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Jaume Llibre - One of the best experts on this subject based on the ideXlab platform.
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phase portraits and bifurcation diagram of the gray scott model
Journal of Mathematical Analysis and Applications, 2021Co-Authors: Ting Chen, Jaume LlibreAbstract:Abstract We give a complete classification of the phase portraits in the Poincar e ´ disk for the Cubic Polynomial systems x ˙ = 1 − x − a x y 2 , y ˙ = − b y + a x y 2 , in R 2 according with the values of its two parameters a and b. These differential systems correspond to the Gray-Scott model. Moreover we provide the bifurcation diagram in the parameter plane ( a , b ) of these systems.
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on the singularities of the planar Cubic Polynomial differential systems and the euler jacobi formula
Qualitative Theory of Dynamical Systems, 2020Co-Authors: Jaume Llibre, Claudia VallsAbstract:Using the Euler–Jacobi formula we obtain an algebraic relation between the singular points of a Polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar Cubic Polynomial differential systems when these systems have nine finite singular points.
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on the centers of Cubic Polynomial differential systems with four invariant straight lines
Topological Methods in Nonlinear Analysis, 2020Co-Authors: Jaume LlibreAbstract:Assume that a Cubic Polynomial differential system in the plane has four invariant straight lines in generic position, i.e., they are not parallel and no more than two straight lines intersect in a point. Then such a differential system only can have $0$, $1$ or $3$ centers.
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Z2-equivariant linear type bi-center Cubic Polynomial Hamiltonian vector fields
Journal of Differential Equations, 2020Co-Authors: Ting Chen, Jaume LlibreAbstract:Abstract We study the global dynamical behavior of Z 2 -equivariant Cubic Hamiltonian vector fields with a linear type bi-center at ( ± 1 , 0 ) . By using a series of symbolic computation tools, we obtain all possible phase portraits of these Z 2 -equivariant Hamiltonian systems.
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limit cycles of discontinuous piecewise quadratic and Cubic Polynomial perturbations of a linear center
Discrete and Continuous Dynamical Systems-series B, 2019Co-Authors: Jaume Llibre, Yilei TangAbstract:We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and Cubic Polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or Cubic Polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise Polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order \begin{document}$ n $\end{document} for \begin{document}$ n = 1, 2, 3, 4, 5 $\end{document} . Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and Cubic Polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations. Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic Polynomials or Cubic quasi-homogenous Polynomials.
Llibre Jaume - One of the best experts on this subject based on the ideXlab platform.
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Cubic homogeneous Polynomial centers
2021Co-Authors: Li Chengzhi, Llibre JaumeAbstract:Agraïments: The first author is partially supported by NSFC-11271027 and NSFC- 11171267.First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all Cubic homogeneous Polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most 1 limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all Cubic Polynomial differential systems. Moreover, there are examples with 1 limit cycles
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Limit cycles bifurcating from the periodic annulus of the weight-homogeneous Polynomial centers of weight-degree 2
2021Co-Authors: Llibre Jaume, Lopes, Bruno D., De Moraes, Jaime R.Abstract:Agraïments: FEDER-UNAB-10-4E-378, and a CAPES grant number 88881. 030454/2013-01 from the program CSF-PVE and CNPq grant "Projeto Universal 472796/2013-5". The second author is supported by CAPES/GDU-7500/13-0.We obtain an explicit Polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of Cubic Polynomial differential centers when it is perturbed inside the class of all Cubic Polynomial differential systems. The family considered is the unique family of weight-homogeneous Polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica
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Limit cycles for a class of continuous and discontinuous Cubic Polynomial differential systems
2021Co-Authors: Llibre Jaume, Lopes, Bruno D., De Moraes, Jaime R.Abstract:Agraïments: FEDER-UNAB10-4E-378. The first and second author are supported by CAPES-MECD grant PHB-2009-0025-PC. The third author is supported by FAPESP-2010/17956-1.We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous Cubic Polynomial centers x˙ = y(−1 + 2αx + 2βx2), y˙ = x + α(y2 − x2) + 2βxy2, α ∈ R, β < 0, when it is perturbed inside the classes of all continuous and discontinuous Cubic Polynomial differential systems. We obtain that the maximum number of limit cycles which can be obtained by the averaging method of first order is 3 for the perturbed continuous systems and for the perturbed discontinuous systems at least 12 limit cycles can appear
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Limit cycles of Cubic Polynomial differential systems with rational first integrals of degree 2
2021Co-Authors: Llibre Jaume, Lopes, Bruno D., De Moraes, Jaime R.Abstract:Agraïments: FEDER-UNAB-10-4E-378, and a CAPES Grant No. 88881. 030454/2013-01 from the program CSF-PVE. The second authors is partially supported by the project CAPES Grant No. 88881.030454/2013-01 from the program CSF-PVE and CNPq grant "Projeto Universal 472796/2013-5". The second author is supported by CAPES/GDU - 7500/13-0. The last author is supported by FAPESP-2010/17956-1.The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the Cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all Cubic Polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Mapl
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Phase portraits for some symmetric Cubic Riccati Polynomial differential equations
2021Co-Authors: Llibre Jaume, Oliveira, Regilene D. S., Valls ClàudiaAbstract:We classify the topological phase portraits in the Poincaré disc of two classes of symmetric Riccati Cubic Polynomial differential systems
Nicolae Vulpe - One of the best experts on this subject based on the ideXlab platform.
