The Experts below are selected from a list of 96 Experts worldwide ranked by ideXlab platform
Arkadiusz Ploski - One of the best experts on this subject based on the ideXlab platform.
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a bound for the milnor number of plane curve Singularities
Open Mathematics, 2014Co-Authors: Arkadiusz PloskiAbstract:Let f = 0 be a plane algebraic curve of degree d > 1 with an Isolated Singular Point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].
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a bound for the milnor number of plane curve Singularities
arXiv: Algebraic Geometry, 2013Co-Authors: Arkadiusz PloskiAbstract:Let $f=0$ be a plane algebraic curve of degree $d>1$ with an Isolated Singular Point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.
Grau Maite - One of the best experts on this subject based on the ideXlab platform.
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Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor
'Springer Science and Business Media LLC', 2011Co-Authors: Garcia, Isaac A., Giacomini Hector, Grau MaiteAbstract:41 pages, no figuresInternational audienceIn this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the Singular Point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue
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Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor
2009Co-Authors: Garcia, Isaac A., Giacomini Hector, Grau MaiteAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the Singular Point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue.Comment: 41 pages, no figure
Maite Grau - One of the best experts on this subject based on the ideXlab platform.
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Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor
Journal of Dynamics and Differential Equations, 2011Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point p _0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p _0 being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p _0 the differential system can always be brought, by means of a change to (generalized) polar coordinates ( r , θ ), to an equation over a cylinder in which the Singular Point p _0 corresponds to a limit cycle γ _0. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ _0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ _0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point p _0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue.
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Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor
Journal of Dynamics and Differential Equations, 2011Co-Authors: Isaac A. Garcia, Hector Giacomini, Maite GrauAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the Singular Point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue.
Garcia, Isaac A. - One of the best experts on this subject based on the ideXlab platform.
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Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor
'Springer Science and Business Media LLC', 2011Co-Authors: Garcia, Isaac A., Giacomini Hector, Grau MaiteAbstract:41 pages, no figuresInternational audienceIn this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the Singular Point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue
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Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor
2009Co-Authors: Garcia, Isaac A., Giacomini Hector, Grau MaiteAbstract:In this paper we study the maximum number of limit cycles that can bifurcate from a focus Singular Point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus Singular Point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the Singular Point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the Singular Point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of Singular Points, namely for the three types of focus considered in the previous paragraph and for any Isolated Singular Point with at least one non-zero eigenvalue.Comment: 41 pages, no figure
M Reyes - One of the best experts on this subject based on the ideXlab platform.
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characterization of a monodromic Singular Point of a planar vector field
Nonlinear Analysis-theory Methods & Applications, 2011Co-Authors: Antonio Algaba, Cristobal Garcia, M ReyesAbstract:The Newton diagram and the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine whether an Isolated Singular Point of the vector field is monodromic or has a characteristic trajectory.
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the center problem for a family of systems of differential equations having a nilpotent Singular Point
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Antonio Algaba, Cristobal Garcia, M ReyesAbstract:Abstract We study the analytic system of differential equations in the plane ( x ˙ , y ˙ ) t = ∑ i = 0 ∞ F q − p + 2 i s , where p , q ∈ N , p ⩽ q , s = ( n + 1 ) p − q > 0 , n ∈ N , and F i = ( P i , Q i ) t are quasi-homogeneous vector fields of type t = ( p , q ) and degree i, with F q − p = ( y , 0 ) t and Q q − p + 2 s ( 1 , 0 ) 0 . The origin of this system is a nilpotent and monodromic Isolated Singular Point. We prove for this system the existence of a Lyapunov function and we solve theoretically the center problem for such system. Finally, as an application of the theoretical procedure, we characterize the centers of several subfamilies.
