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Journal Article
Journal of Differential Equations, ISSN 0022-0396, 04/2016, Volume 260, Issue 7, pp. 5726 - 5760
We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles.... 
Polynomial vector fields | Algebraic limit circles | Planar polynomial differential system | 16th Hilbert's problem | Darboux integrability | Invariant algebraic circles
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 2011, Volume 251, Issue 7, pp. 1778 - 1789
For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound... 
Simple version of the 16th Hilbert problem | Algebraic limit cycles | Holomorphic singular foliations | Polynomial differential systems | Simple version of the 16th hilbert problem | INTEGRALS | MATHEMATICS | HOLOMORPHIC FOLIATIONS | POLYNOMIAL VECTOR-FIELDS | CURVES
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 2006, Volume 221, Issue 2, pp. 309 - 342
The weak Hilbert 16th problem for n = 2 was solved by Horozov and Iliev (Proc. London Math. Soc. 69 (1994) 198–244), Zhang and Li (Adv. in Math. 26 (1997)... 
Centroid curve | Abelian integral | Deformation argument | Weak Hilbert 16th problem | NUMBER | PERTURBATIONS | SEGMENT | QUADRATIC HAMILTONIAN-SYSTEMS | PERIOD ANNULUS | deformation argument | weak Hilbert 16th problem | MATHEMATICS | CYCLICITY | centroid curve | SADDLE-LOOP | ABELIAN-INTEGRALS | LIMIT-CYCLES | ZEROS
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 2010, Volume 248, Issue 6, pp. 1401 - 1409
For a polynomial planar vector field of degree n ⩾ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (... 
Polynomial vector fields | Limit cycles | 16th Hilbert problem | Algebraic limit cycles | MATHEMATICS | INTEGRABILITY | POLYNOMIAL VECTOR-FIELDS | SYSTEMS | CURVES | Teoria de cossos i polinomis | Problems, exercises, etc | Algebra | Àlgebra | Matemàtica | Matemàtiques i estadística | Problemes, exercicis, etc | Polynomials | Mathematics | Àrees temàtiques de la UPC | Polinomis
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 2011, Volume 250, Issue 2, pp. 983 - 999
Journal Article
by Yu, P and Corless, R
Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 12/2009, Volume 14, Issue 12, pp. 4041 - 4056
Journal Article
by YU, P and HAN, M
International Journal of Bifurcation and Chaos, ISSN 0218-1274, 07/2005, Volume 15, Issue 7, pp. 2191 - 2205
In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in... 
The 16th Hilbert problem | Hopf bifurcation | Limit cycle | Normal form | Planar system | limit cycle | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MULTIDISCIPLINARY SCIENCES | planar system | normal form | the 16th Hilbert problem
Journal Article
Electronic Journal of Differential Equations, ISSN 1072-6691, 05/2014, Volume 2014, Issue 120, pp. 1 - 22
In this article, we study the bifurcation of limit cycles from the harmonic oscillator x = y, y = -x in the system x = y, y = -x + epsilon f(y) (1 - x(2)),... 
Sinusoidal-type number | Generalized Van der Pol equation | Hilbert's 16th problem | Sinusoidal-type set | Dependent radius | Lambda-point | Limit cycle | Existence | MATHEMATICS, APPLIED | sinusoidal-type set | NUMBER | existence | MATHEMATICS | limit cycle | LIENARD EQUATIONS | sinusoidal-type number | LIMIT-CYCLES | dependent radius | lambda-point
Journal Article
Qualitative Theory of Dynamical Systems, ISSN 1575-5460, 2012, Volume 11, Issue 1, pp. 61 - 77
This survey on the Hopf bifurcation (in a broad sense) has been written in order to clarify some aspects in the light of recent progresses made in the local... 
Local Hilbert's 16th problem | Dynamical and singular Hopf bifurcation | Poincaré-Andronov-Hopf bifurcation | Bautin theory | MATHEMATICS | MATHEMATICS, APPLIED | Poincare-Andronov-Hopf bifurcation
Journal Article
Journal of Applied Analysis and Computation, ISSN 2156-907X, 2015, Volume 5, Issue 1, pp. 141 - 145
The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form dr/d0 =... 
