Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 05/2017, Volume 449, Issue 2, pp. 1570 - 1580

The Liénard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we...

Sundman transformation | Painlevé–Gambier classification | Liénard equation | Analytical solution | MATHEMATICS | Painleve-Gambier classification | LINEARIZATION | MATHEMATICS, APPLIED | INTEGRABILITY | Lienard equation | BEHAVIOR | SYSTEMS | ORDINARY DIFFERENTIAL-EQUATIONS

Sundman transformation | Painlevé–Gambier classification | Liénard equation | Analytical solution | MATHEMATICS | Painleve-Gambier classification | LINEARIZATION | MATHEMATICS, APPLIED | INTEGRABILITY | Lienard equation | BEHAVIOR | SYSTEMS | ORDINARY DIFFERENTIAL-EQUATIONS

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 06/2017, Volume 27, Issue 6, p. 1750081

In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a...

Gröbner bases | analytic integrability | Center problem | Liénard differential systems | decomposition in prime ideals | Lienard differential systems | NUMBER | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Grobner bases | MULTIDISCIPLINARY SCIENCES | LIMIT-CYCLES | CENTERS | POLYNOMIAL DIFFERENTIAL-SYSTEMS

Gröbner bases | analytic integrability | Center problem | Liénard differential systems | decomposition in prime ideals | Lienard differential systems | NUMBER | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Grobner bases | MULTIDISCIPLINARY SCIENCES | LIMIT-CYCLES | CENTERS | POLYNOMIAL DIFFERENTIAL-SYSTEMS

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 09/2018, Volume 339, pp. 405 - 413

The key purpose of the present work is to constitute a numerical algorithm based on fractional homotopy analysis transform method to study the fractional model...

Laplace transform | Fractional homotopy | Homotopy polynomials | Lienard's equation | Analysis | Algorithms

Laplace transform | Fractional homotopy | Homotopy polynomials | Lienard's equation | Analysis | Algorithms

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 02/2019, Volume 45, pp. 542 - 556

In this paper, the problem of periodic solutions is studied for Liénard equations with anindefinite singularity x′′(t)+f(x(t))x′(t)+φ(t)xm(t)−α(t)xμ(t)=0,where...

Singularity | Continuation theorem | Periodic solution | Liénard equation | EXISTENCE | MATHEMATICS, APPLIED | MOTION | MULTIPLICITY | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | ATOM

Singularity | Continuation theorem | Periodic solution | Liénard equation | EXISTENCE | MATHEMATICS, APPLIED | MOTION | MULTIPLICITY | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | ATOM

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 08/2018, Volume 275, Issue 4, pp. 988 - 1007

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions...

Camassa–Holm equation | Geometric singular perturbation theory | Invariant manifold | Solitary wave solutions | SINGULAR PERTURBATION PROBLEMS | Carnassa-Holm equation | DIFFERENTIAL-EQUATIONS | MODEL | TRAVELING-WAVES | MATHEMATICS | REAL NOISE | LIENARD EQUATIONS | KDV EQUATION | NAGUMO TYPE EQUATIONS | RANDOM DYNAMICAL-SYSTEMS | CANARD CYCLES | Water waves | Differential equations

Camassa–Holm equation | Geometric singular perturbation theory | Invariant manifold | Solitary wave solutions | SINGULAR PERTURBATION PROBLEMS | Carnassa-Holm equation | DIFFERENTIAL-EQUATIONS | MODEL | TRAVELING-WAVES | MATHEMATICS | REAL NOISE | LIENARD EQUATIONS | KDV EQUATION | NAGUMO TYPE EQUATIONS | RANDOM DYNAMICAL-SYSTEMS | CANARD CYCLES | Water waves | Differential equations

Journal Article

6.
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Piecewise smooth localized solutions of Liénard‐type equations with application to NLSE

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 11/2018, Volume 41, Issue 17, pp. 7869 - 7887

In this work, the rapidly convergent approximation method (RCAM) followed by appropriate modifications is applied to obtain piecewise smooth solutions and...

Liénard‐type equation | piecewise smooth solution | RCAM | conserved quantities | generalized nonlinear Schrödinger equation with dual‐power law nonlinearities | Liénard-type equation | generalized nonlinear Schrödinger equation with dual-power law nonlinearities | Generalized nonlinear schrödinger equation with dual-power law nonlinearities | Conserved quantities | Piecewise smooth solution | generalized nonlinear Schrodinger equation with dual-power law nonlinearities | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | LAW | DECOMPOSITION METHOD | DIFFERENTIAL-EQUATIONS | Lienard-type equation | Nonlinear equations | Parameters | Partial differential equations | Nonlinear differential equations | Dependence | Traveling waves | Schroedinger equation | Nonlinear systems

