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1.
18 p, 887.5 KB Nilpotent bi-center in continuous piecewise Z2-equivariant cubic polynomial Hamiltonian systems / Chen, Ting (Guangdong University of Finance and Economics. School of Statistics and Mathematics) ; Li, Shimin (Guangdong University of Finance and Economics. School of Statistics and Mathematics) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
One of the classical and difficult problems in the theory of planar differential systems is to classify their centers. Here we classify the global phase portraits in the Poincaré disk of the class continuous piecewise differential systems separated by one straight line and formed by two cubic Hamiltonian systems with nilpotent bi-center at (± 1, 0). [...]
2022 - 10.1007/s11071-022-07631-z
Nonlinear Dynamics, Vol. 110, Issue 1 (September 2022) , p. 705-721  
2.
24 p, 1.3 MB Nilpotent center in a continuous piecewise quadratic polynomial Hamiltonian vector field / Chen, Ting (Guangdong University of Finance and Economics. School of Statistics and Mathematics (China)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line x = 0, where these kinds of systems have a nilpotent center at (0, 0), which comes from the combination of two cusps of both Hamiltonian systems. [...]
2022 - 10.1142/S0218127422501164
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 32, Issue 8 (June 2022) , art. 2250116  
3.
37 p, 678.9 KB Z2-symmetric planar polynomial Hamiltonian systems of degree 3 with nilpotent centers / Dias, Fabio Scalco (Universidade Federal de Itajubá. Instituto de Matemática e Computação (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)
We provide the normal forms and the global phase portraits in the Poincaré disk of all Z2-symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin.
2019
Electronic journal of differential equations, Vol. 2019, Issue 82 (2019) , p. 1-29
2 documents
4.
14 p, 298.1 KB Nilpotent global centers of linear systems with cubic homogeneous nonlinearities / García-Saldaña, Johanna Denise (Universidad Católica de la Santísima Concepción. Departamento de Matemática y Física Aplicadas (Chile)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
In this paper, we characterize the global nilpotent centers of polynomial differential systems of the linear form plus cubic homogeneous terms.
2020 - 10.1142/S0218127420500108
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 30, Issue 1 (January 2020) , art. 2050010  
5.
16 p, 322.4 KB The center problem for Z2-symmetric nilpotent vector fields / Algaba, Antonio (Universidad de Huelva. Departamento de Ciencias Integradas) ; García, Cristóbal (Universidad de Huelva. Departamento de Ciencias Integradas) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z-symmetric if P(−x,−y)=−P(x,y) and Q(−x,−y)=−Q(x,y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C first integral. [...]
2018 - 10.1016/j.jmaa.2018.05.079
Journal of mathematical analysis and applications, Vol. 466, Issue 1 (October 2018) , p. 183-198  
6.
29 p, 409.2 KB Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers / Dias, Fabio Scalco (Universidade Federal de Itajubá(Brazil). Instituto de Matemática e Computacâo) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the x-axis having a nilpotent center at the origin.
2018 - 10.1016/j.matcom.2017.06.002
Mathematics and computers in simulation, Vol. 144 (Feb. 2018) , p. 60-77  
7.
21 p, 834.1 KB Centers for generalized quintic polynominal differential systems / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matematica)
2017 - 10.1216/RMJ-2017-47-4-1097
The Rocky Mountain Journal of Mathematics, Vol. 47 Núm. 4 (2017) , p. 1097-1120  
8.
15 p, 383.0 KB Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields / Colak, Ilker (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
Following the work done in [8] we provide the bifurcation diagrams for the global phase portraits in the Poincaré disk of all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.
2017 - 10.1016/j.jde.2017.02.001
Journal of differential equations, Vol. 262 (2017) , p. 5518-5533  
9.
24 p, 13.1 MB Reversible nilpotent centers with cubic homogeneous nonlinearities / Dukarić, Maša (University of Maribor(Slovenia). Center for Applied Mathematics and Theoretical Physics) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We provide 13 non-topological equivalent classes of global phase portraits in the Poincaré disk of reversible cubic homogeneous systems with a nilpotent center at origin, which complete the classification of the phase portraits of the nilpotent centers with cubic homogeneous nonlinearities.
2016 - 10.1016/j.jmaa.2015.07.049
Journal of mathematical analysis and applications, Vol. 433 (2016) , p. 305-319  
10.
6 p, 556.2 KB Analytic nilpotent centers as limits of nondegenerate centers revisited / García, Isaac (Universitat de Lleida. Departament de Matemàtica) ; Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers.
2016 - 10.1016/j.jmaa.2016.04.046
Journal of mathematical analysis and applications, Vol. 441 (2016) , p. 893-899  

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