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Jaume Llibre - One of the best experts on this subject based on the ideXlab platform.
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Algebraic and topological classification of Homogeneous quartic vector fields in the plane
Annali di Matematica Pura ed Applicata (1923 -), 2021Co-Authors: Jaume Llibre, Y. Paulina Martínez, Claudio VidalAbstract:We provide canonical forms for the Homogeneous Polynomials of degree five. Then we characterize all the phase portraits in the Poincaré disk for all quartic Homogeneous Polynomial differential systems. More precisely, there are exactly 23 different topological phase portraits for the quartic Homogeneous Polynomial differential systems.
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Centers of discontinuous piecewise smooth quasi–Homogeneous Polynomial differential systems
Discrete & Continuous Dynamical Systems - B, 2019Co-Authors: Hebai Chen, Jaume Llibre, Yilei TangAbstract:In this paper we investigate the center problem for the discontinuous piecewise smooth quasi–Homogeneous but non–Homogeneous Polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi–Homogeneous Polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi–Homogeneous cubic and quartic Polynomial differential systems.
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centers of weight Homogeneous Polynomial vector fields on the plane
Proceedings of the American Mathematical Society, 2016Co-Authors: Jaume Gine, Jaume Llibre, Claudia VallsAbstract:We characterize all centers of a planar weight-Homogeneous Polynomial vector fields. Moreover we classify all centers of a planar weight-Homogeneous Polynomial vector fields of degrees $6$ and $7$.
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global phase portraits of kukles differential systems with Homogeneous Polynomial nonlinearities of degree 6 having a center and their small limit cycles
International Journal of Bifurcation and Chaos, 2016Co-Authors: Jaume Llibre, Mauricio Fronza Da SilvaAbstract:We provide the nine topological global phase portraits in the Poincare disk of the family of the centers of Kukles Polynomial differential systems of the form ẋ = −y, ẏ = x + ax5y + bx3y3 + cxy5, where x,y ∈ ℝ and a,b,c are real parameters satisfying a2 + b2 + c2≠0. Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all Homogeneous Polynomial differential systems of degree 6, and second inside the class of all Polynomial differential systems of degree 6.
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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.
Claudio Altafini - One of the best experts on this subject based on the ideXlab platform.
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CDC - Homogeneous Polynomial Forms for Simultaneous Stabilizability of Families of Linear Control Systems: a Tensor Product Approach
Proceedings of the 45th IEEE Conference on Decision and Control, 2006Co-Authors: Claudio AltafiniAbstract:The paper uses the formalism of tensor products in order to deal with the problem of simultaneous stabilizability of a family of linear control systems by means of Lyapunov functions which are Homogeneous Polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of Homogeneous Polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.
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Homogeneous Polynomial Forms for Simultaneous Stabilizability of Families of Linear Control Systems: A Tensor Product Approach
IEEE Transactions on Automatic Control, 2006Co-Authors: Claudio AltafiniAbstract:This note uses the formalism of tensor products in order to deal with the problem of simultaneous stabilizability of a family of linear control systems by means of Lyapunov functions which are Homogeneous Polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of Homogeneous Polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability
Cristóbal García - One of the best experts on this subject based on the ideXlab platform.
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centers of quasi Homogeneous Polynomial planar systems
Nonlinear Analysis-real World Applications, 2012Co-Authors: Antonio Algaba, N Fuentes, Cristóbal GarcíaAbstract:Abstract In this paper we determine the centers of quasi-Homogeneous Polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers. We find Polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.
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Monodromy, center–focus and integrability problems for quasi-Homogeneous Polynomial systems
Nonlinear Analysis-theory Methods & Applications, 2010Co-Authors: Antonio Algaba, Emilio Freire, E. Gamero, Cristóbal GarcíaAbstract:Abstract This paper deals with planar quasi-Homogeneous Polynomial vector fields, and addresses three major questions: the monodromy, the center–focus and the integrability problems. We characterize the monodromic planar quasi-Homogeneous Polynomial vector fields, and we give a condition to distinguish between a center and a focus in this case. Also, we provide conditions which characterize the integrability of quasi-Homogeneous Polynomial systems under non-resonance conditions. The results obtained allow us to analyse two monodromic planar systems with degenerate linear part: one of them with nilpotent linearization, and another one with null linear part.
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monodromy center focus and integrability problems for quasi Homogeneous Polynomial systems
Nonlinear Analysis-theory Methods & Applications, 2010Co-Authors: Antonio Algaba, Emilio Freire, E. Gamero, Cristóbal GarcíaAbstract:Abstract This paper deals with planar quasi-Homogeneous Polynomial vector fields, and addresses three major questions: the monodromy, the center–focus and the integrability problems. We characterize the monodromic planar quasi-Homogeneous Polynomial vector fields, and we give a condition to distinguish between a center and a focus in this case. Also, we provide conditions which characterize the integrability of quasi-Homogeneous Polynomial systems under non-resonance conditions. The results obtained allow us to analyse two monodromic planar systems with degenerate linear part: one of them with nilpotent linearization, and another one with null linear part.
