47 results on '"Lyapunov constants"'
Search Results
2. Cyclicity of rigid centres on centre manifolds of three-dimensional systems.
- Author
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Pessoa, Claudio, Queiroz, Lucas, and Ribeiro, Jarne D.
- Subjects
LIMIT cycles ,POLYNOMIALS - Abstract
We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems
- Author
-
Fei Luo, Yundong Li, and Yi Xiang
- Subjects
piecewise smooth cubic system ,Lyapunov constants ,limit cycles ,focus-parabolic-type critical point ,Mathematics ,QA1-939 - Abstract
In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at (0,0). Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.
- Published
- 2024
- Full Text
- View/download PDF
4. Lower bounds for the number of limit cycles in a generalised Rayleighâ€"LiĂ©nard oscillator.
- Author
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Euzébio, Rodrigo D, Llibre, Jaume, and Tonon, Durval J
- Subjects
- *
LIMIT cycles , *EXISTENCE theorems - Abstract
In this paper a generalised Rayleighâ€"LiĂ©nard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply PoincarĂ©â€"Bendixson theorem for assure the existence of limit cycles. In particular, the algorithm for obtaining the limit cycles through the referred approach is explicitly exhibited. The main techniques applied in this study are the Lyapunov constants and the Melnikov method. The obtained results contemplate the simultaneity of limit cycles of small amplitude and medium amplitude, the former emerging from a weak focus equilibrium and the latter from homoclinic or heteroclinic saddle connections. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
5. The local cyclicity problem: Melnikov method using Lyapunov constants.
- Author
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Gouveia, Luiz F. S. and Torregrosa, Joan
- Abstract
In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so $\mathcal {M}(6) \geq 44$. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that $\mathcal {M}^{c}_{p}(4) \geq 43$ and $\mathcal {M}^{c}_{p}(5) \geq 65.$ [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. Lower bounds for the cyclicity of centers of quadratic three-dimensional systems.
- Author
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Gouveia, Luiz F.S. and Queiroz, Lucas
- Published
- 2024
- Full Text
- View/download PDF
7. Lower bounds for the local cyclicity for families of centers.
- Author
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Giné, Jaume, Gouveia, Luiz F.S., and Torregrosa, Joan
- Subjects
- *
LIMIT cycles , *POLYNOMIALS - Abstract
In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21] , to prove that there exists a family of cubic centers, labeled by C D 31 12 in [25] , with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
8. Lower bounds for the local cyclicity of centers using high order developments and parallelization.
- Author
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Gouveia, Luiz F.S. and Torregrosa, Joan
- Subjects
- *
VECTOR fields , *HOPF bifurcations , *LIMIT cycles , *POLYNOMIALS - Abstract
We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M (n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M (4) ≥ 20 , M (5) ≥ 33 , M (7) ≥ 61 , M (8) ≥ 76 , and M (9) ≥ 88. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
9. Limit cycles in a quartic system with a third-order nilpotent singular point
- Author
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Xinli Li
- Subjects
Quartic system ,Nilpotent critical point ,Lyapunov constants ,Bifurcation of limit cycles ,Mathematics ,QA1-939 - Abstract
Abstract In this paper, limit cycles bifurcating from a third-order nilpotent critical point in a class of quartic planar systems are studied. With the aid of computer algebra system MAPLE, the first 12 Lyapunov constants are deduced by the normal form method. As a result, sufficient and necessary center conditions are derived, and the fact that there exist 12 or 13 limit cycles bifurcating from the nilpotent critical point is proved by different perturbations. The result in [Qiu et al. in Adv. Differ. Equ. 2015(1):1, 2015] is improved.
