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# Abstracts

## Global bifurcations in multi-parameter dynamical systems

Gaiko V.A.

United Institute of Informatics Problems, National Academy of Sciences of Belarus, Leonid Beda Str. 6-4, Minsk 220040, Belarus, Phone: +375-29-7646662, E-mail: valery.gaiko@gmail.com

1 pp. (accepted)

We carry out the global bifurcation analysis of multi-parameter polynomial dynamical systems. To

control all of the limit cycle bifurcations in such systems, especially, bifurcations of multiple

limit cycles, it is necessary to know the properties and combine the effects of all their rotation

parameters. It can be done by means of the development of new bifurcational geometric methods based

on the well-known Weierstrass preparation theorem and the Perko planar termination principle

stating that the maximal one-parameter family of multiple limit cycles terminates either at a

singular point which is typically of the same multiplicity (cyclicity) or on a separatrix cycle

which is also typically of the same multiplicity (cyclicity). This principle is a consequence of

the principle of natural termination which was stated for higher-dimensional dynamical systems by

A.\,Wintner who studied one-parameter families of periodic orbits of the restricted three-body

problem and used Puiseux series to show that in the analytic case any one-parameter family of

periodic orbits can be uniquely continued through any bifurcation except a period-doubling

bifurcation. Such a bifurcation can happen, e.\,g., in a three-dimensional Lorenz system. But this

cannot happen for planar systems. That is why the Wintner--Perko termination principle is applied

for studying multiple limit cycle bifurcations of planar polynomial dynamical systems. If we do not

know the cyclicity of the termination points, then, applying canonical systems with field rotation

parameters, we use geometric properties of the spirals filling the interior and exterior domains of

limit cycles. Applying this method, we have solved, e.\,g., Smale's Thirteenth Problem for the

classical Li\'{e}nard system. Generalizing the obtained results, we have solved Hilbert's Sixteenth

Problem on the maximum number and distribution of limit cycles for the Kukles cubic-linear system

and for the general Li\'{e}nard polynomial system with an arbitrary number of singular points. We

have completed the global qualitative analysis of Holling-type and Leslie--Gower systems which

model the dynamics of the populations of predators and their prey in a given ecological or

biomedical system. Finally, applying a similar approach, we consider various applications of

three-dimensional polynomial dynamical systems and, in particular, complete the strange attractor

bifurcation scenario in the classical Lorenz system globally connecting the homoclinic,

period-doubling, Andronov--Shilnikov, and period-halving bifurcations of its limit cycles.

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