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1.
30 p, 951.8 KB On the configurations of centers of planar Hamiltonian Kolmogorov cubic polynomial differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Xiao, Dongmei (Shanghai Jiao Tong University. School of Mathematical Sciences (China))
We study the kind of centers that Hamiltonian Kolmogorov cubic polynomial differential systems can exhibit. Moreover, we analyze the possible configurations of these centers with respect to the invariant coordinate axes, and obtain that the real algebraic curve xy(a+bx+cy+dx2+exy+fy2)=h has at most four families of level ovals in R2 for all real parameters a,b,c,d,e,f and h.
2020 - 10.2140/pjm.2020.306.611
Pacific Journal of Mathematics, Vol. 306, Issue 2 (2020) , p. 611-644  
2.
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
3.
10 p, 661.5 KB On the integrability of Hamiltonian systems with d degrees of freedom and homogenous polynomial potential of degree n / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Instituto Superior Técnico. Departamento de Matemática)
We consider Hamiltonian systems with d degrees of freedom and a Hamiltonian of the form H = 1/2 d∑i=1 p21+V(q1,. . . ,qd), where V is a homogenous polynomial of degree n ≥ 3. We prove that such Hamiltonian systems with n odd or n = 4m, have a Darboux first integral if and only if they have a polynomial first integral.
2018 - 10.1142/S0219199717500456
Communications in Contemporary Mathematics, Vol. 20, Issue 8 (December 2018) , art. 1750045  
4.
7 p, 285.2 KB New classes of polynomial maps satisfying the real Jacobian conjecture in R2 / Itikawa, Jackson (Universidade Federal de Rondônia. Departamento de Matemática (Brazil)) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We present two new classes of polynomial maps satisfying the real Jacobian conjecture in ℝ2. The first class is formed by the polynomials maps of the form (q(x) - p(y),q(y) + p(x)): ℝ2 → ℝ2 such that p and q are real polynomials satisfying p'(x)q'(x) ≠ 0. [...]
2019 - 10.1590/0001-3765201920170627
Anais da Academia Brasileira de Ciencias, Vol. 91, Issue 2 (2019) , art. e20170627
2 documents
5.
Global phase portraits of ℤ2-symmetric planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin / Corbera Subirana, Montserrat (Universitat de Vic, Universitat Central de Catalunya (UVic-UCC)) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemàtica)
We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and ℤ2-symmetric with (x,y)→(-x,y).
2020 - 10.1142/S0218127420500066
International Journal of Bifurcation and Chaos, Vol. 30, Issue 1 (January 2020) , art. 2050006  
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.
36 p, 1.2 MB Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Martinez Mancilla, Yohanna Paulina (Universidad del Bío-Bío. Departamento de Matemática) ; Vidal, Claudio (Universidad del Bío-Bío. Departamento de Matemática)
We provide normal forms and the phase portraits in the Poincaré disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis.
2018 - 10.3934/dcdsb.2018047
Discrete and continuous dynamical systems. Series B, Vol. 23, issue 2 (2018) , p. 887-912  
8.
14 p, 708.3 KB On the integrability of the 5-dimensional Lorenz system for the gravity-wave activity / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
We consider the 5-dimensional Lorenz system \[ U' &= -V W b V Z, \\ V' &= UW-b UZ, \\ W'&= -U V,\\ X' &= -Z, \\ Z'&=b UV X \] where b \R \0\ and the derivative is with respect to T. [...]
2017
Proceedings of the American Mathematical Society, Vol. 145 Núm. 2 (2017) , p. 665-679  
9.
13 p, 669.2 KB Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems / Braun, Francisco (Universidade Federal de Sâo Carlos Rod(Brasil). Departamento de Matemática) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Mereu, Ana Cristina (UFSCar(Brazil). Department of Physics, Chemistry and Mathematics)
In this paper we completely characterize trivial isochronous centers of degrees 5 and 7. Precisely, we provide formulas, up to linear change of coordinates, for the Hamiltonian H of the isochronous centers such that H =(f_1^2 f_2^2)/2 has degrees 6 and 8, and f = (f_1, f_2): R^2 R^2 is a polynomial map with D f = 1 and f(0,0) = (0,0).
2016 - 10.3934/dcds.2016029
Discrete and continuous dynamical systems. Series A, Vol. 36 Núm. 10 (2016) , p. 5245-5255  
10.
14 p, 706.0 KB Darboux integrability of 2-dimensional Halmiltonian systems with homogeneous potentials of degree 3 / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade Técnica de Lisboa. Departamento de Matemática)
We provide a characterization of all Hamiltonian systems of the form H = (p21 + p22)/2 + V (q1, q2) where V is a homogenous polynomial of degree 3 which are completely integrable with Darboux first integrals.
2014 - 10.1063/1.4868701
Journal of Mathematical Physics, Vol. 55 (2014) , p. 33507  

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