- Published
- 2018
- Full Text
- View/download PDF
10. Higher Eigenvalues
- Author
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Cañada, Antonio, Villegas, Salvador, Bellomo, Nicola, Series editor, Benzi, Michele, Series editor, Jorgensen, Palle, Series editor, Li, Tatsien, Series editor, Melnik, Roderick, Series editor, Scherzer, Otmar, Series editor, Steinberg, Benjamin, Series editor, Reichel, Lothar, Series editor, Tschinkel, Yuri, Series editor, Yin, George, Series editor, Zhang, Ping, Series editor, Cañada, Antonio, and Villegas, Salvador
- Published
- 2015
- Full Text
- View/download PDF
11. A Variational Characterization of the Best Lyapunov Constants
- Author
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Cañada, Antonio, Villegas, Salvador, Bellomo, Nicola, Series editor, Benzi, Michele, Series editor, Jorgensen, Palle, Series editor, Li, Tatsien, Series editor, Melnik, Roderick, Series editor, Scherzer, Otmar, Series editor, Steinberg, Benjamin, Series editor, Reichel, Lothar, Series editor, Tschinkel, Yuri, Series editor, Yin, George, Series editor, Zhang, Ping, Series editor, Cañada, Antonio, and Villegas, Salvador
- Published
- 2015
- Full Text
- View/download PDF
12. Systems of Equations
- Author
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Cañada, Antonio, Villegas, Salvador, Bellomo, Nicola, Series editor, Benzi, Michele, Series editor, Jorgensen, Palle, Series editor, Li, Tatsien, Series editor, Melnik, Roderick, Series editor, Scherzer, Otmar, Series editor, Steinberg, Benjamin, Series editor, Reichel, Lothar, Series editor, Tschinkel, Yuri, Series editor, Yin, George, Series editor, Zhang, Ping, Series editor, Cañada, Antonio, and Villegas, Salvador
- Published
- 2015
- Full Text
- View/download PDF
13. Lower bounds for the number of limit cycles in a generalised Rayleigh–Liénard oscillator
- Author
-
Rodrigo D Euzébio, Jaume Llibre, and Durval J Tonon
- Subjects
Limit cycles ,Lyapunov constants ,Melnikov function ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Rayleigh Liénard oscillator ,Mathematical Physics - Abstract
In this paper a generalised Rayleigh–Liénard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply Poincaré–Bendixson theorem for assure the existence of limit cycles. In particular, the algorithm for obtaining the limit cycles through the referred approach is explicitly exhibited. The main techniques applied in this study are the Lyapunov constants and the Melnikov method. The obtained results contemplate the simultaneity of limit cycles of small amplitude and medium amplitude, the former emerging from a weak focus equilibrium and the latter from homoclinic or heteroclinic saddle connections.
- Published
- 2022
14. The local cyclicity problem: Melnikov method using Lyapunov constants
- Author
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Luiz Fernando Gouveia and Joan Torregrosa
- Subjects
Lyapunov constants ,Local cyclicity ,General Mathematics ,Melnikov theory - Abstract
In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so $\mathcal {M}(6) \geq 44$. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that $\mathcal {M}^{c}_{p}(4) \geq 43$ and $\mathcal {M}^{c}_{p}(5) \geq 65.$
- Published
- 2022
15. Hopf bifurcation in 3-dimensional polynomial vector fields
- Author
-
Sanchez Sanchez, Ivan, Torregrosa, Joan, Sanchez Sanchez, Ivan, and Torregrosa, Joan
- Abstract
In this work we study the local cyclicity of some polynomial vector fields in R3. In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees.
- Published
- 2022
16. Bi-center problem for some classes of [formula omitted]-equivariant systems.
- Author
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Romanovski, Valery G., Fernandes, Wilker, and Oliveira, Regilene
- Subjects
- *
PROBLEM solving , *SET theory , *EXISTENCE theorems , *ANALYSIS of variance , *SYSTEMS theory - Abstract
We investigate the simultaneous existence of two centers and their isochronicity for two families of planar Z 2 -equivariant differential systems. First, for a family studied by Liu and Li (2011) we present the necessary and sufficient conditions for the existence of an isochronous bi-center. Next, we give conditions for the existence of a bi-center and study its isochronicity for a planar Z 2 -equivariant quintic system having two weak foci or centers. We also give two examples of non-Hamiltonian cubic systems with three isochronous centers. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