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First Integrals and Phase Portraits of Planar Polynomial Differential Cubic Systems with the Maximum Number of Invariant Straight Lines
Qualitative Theory of Dynamical Systems, 2016Co-Authors: Cristina Bujac, Jaume Llibre, Nicolae VulpeAbstract:In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006 ) the family of Cubic Polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a Cubic Polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006 ) by adding a new class of such Cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of Cubic systems with maximum number of invariant straight lines when this number is finite.
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phase portraits and invariant straight lines of Cubic Polynomial vector fields having a quadratic rational first integral
Rocky Mountain Journal of Mathematics, 2011Co-Authors: Jaume Llibre, Adam Mahdi, Nicolae VulpeAbstract:In this paper we classify all Cubic Polynomial dierential systems having a first integral of degree two. In other words we characterize all the global phase portraits of the Cubic Polynomial dierential systems having all their orbits contained in conics. We also determine their configurations of invariant straight lines. We show that there are exactly 36 topologically dierent phase
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quadratic systems with a Polynomial first integral a complete classification in the coefficient space r12
Journal of Differential Equations, 2009Co-Authors: Joan C Artes, Jaume Llibre, Nicolae VulpeAbstract:Abstract In this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non-degenerate quadratic system in order to determine if it has or not a Polynomial first integral without using any normal form. We obtain that the existence of Polynomial first integral is directly related with the fact that all the roots of a convenient Cubic Polynomial are rational and negative. The coefficients of this Cubic Polynomial are invariants related with some geometric properties of the system.
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planar Cubic Polynomial differential systems with the maximum number of invariant straight lines
arXiv: Dynamical Systems, 2004Co-Authors: Jaume Llibre, Nicolae VulpeAbstract:We classify all Cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every class we provide the configuration of its invariant straight lines in the Poincare disc. Moreover, every class is characterized by a set of affine invariant conditions.
J R De Moraes - One of the best experts on this subject based on the ideXlab platform.
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limit cycles bifurcating from the periodic annulus of the weight homogeneous Polynomial centers of weight degree 2
Applied Mathematics and Computation, 2016Co-Authors: Jaume Llibre, B D Lopes, J R De MoraesAbstract:We obtain an explicit Polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of Cubic Polynomial differential centers when it is perturbed inside the class of all Cubic Polynomial differential systems. The family considered is the unique family of weight-homogeneous Polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.
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limit cycles of Cubic Polynomial differential systems with rational first integrals of degree 2
Applied Mathematics and Computation, 2015Co-Authors: Jaume Llibre, B D Lopes, J R De MoraesAbstract:The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the Cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all Cubic Polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Maple.
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limit cycles for a class of continuous and discontinuous Cubic Polynomial differential systems
Qualitative Theory of Dynamical Systems, 2014Co-Authors: Jaume Llibre, B D Lopes, J R De MoraesAbstract:We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous Cubic Polynomial centers $$\begin{aligned} \dot{x}=y(-1+2\alpha x+2\beta x^2),\quad \dot{y}=x+\alpha (y^2-x^2)+2\beta xy^2, \quad \alpha \in \mathbb {R},\,\beta <0, \end{aligned}$$ when it is perturbed inside the classes of all continuous and discontinuous Cubic Polynomial differential systems with two zones of discontinuity separated by a straight line. We obtain that this number is 3 for the perturbed continuous systems and at least 12 for the discontinuous ones using the averaging method of first order.
Wentao Huang - One of the best experts on this subject based on the ideXlab platform.
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center problem and the bifurcation of limit cycles for a Cubic Polynomial system
Applied Mathematical Modelling, 2015Co-Authors: Wentao Huang, Qi ZhangAbstract:Abstract In this study, we consider the center problem and the bifurcation of limit cycles for a Cubic system that lies in a symmetrical vector field about the origin. By analyzing and calculating the focal values (or the Lyapunov constant), we obtain the conditions where two equilibrium points, (1, 1) and (−1, −1), become a pair of simultaneous centers. Moreover, six limit cycles, including three stable limit cycles, can bifurcate from (1, 1) under a specific condition. From the symmetric quality, (−1, −1) can also bifurcate into six limit cycles by simultaneous Hopf bifurcation, which is an interesting result.
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seven large amplitude limit cycles in a Cubic Polynomial system
International Journal of Bifurcation and Chaos, 2006Co-Authors: Yirong Liu, Wentao HuangAbstract:In this paper, the problem of limit cycles bifurcated from the equator for a Cubic Polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a Cubic Polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.
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a Cubic system with twelve small amplitude limit cycles
Bulletin Des Sciences Mathematiques, 2005Co-Authors: Yirong Liu, Wentao HuangAbstract:In this paper, the bifurcation of limit cycles for a Cubic Polynomial system is investigated. By the computation of the singular point values, we prove that the system has 12 small amplitude limit cycles. The process of the proof is algebraic and symbolic.