Results 211 to 220 of about 3,187 (232)

Algebraic limit cycles in polynomial systems of differential equations

Journal of Physics A: Mathematical and Theoretical, 2007
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree
Jaume Llibre, Yulin Zhao
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Polynomial Vector Fields with Prescribed Algebraic Limit Cycles

Geometriae Dedicata, 2001
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Algebraical Limit Cycles of Polynomial Vector Fields on the Plane

Differential Equations, 2001
The existence of limit cycles of algebraic differential equations has been studied and their number has been estimated in numerous papers. In recent decades, much attention was paid to polynomial vector fields with invariant algebraic curves. The statement of the problem goes back to Darboux, Poincaré, and Erugin.
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Hilbert’s 16th Problem for Algebraic Limit Cycles

2016
In this chapter we state Hilbert’s 16th problem restricted to algebraic limit cycles. Namely, consider the set Σ’ n of all real polynomial vector fields \( \chi = \left( {P,\,Q} \right)\) of degree n having real irreducible \( \left( {{\rm on}\, \mathbb{R}\left[ {x,\,y} \right]} \right)\) invariant algebraic curves.
Jaume Llibre, Rafael Ramírez
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Planar Polynomial Systems with Non-Algebraic Limit Cycles

AIP Conference Proceedings, 2009
In this paper, we study the existence of the non‐algebraic limit cycles of the systems dxdt = Pn(x,y)+xRm(x,y) dydt = Qn(x,y)+yRm(x,y) where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with ...
Khalil I. T. Al-Dosary, George Maroulis, Theodore E. Simos   +2 more
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NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

International Journal of Mathematics, 2007
In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y)
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Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations

Nonlinearity, 2006
We give a family of planar polynomial differential systems whose limit cycles can be explicitly described using polar coordinates. Moreover, we characterize the multiplicity of each one of the limit cycles whenever they exist. The given family of planar polynomial differential systems can have at most two limit cycles, counted with multiplicity.As an ...
Jaume Giné, Maite Grau
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Coexistence of limit cycles and invariant algebraic curves for a Kukles system

Nonlinear Analysis: Theory, Methods & Applications, 2004
This paper deals with the so-called Kukles systems, \[ \dot x=-y,\qquad \dot y=f(x,y), \] \(f\) being a polynomial of degree \(d.\) Firstly, the authors study the number and distribution of invariant straight lines that this type of systems can have.
Chavarriga, J., Sáez, E., Szántó, I., Grau, M.   +3 more
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Homology for operator algebras. IV. On the regular classification of limits of 4-cycle algebras

1997
According with the terminology and notations from the research notes ``Limit algebras'' [Pitman Research Notes in Math. 278, London (1993)] of the second author, the subject of the present paper is related to the 4-cycle algebras, that is a CSL-algebra whose reduced digraph is a 4-cycle, and to the regular direct systems of 4-cycle algebras with rigid ...
Power, Stephen C., Donsig, A. P.
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On the cubic Kukles systems with an algebraic limit cycle of degree two

Journal of Mathematical Analysis and Applications, 2023
Yulin Zhao
exaly