17. Integrability conditions for complex kukles systems.
- Author
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Giné, Jaume and Valls, Claudia
- Subjects
- *
INTEGRALS , *SYSTEMS theory , *EXISTENCE theorems , *DYNAMICAL systems , *LYAPUNOV functions - Abstract
In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
18. A Class of Three-Dimensional Quadratic Systems with Ten Limit Cycles.
- Author
-
Du, Chaoxiong, Liu, Yirong, and Huang, Wentao
- Subjects
- *
QUADRATIC differentials , *LIMIT cycles , *MATHEMATICAL singularities , *MATHEMATICAL notation , *SINGULAR value decomposition , *HOPF bifurcations - Abstract
Our work is concerned with a class of three-dimensional quadratic systems with two symmetric singular points which can yield ten small limit cycles. The method used is singular value method, we obtain the expressions of the first five focal values of the two singular points that the system has. Both singular symmetric points can be fine foci of fifth order at the same time. Moreover, we obtain that each one bifurcates five small limit cycles under a certain coefficient perturbed condition, consequently, at least ten limit cycles can appear by simultaneous Hopf bifurcation. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
19. Lower bounds for the local cyclicity for families of centers
- Author
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Jaume Giné, Luiz F.S. Gouveia, Joan Torregrosa, Univ Lleida, Univ Autonoma Barcelona, Universidade Estadual Paulista (Unesp), and Ctr Recerca Matemat
- Subjects
Pure mathematics ,Class (set theory) ,Lyapunov constants ,Applied Mathematics ,Small-amplitude limit cycle ,010102 general mathematics ,Holomorphic function ,Higher order developments and parallelization ,Higher-order developments and parallelization ,Center (group theory) ,01 natural sciences ,010101 applied mathematics ,Polynomial vector field ,Quartic function ,Center cyclicity ,Vector field ,Limit (mathematics) ,0101 mathematics ,Cubic function ,Analysis ,Bifurcation ,Mathematics - Abstract
Made available in DSpace on 2021-06-25T12:30:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-02-25 Catalan AGAUR Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion European Community Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by C D-31(12) in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. (C) 2020 Elsevier Inc. All rights reserved. Univ Lleida, Dept Matemat, Avda Jaume II 69, Lleida 6925001, Catalonia, Spain Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Ctr Recerca Matemat, Campus Bellaterra, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Catalan AGAUR: 2017SGR1617 Catalan AGAUR: 2017SGR127 Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: MTM2017-84383-P Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: PID2019-104658GB-I00 European Community: H2020-MSCA-RISE-2017-777911 CNPq: 200484/2015-0 FAPESP: 2020/04717-0
- Published
- 2021
20. A method for characterizing nilpotent centers
- Subjects
Lyapunov constants ,Nilpotent center ,Degenerate center ,Bautin method - Published
- 2021
21. Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields
- Subjects
Lyapunov constants ,Parallelization ,Center cyclicity ,Planar polynomial vector field - Published
- 2021
22. Centers for the Kukles homogeneous systems with odd degree
- Subjects
Lyapunov constants ,Complex center-focus problem ,Bautin method - Published
- 2021
23. Centers for a class of generalized quintic polynomial differential systems
- Subjects
Lyapunov constants ,Nilpotent center ,Degenerate center ,Bautin method - Published
- 2021
24. Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields
- Author
-
Haihua Liang and Joan Torregrosa
- Subjects
Lyapunov function ,Lyapunov constants ,Applied Mathematics ,Computation ,Holomorphic function ,Parallelization ,Parallel computing ,Upper and lower bounds ,Planar polynomial vector field ,Combinatorics ,symbols.namesake ,Corollary ,Singularity ,Planar ,symbols ,Center cyclicity ,Hamiltonian (quantum mechanics) ,Analysis ,Mathematics - Abstract
Agraïments: The first author is supported by the NSF of China (No. 11201086, No.11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guang-dong Province, China (No. Yq2013107). Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms. To show the power of this approach, we study the cyclicity of the holomorphic center =iz z^2 z^3 z^n under general polynomial perturbations of degree n, for n 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n=4,5, 13, we prove that the cyclicity of the holomorphic center is at least n^2 n-2. This result give the highest lower bound for M(6), M(7), M(13) among the existing results, where M(n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6) 40, H(8) 70, and H(10) 108, because until now the best result was H(6) 39, H(8) 67, and H(10) 100.
