratov, Russia
One of the most effective ways of analysys of the complicated dynamic systems is the construction of two-parameter bifurcation diagrams. Traditionally, two-parameter bifurcation diagram is a result of search for bifurcation values of parameters on experimental data. Automatical parameter plane plotting takes place only for maps and only for periodic regimes. In the case of ordinary differential equation systems, automatical two-parameter bifurcation diagram plotting is associated with some difficulties, because there are no sufficiently effective methods for detecting of the oscillation period and, consequently, for the automatic determinations of bifurcation points.
A reliable procedure of the oscillation period evaluation, proposed in this work, allows to identify periodic and complex periodic (i.e. containes a subharmonic components) oscillation regimes. A block of digital filters is used for the extraction of subharmonic components from the signal spectrum. The number of subharmonic at the specified level determines the period of testing oscillations. To optimize calculation time, the filtering frequency and fundamental frequency of analysing oscillations can be fitted (adapted) for each parameter plane point.
Efficiency and reliability of developed technique is supported by the construction of two-parameter bifurcation diagram for Rössler and Lorenz systems.