Real plane algebraic curves with many singularities

2006 ◽  
pp. 389-401
Author(s):  
E. I. Shustin
Keyword(s):  
Algebraic Curves ◽  
Real Plane

scholarly journals Real plane algebraic curves with prescribed singularities

Topology ◽  
1993 ◽  
Vol 32 (4) ◽  
pp. 845-856 ◽  
Author(s):  
Eugenii Shustin
Keyword(s):  
Algebraic Curves ◽  
Real Plane

scholarly journals Dulac – Cherkas functions for systems equivalent to the van der Pol equation

2020 ◽  
Vol 56 (3) ◽  
pp. 275-286
Author(s):  
A. A. Hryn
Keyword(s):  
Limit Cycle ◽  
Phase Plane ◽  
Algebraic Curves ◽  
Real Parameter ◽  
Real Plane ◽  
Singularly Perturbed Systems ◽  
Van Der Pol ◽  
The Subject ◽  
Equivalent Systems ◽  
Parameter Values

The object of this study is an autonomous van der Pol system on a real plane. The subject of the study is the properties of the limit cycle of this system. The main purpose of this paper is to find the localization of the limit cycle on the phase plane and establish its shape for various values of the real parameter of the van der Pol system. Our approach is based on the use of transverse curves related to the Dulac – Cherkas functions and approximating the location of the limit cycle. As the first step, five topologically equivalent systems, including systems with a parameter rotating the vector field, as well as singularly perturbed systems are determined for the van der Pol system. Then, applying the previously elaborated method, we constructed two polynomial Dulac – Cherkas functions for each of three systems from the considered ones in the phase plane for all real nonzero values of the parameter. Using them, transverse curves forming the boundaries of the localization regions of the limit cycle for the van der Pol system are found. Thus, the constructed Dulac – Cherkas functions allow us to determine the location of the limit cycle on the basis of algebraic curves for all real parameter values, including values close to the bifurcation of a limit cycle from the center ovals, the Andronov – Hopf bifurcation, and the bifurcation from a closed trajectory related to a discontinuous periodic solution.

A Note Concerning the Real Inflexions of Real, Plane, Algebraic Curves

1974 ◽  
Vol 17 (3) ◽  
pp. 411-412
Author(s):  
Gareth J. Griffith
Keyword(s):  
Algebraic Curve ◽  
Double Point ◽  
Algebraic Curves ◽  
Real Plane ◽  
The Real ◽  
Plane Algebraic Curve

Theorem. “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double point.”

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