- Published
- 2021
25. Lower bounds for the local cyclicity of centers using high order developments and parallelization
- Author
-
Luiz F.S. Gouveia, Joan Torregrosa, Univ Autonoma Barcelona, and Universidade Estadual Paulista (Unesp)
- Subjects
Hopf bifurcation ,Equilibrium point ,Polynomial ,Lyapunov constants ,Degree (graph theory) ,Applied Mathematics ,Small-amplitude limit cycle ,010102 general mathematics ,Degenerate energy levels ,Higher-order developments and parallelization ,Parallel computing ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Polynomial vector field ,symbols ,Order (group theory) ,Center cyclicity ,Vector field ,Limit (mathematics) ,0101 mathematics ,Analysis ,Mathematics - Abstract
Made available in DSpace on 2021-06-25T12:27:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-01-15 Catalan AGAUR Spanish Ministerio de Ciencia, Innovacion y Universidades - Agencia estatal de investigacion European Community Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M(n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M(4) >= 20, M(5) >= 33, M(7) >= 61, M(8) >= 76, and M(9) >= 88. (C) 2020 Elsevier Inc. All rights reserved. Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Catalan AGAUR: 2017SGR1617 Spanish Ministerio de Ciencia, Innovacion y Universidades - Agencia estatal de investigacion: MTM201677278-P European Community: H2020-MSCA-RISE-2017-777911 CNPq: 200484/2015-0 FAPESP: 2020/04717-0
- Published
- 2021
26. Centers for a class of generalized quintic polynomial differential systems
- Author
-
Claudia Valls, Jaume Llibre, and Jaume Giné
- Subjects
Discrete mathematics ,Computational Mathematics ,Polynomial ,Lyapunov constants ,Nilpotent center ,Degenerate center ,Degree (graph theory) ,Applied Mathematics ,Bautin method ,Differential systems ,Mathematics ,Quintic function - Abstract
We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C and either A = Re(D) = 0, or A = Im(D) = 0, or Re(A) = D = 0, or Im(A) = D = 0. The first author is partially supported by a MINECO/ FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2014SGR 568, ICREA Academia, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378. The third author is supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD.
- Published
- 2021
27. Hopf bifurcation in 3-dimensional polynomial vector fields
- Author
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Iván Sánchez-Sánchez and Joan Torregrosa
- Subjects
Hopf bifurcation ,Lyapunov function ,Lyapunov constants ,Numerical Analysis ,Work (thermodynamics) ,Applied Mathematics ,Hopf bifurcation in dimension three ,Mathematical analysis ,Degenerate energy levels ,Type (model theory) ,Quintic function ,Limit cycles ,symbols.namesake ,Modeling and Simulation ,Quartic function ,symbols ,Limit (mathematics) ,Mathematics - Abstract
In this work we study the local cyclicity of some polynomial vector fields in R 3 . In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R 3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees.
- Published
- 2022
28. Cyclicity versus Center Problem.
- Author
-
Gasull, Armengol and Giné, Jaume
- Abstract
We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate this phenomenon in several examples for which this cyclicity is computed. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
29. On the centers of the weight-homogeneous polynomial vector fields on the plane
- Author
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Llibre, Jaume and Pessoa, Claudio
- Subjects
- *
POLYNOMIALS , *VECTOR fields , *PLANE geometry , *LYAPUNOV functions , *MATHEMATICAL constants , *NUMERICAL analysis - Abstract
Abstract: We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
30. On persistent centers
- Author
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Cima, Anna, Gasull, Armengol, and Medrado, João C.
