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13 p, 663.8 KB |
Phase portraits of uniform isochronous centers with homogeneous nonlinearities
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Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática (Portugal))
We classify the phase portraits in the Poincaré disc of the differential equations of the form x' = − y + xf(x, y), ẏ = x + yf(x, y) where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2, 3, 4, 5, and f has only simple zeroes. [...]
2021 -  10.1007/s10883-021-09529-2
Journal of Dynamical and Control Systems, Vol. 28 (February 2021) , p. 319-332
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17 p, 362.7 KB |
A new result on averaging theory for a class of discontinuous planar differential systems with applications
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Itikawa, Jackson (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Novaes, Douglas D. (Universidade Estadual de Campinas. Departamento de Matemática (Brazil))
We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form dr/dθ= r'={F+(θ, r, ϵ) if 0≤ θ ≤ α, F-(θ, r, ϵ) if α ≤ θ ≤ 2π, where F±(θ, r, ϵ) = Σk i=1 ϵiF± i (θ, r) + ϵk+1R ± (θ, r, ϵ) with θ ϵ S and r ϵ D, where D is an open interval of ℝ+, and ϵ is a small real parameter. [...]
2017 - 10.4171/rmi/970
Revista Matemática Iberoamericana, Vol. 33, Issue 4 (2017) , p. 1247-1265
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14 p, 332.1 KB |
Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones
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Itikawa, Jackson (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação. Departamento de Matemática (Brazil)) ;
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Mereu, Ana Cristina (Universidade Federal de São Carlos. Departamento de Física, Química e Matemática (Brazil)) ;
Oliveira, Regilene (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação. Departamento de Matemática (Brazil))
We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems ẋ = -y+x, y = x + xy, and ẋ = -y + xy, y = x + xy, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. [...]
2017 - 10.3934/dcdsb.2017136
Discrete and continuous dynamical systems. Series B, Vol. 22, Issue 9 (November 2017) , p. 3259-3272
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15 p, 625.2 KB |
Centers and uniform isochronous centers of planar polynomial differential systems
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Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Ramírez, Rafael Orlando (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques) ;
Ramírez, Valentín (Universitat de Barcelona) ;
Sadovskaia, Natalia (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II)
For planar polynomial vector fields of the form \[ (-y X(x,y)) x (x Y(x,y)) y, \] where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.
2018 - 10.1007/s10884-018-9672-0
Journal of dynamics and differential equations, Vol. 30, issue 3 (Sep. 2018) , p. 1295-1310
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37 p, 871.8 KB |
An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities
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Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Ramírez, Rafael Orlando (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques) ;
Ramírez, Valentín (Universitat de Barcelona)
We consider polynomial vector fields of the form \[ \X=(-y X_m) x (x Y_m) y, \] where X_m=X_m(x,y) and Y_m=Y_m(x,y) are homogenous polynomials of degree m. It is well--known that \X has a center at the origin if and only if \X has an analytic first integral of the form \[ H=12(x^2 y^2) _j=3^ H_j, \] where H_j=H_j(x,y) is a homogenous polynomial of degree j. [...]
2017 - 10.1016/j.jde.2017.04.030
Journal of differential equations, Vol. 263 (2017) , p. 3327-3369
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13 p, 310.1 KB |
Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities
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Itikawa, Jackson (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ;
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We 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. [...]
2016 - 10.3934/dcdsb.2016.21.121
Discrete and continuous dynamical systems. Series B, Vol. 21 Núm. 1 (2016) , p. 121-131
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