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- Valery A. Gaiko
- 2000

Two-dimensional polynomial dynamical systems are mainly considered. We develop Erugin's two-isocline method for the global analysis of such systems, construct canonical systems with field-rotation parameters and study various limit cycle bifurcations. In particular, we show how to carry out the classification of separatrix cycles and consider the most… (More)

In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary (but finite) number of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections… (More)

- Valery A. Gaiko, Wim T. van Horssen
- I. J. Bifurcation and Chaos
- 2009

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve first the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial system. Then, by means of the same bifurcationally… (More)

In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.

- Valery A. Gaiko
- 2008

In this paper, a quadratic system with two parallel straight line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [13]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles,… (More)

In this paper we complete the global qualitative analysis of a quartic ecological model. In particular, studying global bifurcations of singular points and limit cycles, we prove that the corresponding dynamical system has at most two limit cycles.

No part of the Journal may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from Department of Applied Mathematical Analysis, Abstract In this paper, a quadratic system with two parallel straight line-isoclines is… (More)

In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number (but finite) of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifur-cations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections… (More)

- Valery A. Gaiko
- 2006

In this paper, using geometric properties of the field rotation parameters, we present a solution of Smale's Thirteenth Problem on the maximum number of limit cycles for Liénard's polynomial system. We also generalize the obtained result and present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles surrounding a singular point… (More)