International Journal of Bifurcation and Chaos, ISSN 0218-1274, 01/2013, Volume 23, Issue 1, pp. 1330002 - 1330069

From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors...

drilling system | quadratic system | induction motor | Aizerman conjecture | absolute stability | describing function method | Chua circuits | harmonic balance | Hidden oscillation | phase-locked loop (PLL) | Kalman conjecture | 16th Hilbert problem | Lienard equation | Lyapunov focus values (Lyapunov quantities Poincaré-Lyapunov constants Lyapunov coefficients) | large (normal amplitude) and small limit cycle | hidden attractor | nonlinear control system | SIZE LIMIT-CYCLES | BIFURCATIONS | ORDER 3 | ASYMPTOTIC STABILITY | COMPUTATION | LYAPUNOV QUANTITIES | MULTIDISCIPLINARY SCIENCES | 16TH PROBLEM | QUADRATIC DIFFERENTIAL-SYSTEMS | WEAK FOCUS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lyapunov focus values (Lyapunov quantities, Poincare-Lyapunov constants, Lyapunov coefficients) | Control systems | Analysis | Crashes | Computation | Circuits | Chaos theory | Mathematical analysis | Oscillations | Dynamical systems | Standards

drilling system | quadratic system | induction motor | Aizerman conjecture | absolute stability | describing function method | Chua circuits | harmonic balance | Hidden oscillation | phase-locked loop (PLL) | Kalman conjecture | 16th Hilbert problem | Lienard equation | Lyapunov focus values (Lyapunov quantities Poincaré-Lyapunov constants Lyapunov coefficients) | large (normal amplitude) and small limit cycle | hidden attractor | nonlinear control system | SIZE LIMIT-CYCLES | BIFURCATIONS | ORDER 3 | ASYMPTOTIC STABILITY | COMPUTATION | LYAPUNOV QUANTITIES | MULTIDISCIPLINARY SCIENCES | 16TH PROBLEM | QUADRATIC DIFFERENTIAL-SYSTEMS | WEAK FOCUS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lyapunov focus values (Lyapunov quantities, Poincare-Lyapunov constants, Lyapunov coefficients) | Control systems | Analysis | Crashes | Computation | Circuits | Chaos theory | Mathematical analysis | Oscillations | Dynamical systems | Standards

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

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

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

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

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

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has...

Polynomial vector fields | Limit cycles | 16th Hilbert problem | Algebraic limit cycles | MATHEMATICS | INTEGRABILITY | INVERSE PROBLEMS | Equacions diferencials ordinàries | Vector fields | 34 Ordinary differential equations | Camps vectorials | Classificació AMS | Equacions diferencials i integrals | Polinomis | 34C Qualitative theory | Cicles límits | 34A General theory | Matemàtiques i estadística | Polynomials | Àrees temàtiques de la UPC

Polynomial vector fields | Limit cycles | 16th Hilbert problem | Algebraic limit cycles | MATHEMATICS | INTEGRABILITY | INVERSE PROBLEMS | Equacions diferencials ordinàries | Vector fields | 34 Ordinary differential equations | Camps vectorials | Classificació AMS | Equacions diferencials i integrals | Polinomis | 34C Qualitative theory | Cicles límits | 34A General theory | Matemàtiques i estadística | Polynomials | Àrees temàtiques de la UPC

Journal Article

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 12/2009, Volume 14, Issue 12, pp. 4041 - 4056

This paper is concerned with the practical complexity of the symbolic computation of limit cycles associated with Hilbert's 16th problem. In particular, in...

Bifurcation | Focus value | Hilbert's 16th problem | Limit cycle | Normal form | Maple | CUBIC SYSTEM | HOPF CYCLICITY | MATHEMATICS, APPLIED | CONSTANTS | PHYSICS, FLUIDS & PLASMAS | FOCUS | VECTOR-FIELDS | PHYSICS, MATHEMATICAL | DIFFERENTIAL-SYSTEMS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SINGULAR POINT | Algebra | Computer simulation | Computation | Mathematical analysis | Nonlinearity | Computational efficiency | Dynamical systems | Complexity

Bifurcation | Focus value | Hilbert's 16th problem | Limit cycle | Normal form | Maple | CUBIC SYSTEM | HOPF CYCLICITY | MATHEMATICS, APPLIED | CONSTANTS | PHYSICS, FLUIDS & PLASMAS | FOCUS | VECTOR-FIELDS | PHYSICS, MATHEMATICAL | DIFFERENTIAL-SYSTEMS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | SINGULAR POINT | Algebra | Computer simulation | Computation | Mathematical analysis | Nonlinearity | Computational efficiency | Dynamical systems | Complexity

Journal Article

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

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

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

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

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

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

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

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 sint, and A2,A4 the same signs as cost, 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

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

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

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

18.
Full Text
Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 08/2014, Volume 19, Issue 8, pp. 2690 - 2705

•Studying bifurcation of limit cycles related to Hilbert’s 16th problem.•Showing the existence of 12 limit cycles around a singular point in a planar cubic...

Bifurcation | Center | Cubic planar system | Focus value | Limit cycle | Hilbert’s 16th problem | Hilbert's 16th problem | MATHEMATICS, APPLIED | BIFURCATIONS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | HILBERTS 16TH PROBLEM | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | COMPUTATION | Nonlinearity | Mathematical models | Polynomials | Fields (mathematics) | Computer simulation

Bifurcation | Center | Cubic planar system | Focus value | Limit cycle | Hilbert’s 16th problem | Hilbert's 16th problem | MATHEMATICS, APPLIED | BIFURCATIONS | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | HILBERTS 16TH PROBLEM | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | COMPUTATION | Nonlinearity | Mathematical models | Polynomials | Fields (mathematics) | Computer simulation

Journal Article

WSEAS Transactions on Systems and Control, ISSN 1991-8763, 02/2011, Volume 6, Issue 2, pp. 54 - 67

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

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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