Liénard‐type equation | piecewise smooth solution | RCAM | conserved quantities | generalized nonlinear Schrödinger equation with dual‐power law nonlinearities | Liénard-type equation | generalized nonlinear Schrödinger equation with dual-power law nonlinearities | Generalized nonlinear schrödinger equation with dual-power law nonlinearities | Conserved quantities | Piecewise smooth solution | generalized nonlinear Schrodinger equation with dual-power law nonlinearities | MATHEMATICS, APPLIED | TRAVELING-WAVE SOLUTIONS | LAW | DECOMPOSITION METHOD | DIFFERENTIAL-EQUATIONS | Lienard-type equation | Nonlinear equations | Parameters | Partial differential equations | Nonlinear differential equations | Dependence | Traveling waves | Schroedinger equation | Nonlinear systems

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 07/2016, Volume 57, pp. 114 - 120

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has...

The Liénard equation | Nonlocal transformations | Elliptic functions | Integrability conditions | General solutions | LINEARIZATION | MATHEMATICS, APPLIED | BURGERS-HUXLEY EQUATION | The Lienard equation | GENERALIZED FISHER EQUATION | ORDINARY DIFFERENTIAL-EQUATIONS | Integral calculus | Criteria | Transformations (mathematics) | Integral equations | Mathematical analysis

The Liénard equation | Nonlocal transformations | Elliptic functions | Integrability conditions | General solutions | LINEARIZATION | MATHEMATICS, APPLIED | BURGERS-HUXLEY EQUATION | The Lienard equation | GENERALIZED FISHER EQUATION | ORDINARY DIFFERENTIAL-EQUATIONS | Integral calculus | Criteria | Transformations (mathematics) | Integral equations | Mathematical analysis

Journal Article

8.
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On the uniqueness of the limit cycle for the Liénard equation with f(x) not sign-definite

Applied Mathematics Letters, ISSN 0893-9659, 02/2018, Volume 76, pp. 208 - 214

The problem of uniqueness of limit cycles for the Liénard equation ẍ+f(x)ẋ+g(x)=0 is investigated. The classical assumption of sign-definiteness of f(x) is...

Liénard equation | Limit cycles | Uniqueness | EXISTENCE | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | Lienard equation | THEOREM | SYSTEMS

Liénard equation | Limit cycles | Uniqueness | EXISTENCE | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | Lienard equation | THEOREM | SYSTEMS

Journal Article

Positivity, ISSN 1385-1292, 9/2019, Volume 23, Issue 4, pp. 779 - 787

In this paper, we investigate the existence of a positive periodic solution for the following Liénard equation with a indefinite singularity $$\begin{aligned}...

Positive periodic solution | 34B16 | Operator Theory | 34C25 | Fourier Analysis | Potential Theory | Calculus of Variations and Optimal Control; Optimization | Mathematics | Indefinite singularity | Econometrics | Liénard equation | MATHEMATICS | SUBHARMONIC SOLUTIONS | MULTIPLICITY | Lienard equation | DIFFERENTIAL-EQUATIONS | DUFFING EQUATION | Computer science | Information science

Positive periodic solution | 34B16 | Operator Theory | 34C25 | Fourier Analysis | Potential Theory | Calculus of Variations and Optimal Control; Optimization | Mathematics | Indefinite singularity | Econometrics | Liénard equation | MATHEMATICS | SUBHARMONIC SOLUTIONS | MULTIPLICITY | Lienard equation | DIFFERENTIAL-EQUATIONS | DUFFING EQUATION | Computer science | Information science

Journal Article

Positivity, ISSN 1385-1292, 04/2019, Volume 23, Issue 2, pp. 431 - 444

In this paper, we consider the following quasilinear Lienard equation with a singularity where g has a attractive singularity at the origin and satisfies...

Superlinear | Attractive singularity | Periodic solution | Liénard equation | MATHEMATICS | PERIODIC-SOLUTIONS | Lienard equation | Information science

Superlinear | Attractive singularity | Periodic solution | Liénard equation | MATHEMATICS | PERIODIC-SOLUTIONS | Lienard equation | Information science

Journal Article

Theoretical and Mathematical Physics, ISSN 0040-5779, 8/2018, Volume 196, Issue 2, pp. 1230 - 1240

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between...

Liénard-type equation | nonlocal transformation | Theoretical, Mathematical and Computational Physics | general solution | Applications of Mathematics | Painlevé–Gambier equation | Physics | JACOBI | Painleve-Gambier equation | 2ND-ORDER | SYMMETRIES | PHYSICS, MULTIDISCIPLINARY | LAGRANGIANS | PHYSICS, MATHEMATICAL | Lienard-type equation | ORDINARY DIFFERENTIAL-EQUATIONS

Liénard-type equation | nonlocal transformation | Theoretical, Mathematical and Computational Physics | general solution | Applications of Mathematics | Painlevé–Gambier equation | Physics | JACOBI | Painleve-Gambier equation | 2ND-ORDER | SYMMETRIES | PHYSICS, MULTIDISCIPLINARY | LAGRANGIANS | PHYSICS, MATHEMATICAL | Lienard-type equation | ORDINARY DIFFERENTIAL-EQUATIONS

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 10/2018, Volume 84, pp. 124 - 129

A. M. Lyapunov proved the inequality that makes it possible to estimate the distance between two consecutive zeros a and b of solutions of a linear...

