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We carry out the global bifurcation analysis of the classical Lorenz system. For many years, this system has been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question at present is to understand the bifurcation scenario of chaos transition in this… (More)

- 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, 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)

- 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 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
- Appl. Math. Lett.
- 2012

In this work, 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 the limit cycle problem for a general Liénard polynomial systemwith an arbitrary (but finite) number of singular points. © 2012 Elsevier Ltd. All rights reserved.

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

- Valery A. Gaiko
- 2006

In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present… (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)