- Subjects
- *
DIFFERENTIAL equations , *BOUNDARY value problems , *PERIODIC functions , *LYAPUNOV functions , *MATHEMATICAL constants , *MATHEMATICAL proofs - Abstract
Abstract: For real planar autonomous analytic differential equations we introduce the notion of persistent center and show a list of equations with this property. We face the problem of whether our list is exhaustive or not and we prove that it is for several families of planar systems, like cubic or rigid systems. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
31. On the focus order of planar polynomial differential equations
- Author
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Qiu, Yu and Yang, Jiazhong
- Subjects
- *
DIFFERENTIAL equations , *POLYNOMIALS , *LYAPUNOV functions , *MATHEMATICAL constants , *MATHEMATICAL analysis , *NUMBER theory - Abstract
Abstract: This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where is a prime p or a power of a prime , and show that these systems can have a focus order ; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
32. Generalized Hopf bifurcation analysis of a towed caster wheel system.
- Author
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Wang, Fanrui, Liu, Haozhe, Wei, Zhouchao, and Moroz, Irene
- Subjects
- *
HOPF bifurcations , *DYNAMICAL systems , *WHEELS , *BIFURCATION diagrams , *COMPUTER simulation - Abstract
This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system, in which the non-smoothness factor is considered. The analysis can be divided into two cases by the coefficient of quadratic term. If the coefficient of quadratic term is equal to zero, the type of generalized Hopf bifurcation can be determined by the stability of first-order fine focus. Otherwise, we have to construct a Poincaré map for illustrating the type of generalized Hopf bifurcation, which is the heart of our discussion. Besides, we put forward another convenient way to judge the bifurcation types due to the symmetrical characteristic of this wheel system. Numerical simulations show the feasibility of theoretical analysis. • This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. • Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system. • The type of generalized Hopf bifurcation could be judged by the stability of first-order fine focus or Poincare map. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
33. A quartic system and a quintic system with fine focus of order 18
- Author
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Huang, Jing, Wang, Fang, Wang, Lu, and Yang, Jiazhong
- Subjects
- *
ALGORITHMS , *POLYNOMIALS , *QUARTIC surfaces , *QUINTIC equations - Abstract
Abstract: By using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems : , where is a homogeneous polynomial of degree n, in this note we construct two concrete examples, and , such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations: where , and where c is the root between of the equation [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
34. ON THE CENTERS OF PLANAR ANALYTIC DIFFERENTIAL SYSTEMS.
- Author
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GINÉ, JAUME
- Subjects
- *
EXTERIOR differential systems , *CALCULUS of variations , *DIFFERENTIABLE manifolds , *STOCHASTIC processes , *PATTERN recognition systems - Abstract
In this work we continue the study of the centers which are limits of linear type centers. It is proved that if a degenerate center has an inverse integrating factor V(x, y) with V(0, 0) ≠ 0, then this degenerate center is also the limit of linear type centers. Moreover, we show that the degenerate centers with characteristic directions that are the limits of degenerate centers without characteristic directions are also detectable, from a theoretical point of view, with the Bautin method. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
35. Hopf bifurcation in 3-dimensional polynomial vector fields.
- Author
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Sánchez-Sánchez, Iván and Torregrosa, Joan
- Subjects
- *
HOPF bifurcations , *LIMIT cycles , *POLYNOMIALS , *VECTOR fields - Abstract
In this work we study the local cyclicity of some polynomial vector fields in R 3. In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R 3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees. • High local cyclicity values of some polynomial vector fields in R 3. • Improvement of the current lower bound for the quadratic family. • First lower bounds for degrees 3, 4, and 5. • Implementation of a highly efficient parallelization approach. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
36. Limit cycles in a quartic system with a third-order nilpotent singular point
- Author
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Li, Xinli
- Published
- 2018
- Full Text
- View/download PDF
37. Centers for generalized quintic polynomial differential systems
- Author
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Jaume Giné, Claudia Valls, and Jaume Llibre
- Subjects
Polynomial (hyperelastic model) ,Discrete mathematics ,Lyapunov constants ,Nilpotent center ,37C10 ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,34C05 ,Differential systems ,01 natural sciences ,Quintic function ,010101 applied mathematics ,Combinatorics ,Degenerate center ,degenerate center ,Bautin method ,0101 mathematics ,Matemàtica ,Mathematics - Abstract
We classify the centers of polynomial differential systems in $R^2$ of odd degree $d \ge 5$, in complex notation, as $\dot{z} = iz + (z \bar z)^(d-5)/2(A z^5 + B z^4 \bar z + C z^3 \bar z^2 + D z^2 \bar z^3 + E z \bar z^4 + F \bar z^5)$, where $A,B,C,D,E, F \in mathbb{C}$ and either $A = Re(D) = 0$, $A = Im(D) = 0$, $Re(A) = D = 0$ or $Im(A) = D = 0$. We thank to Professor Colin Christopher for his help in the proof of statement (g) of Theorem 3. The first author is partially supported by a MINECO/ FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2009SGR 410, ICREA Academia, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDERUNAB10-4E-378. The third author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013.
