Bifurcations of Heteroclinic Contours in Two-Parameter Planar Systems: Overview and Explicit Examples

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of known results on bifurcations of these contours. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software matcont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

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AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405012442 ◽

2005 ◽

Vol 15 (03) ◽

pp. 1033-1044 ◽

Cited By ~ 4

Author(s):

GRZEGORZ ŚWIRSZCZ

Keyword(s):

Differential Equations ◽

Algebraic Curve ◽

Vector Fields ◽

Algebraic Curves ◽

Polynomial System ◽

Polynomial Systems ◽

Invariant Algebraic Curves ◽

Invariant Algebraic Curve ◽

Planar Vector Fields ◽

Planar Vector

Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

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Normal Forms and Bifurcation of Planar Vector Fields

10.1017/cbo9780511665639 ◽

1994 ◽

Cited By ~ 232

Author(s):

Shui-Nee Chow ◽

Chengzhi Li ◽

Duo Wang

Keyword(s):

Vector Fields ◽

Normal Forms ◽

Planar Vector Fields ◽

Planar Vector

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Investigation of Global Bifurcations in Planar Vector Fields

10.21236/ada204159 ◽

1988 ◽

Author(s):

John Guckenheimer

Keyword(s):

Vector Fields ◽

Global Bifurcations ◽

Planar Vector Fields ◽

Planar Vector

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A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01820-7 ◽

2021 ◽

Vol 18 (5) ◽

Author(s):

Antonio Algaba ◽

Cristóbal García ◽

Jaume Giné

Keyword(s):

Normal Form ◽

Vector Fields ◽

New Technique ◽

Center Problem ◽

New Normal ◽

Planar Vector Fields ◽

Planar Vector ◽

A New Technique

AbstractIn this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

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