Periodic orbit | Hilbert number | Trigonometric polynomial | Averaging theory | MATHEMATICS, APPLIED | averaging theory | trigonometric polynomial | 16TH PROBLEM
Journal Article
International Journal of Bifurcation and Chaos, ISSN 0218-1274, 03/2008, Volume 18, Issue 3, pp. 877 - 884
Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of... 
Cycles | Quadratic system | Hilbert's 16th problem | Lienard equation | quadratic system | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MULTIDISCIPLINARY SCIENCES | cycles | LIMIT-CYCLES | Mathematics - Dynamical Systems
Journal Article
Applied Mathematics and Computation, ISSN 0096-3003, 01/2020, Volume 365, p. 124659
The aim of this paper is to study the following inverse problem of ordinary differential equations: For a given set of analytic functions ω={z1(t),…,zr(t)},... 
Inverse problem for ordinary differential equations | Abel equation | Ricatti equation | Planar differential system | First integral | MATHEMATICS, APPLIED | 16TH HILBERT PROBLEM | INVERSE APPROACH
Journal Article
Bulletin des sciences mathematiques, ISSN 0007-4497, 2007, Volume 131, Issue 3, pp. 242 - 257
We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding... 
Abelian integral | Infinitesimal 16th Hilbert problem | MONODROMY | FIELDS | MATHEMATICS, APPLIED | NUMBER | MODULES | ABELIAN-INTEGRALS | VARIETIES | infinitesimal 16th Hilbert problem
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 09/2017, Volume 453, Issue 1, pp. 485 - 501
Given trigonometric monomials A1,A2,A3,A4, such that A1,A3 have the same signs as sin⁡t, and A2,A4 the same signs as cos⁡t, and natural numbers n,m>1, we study... 
Centers | Abel equation | Limit cycles | Periodic solutions | Smale–Pugh problem | Hilbert 16th problem | Smale-Pugh problem | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | NUMBER | DIFFERENTIAL-EQUATIONS | UNIQUENESS | MATHEMATICS | SYSTEMS
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 08/2016, Volume 261, Issue 3, pp. 2141 - 2167
We consider perturbed pendulum-like equations on the cylinder of the form x¨+sin⁡(x)=ε∑s=0mQn,s(x)x˙s where Qn,s are trigonometric polynomials of degree n, and... 
Infinitesimal Sixteenth Hilbert problem | Limit cycles | Perturbed pendulum equation | Abelian integrals | Secondary | Primary | MATHEMATICS | JOSEPHSON-EQUATION | BIFURCATIONS | PROPERTY | ABELIAN-INTEGRALS | MODEL | HILBERTS 16TH PROBLEM | ZEROS
Journal Article
Applied Mathematics and Computation, ISSN 0096-3003, 04/2015, Volume 256, pp. 334 - 343
Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their... 
Lu system | Chaotic analog of 16th Hilbert problem | Chen system | Lorenz system | Lorenz-like systems | Lyapunov exponent | HAUSDORFF DIMENSION | EXISTENCE | INVARIANT ALGEBRAIC-SURFACES | MATHEMATICS, APPLIED | MODEL | LYAPUNOV DIMENSION | HIDDEN ATTRACTORS | DYNAMICS | HOMOCLINIC ORBITS | BIFURCATION | Physics - Chaotic Dynamics
Journal Article
Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 08/2014, Volume 19, Issue 8, pp. 2690 - 2705
Journal Article
Journal Article
Applied Mathematics and Computation, ISSN 0096-3003, 02/2014, Volume 228, pp. 329 - 335
An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=∮Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an... 
Quartic Hamiltonian | Weakened Hilbert 16th problem | Poincare bifurcation | Abelian integrals | MATHEMATICS, APPLIED | LINEAR ESTIMATE | COMPLEX ZEROS | LIMIT-CYCLES | HILBERT PROBLEM | CENTERS | Intervals | Algebra | Upper bounds | Computation | Integrals | Mathematical models | Polynomials
Journal Article
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