- Published
- 2017
38. Weak center problem and bifurcation of critical periods for a Z 4 -equivariant cubic system
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Du, Chaoxiong, Liu, Yirong, and Liu, Canhui
- Published
- 2013
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39. A method for characterizing nilpotent centers
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Jaume Llibre and Jaume Giné
- Subjects
Pure mathematics ,Lyapunov constants ,Nilpotent center ,Applied Mathematics ,010102 general mathematics ,Annulus (mathematics) ,Singular point of a curve ,01 natural sciences ,Effective algorithm ,010101 applied mathematics ,Algebra ,Nilpotent ,Degenerate center ,Liapunov constants ,Bautin method ,Center (algebra and category theory) ,Limit (mathematics) ,0101 mathematics ,Focus (optics) ,Analysis ,Bifurcation ,Mathematics - Abstract
To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and after for studying the bifurcation of limit cycles from it or from its period annulus. We give an effective algorithm in the search of necessary conditions for detecting nilpotent centers based in recent developments. Moreover we survey the last results on this problem and illustrate our approach by means of examples. The first author is partially supported by a MINECO/ FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2009SGR 381. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2009SGR 410, ICREA Academia and two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999.
- Published
- 2014
40. On persistents centers
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ROCHA, Valdomiro and MEDRADO, João Carlos da Rocha
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problema centro-foco ,Equações diferenciais ,ponto singular ,constantes de Lyapunov ,center-focus problem ,Differential Equations ,singular point ,Lyapunov constants ,CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA [CNPQ] - Abstract
O problema de decidir se um ponto singular monodrômico com autovalores imaginários para uma família analítica de um campo de vetores planares é um centro ou um foco foi resolvido por Lyapunov. Este é o famoso problema centro- foco, que foi resolvido calculando as chamadas constantes de Lyapunov e verificar se elas são ou não nulas. Existem métodos diferentes de calculá-las dependendo da aproximação a ser utilizada: cálculo da função de Lyapunov; uso de formas normais; cálculo da potência na expansão da solução em coordenadas polares; uso da estrutura algébrica das constantes de Lyapunov; método de Lyapunov-Schmit e funções de Melnikov. Apesar de todos os métodos acima o problema centro-foco para uma família simples, como a cúbica, tem resisitido a todas as tentativas de solução, por isto classificamos os centros em três níveis para tornar o problema mais viável. The problem of destingnishing whether a monodromic critical point with imaginary eigenvalues of a family of a planar analitical vector field is a center or a focus was already solved by Lyapunov. This is the famous center-focus problem which was solved by calculating the so-called Lyapunov constants and see whether or not they are zero. We present a few ways to calculate them acording the approaches that they use: camputation of a Lyapunov function; use of normal forms; computation of the power of expansion of a solution in polar coordinates; use of the algebraic structure of Lyapunov constants; method of Lyapunov-Schmit and Melnikov functions. Despite all of the above the centerfocus problem for a simple family as the cube is resisting all attempts at solution. For this reason the centers, we propose to grade the in three levels in order to make the problem more feasible.