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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Centers of discontinuous piecewise smooth quasi-Homogeneous Polynomial differential systems
2021Co-Authors: Chen Hebai, Llibre Jaume, Tang YileiAbstract:In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-Homogeneous but non-Homogeneous Polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi-Homogeneous Polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi-Homogeneous cubic and quartic Polynomial differential systems
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Polynomial and rational first integrals for planar quasi-Homogeneous Polynomial differential sytems
2021Co-Authors: Giné Jaume, Grau Maite, Llibre JaumeAbstract:In this paper we find necessary and sufficient conditions in order that a planar quasi-Homogeneous Polynomial differential system has a Polynomial or a rational first integral. We also prove that any planar quasi-Homogeneous Polynomial differential system can be transformed into a differential system of the form u˙ = uf(v), ˙v = g(v) with f(v) and g(v) Polynomials, and vice versa
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Centers of quasi-Homogeneous Polynomial differential equations of degree three
2021Co-Authors: Aziz Waleed, Llibre Jaume, Pantazi CharaAbstract:Agraïments: W. Aziz is financially supported by Ministry of Higher Education and Scientific Research-Iraq.We characterize the centers of the quasi-Homogeneous planar Polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic Homogeneous Polynomial system using the averaging theory of first order
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Global phase portraits of uniform isochronous centers with quartic Homogeneous Polynomial nonlinearities
2021Co-Authors: Itikawa Jackson, Llibre JaumeAbstract:Agraïments: The first author is is supported by a Ciência sem Fronteiras-CNPq grant number 201002/ 2012-4. A CAPES grant number 88881.030454/2013-01 from the program CSF-PVEWe classify the global phase portraits in the Poincar\'e disc of the differential systems =-y xf(x,y), =x yf(x,y), where f(x,y) is a Homogeneous Polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in IL2 completes the classification of the global phase portraits in the Poincar\'e disc of all quartic Polynomial differential systems with a uniform isochronous center at the origin
Xian Zhang - One of the best experts on this subject based on the ideXlab platform.
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Asynchronous Filtering for Delayed Markovian Jump Systems via Homogeneous Polynomial Approach
IEEE Transactions on Automatic Control, 2020Co-Authors: Xian Zhang, Chunhua YangAbstract:In this article, the asynchronous filtering scheme for Markovian jump systems (MJSs) subjected to time-varying delays and infinite distributed delays, is investigated based on the Homogeneous Polynomial approach. First, sufficient conditions are proposed to ensure that the filtering error system is exponentially stabile in mean square and satisfies a given performance index simultaneously. Second, the asynchronous filter is synthesized for MJSs with certain transition probability. Moreover, one so-called Homogeneous Polynomial method is developed to design the asynchronous filter in the case of uncertain transition probabilities. During the analysis, a Homogeneous Polynomial matrix is introduced in the parameter-dependent Lyapunov function that can reduce the conservatism of the results. Finally, a numerical example is given to support the feasibility and effectiveness of the proposed theory.
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exponential stability analysis for delayed semi markovian recurrent neural networks a Homogeneous Polynomial approach
IEEE Transactions on Neural Networks, 2018Co-Authors: Xian Zhang, Chunhua Yang, Weihua GuiAbstract:This paper investigates the exponential stability analysis issue for a class of delayed recurrent neural networks (RNNs) with semi-Markovian parameters. By constructing a stochastic Lyapunov functional and using some zoom techniques to estimate its weak infinitesimal operator, the exponential mean square stability criteria have been proposed for the Markovian neural networks with certain transition probabilities. We then generalize the Homogeneous Polynomial approach for the delayed Markovian RNNs with uncertain transition probabilities during the stability analysis. Theoretical results have obtained by introducing an appropriate technique for dealing with a large number of complex Homogeneous Polynomial matrix inequalities. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed technique.
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Stability Analysis for Discrete-Time Markovian Jump Systems With Time-Varying Delay: A Homogeneous Polynomial Approach
IEEE Access, 2017Co-Authors: Xin Li, Xian Zhang, Xin WangAbstract:This paper is concerned with the stochastic stability problem for discrete-time Markovian jump systems (DTMJSs) with time-varying delays. Two cases are discussed in the main results. On the one hand, it is assumed that the transition probability can be known. In this case, by constructing the Lyapunov-Krasovskii functional and using some summation inequalities to estimate its forward difference, a new stochastic stability criterion of DTMJSs can be obtained, which has less conservatism than existing results. On the other hand, by utilizing the Homogeneous Polynomial approach, the novel delay-dependent stability condition of DTMJSs with uncertain transition probability is proposed. Finally, the results of numerical simulations demonstrate the effectiveness of the proposed methods.