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- 2010
41. A quartic system and a quintic system with fine focus of order 18
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Fang Wang, Jing Huang, Lu Wang, and Jiazhong Yang
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Polynomial ,Mathematics(all) ,Lyapunov constants ,Differential equation ,General Mathematics ,Center (category theory) ,Geometry ,Center–focus ,Quintic function ,Homogeneous polynomial ,Ordinary differential equation ,Quartic function ,Order (group theory) ,Quintic systems ,Quartic systems ,Fine focus ,Mathematics ,Mathematical physics - Abstract
By using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems En: z˙=iz+Rn(z,z¯), where Rn is a homogeneous polynomial of degree n, in this note we construct two concrete examples, E4 and E5, such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations: E4:z˙=iz+2iz4+izz¯3+5227820723eiθz¯4, where θ∉{kπ±π6,kπ+π2,k∈Z}, and E5:z˙=iz+3z5+20(c+3)9c2−15z4z¯+zz¯4+20(c+3)c29c2−15z¯5, where c is the root between (−3,−5/3) of the equation 4155c6−10716c5−63285c4−18070c3+168075c2+205450c+60375=0.
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42. On persistent centers
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Anna Cima, Joao C. Medrado, and Armengol Gasull
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Algebra ,Mathematics(all) ,Lyapunov constants ,Property (philosophy) ,Planar ,Differential equation ,General Mathematics ,Face (geometry) ,Calculus ,Center (algebra and category theory) ,Center-focus problem ,Melnikov functions ,Mathematics - Abstract
For real planar autonomous analytic differential equations we introduce the notion of persistent center and show a list of equations with this property. We face the problem of whether our list is exhaustive or not and we prove that it is for several families of planar systems, like cubic or rigid systems.
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43. On the centers of the weight-homogeneous polynomial vector fields on the plane
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Jaume Llibre and Claudio Pessoa
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Polynomial ,Lyapunov constants ,Invariant polynomial ,Degree (graph theory) ,Plane (geometry) ,Centers ,Applied Mathematics ,Mathematical analysis ,Dynamical Systems (math.DS) ,Reciprocal polynomial ,Planar ,Weight-homogeneous polynomial differential systems ,Homogeneous polynomial ,FOS: Mathematics ,Vector field ,Mathematics - Dynamical Systems ,34C35, 58F09, 34D30 ,Analysis ,Mathematics - Abstract
We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.
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44. Cyclicity versus center problem
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Armengol Gasull and Jaume Giné
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Lyapunov constants ,Focal quantities ,Differential equation ,Applied Mathematics ,Polynomial differential system ,Trivial solution ,Phenomenon ,Calculus ,Discrete Mathematics and Combinatorics ,Center (algebra and category theory) ,Cyclicity ,Polinomis ,Focus (optics) ,Mathematics - Abstract
We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate this phenomenon in several examples for which this cyclicity is computed.
45. Centers for the Kukles homogeneous systems with odd degree
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Jaume Giné, Jaume Llibre, and Claudia Valls
- Subjects
Discrete mathematics ,Polynomial ,Lyapunov constants ,Degree (graph theory) ,General Mathematics ,Complex center-focus problem ,010102 general mathematics ,Center (group theory) ,Integrability ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Homogeneous ,Homogeneous polynomial ,Bautin method ,Vector field ,0101 mathematics ,Matemàtica ,Mathematics - Abstract
For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n ≥ 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all n odd. The first author is partially supported by a MINECO/FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/FEDER grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014 SGR568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378. The third author is supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD
46. Integrability conditions for complex kukles systems
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Claudia Valls and Jaume Giné
- Subjects
010101 applied mathematics ,Combinatorics ,Pure mathematics ,Lyapunov constants ,General Mathematics ,010102 general mathematics ,First integrals ,0101 mathematics ,Integrability ,01 natural sciences ,Computer Science Applications ,Mathematics ,Complex centre-focus problem - Abstract
In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems. The first author is partially supported by a MINECO/ FEDER [grant number MTM2014-53703-P]; an AGAUR (Generalitat de Catalunya) [grant number 2014SGR 1204]. The second author is supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD.
47. CENTERS FOR GENERALIZED QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS
- Author
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GINÉ, JAUME, LLIBRE, JAUME, and VALLS, CLAUDIA
- Published
- 2017
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