The Van der Pol equation with relativistic acceleration | Relativistic equation | Lyapunov-type inequality | EXISTENCE | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | LIENARD EQUATIONS | PENDULUM | PHI-LAPLACIAN | Differential equations

The Van der Pol equation with relativistic acceleration | Relativistic equation | Lyapunov-type inequality | EXISTENCE | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | LIENARD EQUATIONS | PENDULUM | PHI-LAPLACIAN | Differential equations

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 06/2019, Volume 29, Issue 6, p. 1950074

A dynamical system possessing an equilibrium point with two zero eigenvalues is considered. We assume that a degenerate Bogdanov–Takens bifurcation with...

Heteroclinic orbit | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | double-zero bifurcation with symmetry of order two | Lienard equation | MULTIDISCIPLINARY SCIENCES | regular perturbation method | CONNECTIONS

Heteroclinic orbit | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | double-zero bifurcation with symmetry of order two | Lienard equation | MULTIDISCIPLINARY SCIENCES | regular perturbation method | CONNECTIONS

Journal Article

Regular and Chaotic Dynamics, ISSN 1560-3547, 7/2015, Volume 20, Issue 4, pp. 486 - 496

The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been...

34A05 | 34A34 | general solution | Mathematics | quadratic lienard equation | nonlocal transformations | Dynamical Systems and Ergodic Theory | 33E05 | elliptic functions | MATHEMATICS, APPLIED | MECHANICS | SOLITONS | BUBBLE DYNAMICS | PHYSICS, MATHEMATICAL | Differential equations, Linear | Transformations (Mathematics) | Research | Functions, Elliptic | Mathematical research | Equations, Quadratic

34A05 | 34A34 | general solution | Mathematics | quadratic lienard equation | nonlocal transformations | Dynamical Systems and Ergodic Theory | 33E05 | elliptic functions | MATHEMATICS, APPLIED | MECHANICS | SOLITONS | BUBBLE DYNAMICS | PHYSICS, MATHEMATICAL | Differential equations, Linear | Transformations (Mathematics) | Research | Functions, Elliptic | Mathematical research | Equations, Quadratic

Journal Article

TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, ISSN 1230-3429, 09/2019, Volume 54, Issue 1, pp. 203 - 218

In this paper, the existence of positive periodic solutions is studied for a singular Lienard equation where the weight function has an indefinite sign. Due to...

periodic solution | EXISTENCE | MATHEMATICS | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | Mawhin's continuation theorem | singularity

periodic solution | EXISTENCE | MATHEMATICS | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | Mawhin's continuation theorem | singularity

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2011, Volume 250, Issue 4, pp. 2162 - 2176

Based on geometric singular perturbation theory we prove the existence of classical Liénard equations of degree 6 having 4 limit cycles. It implies the...

Singular perturbations | Relaxation oscillation | Limit cycles | Classical Liénard equations | Slow–fast system | Slow-fast system | MATHEMATICS | Classical Lienard equations

Singular perturbations | Relaxation oscillation | Limit cycles | Classical Liénard equations | Slow–fast system | Slow-fast system | MATHEMATICS | Classical Lienard equations

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2019, Volume 346, pp. 183 - 192

A periodic problem of Ambrosetti–Prodi type is studied in this paper for the Liénard equation with a singularity of attractive...

Multiplicity result | Singularity | Periodic solutions | Liénard equation | Upper and lower functions | MATHEMATICS, APPLIED | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | NONLINEAR PERTURBATIONS

Multiplicity result | Singularity | Periodic solutions | Liénard equation | Upper and lower functions | MATHEMATICS, APPLIED | Lienard equation | 2ND-ORDER DIFFERENTIAL-EQUATIONS | NONLINEAR PERTURBATIONS

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 02/2015, Volume 25, Issue 2, pp. 1550032 - 1-1550032-11

In this paper, we prove the existence of pullback and uniform attractors for a nonautonomous Liénard equation. The relation among these attractors is also...

Hausdorff dimension | nonautonomous equation | pullback attractor | uniform attractor | Liénard equation | LOCALIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lienard equation | PULLBACK ATTRACTORS | DYNAMICAL-SYSTEMS | DIMENSION | MULTIDISCIPLINARY SCIENCES | Periodic functions | Computer simulation | Chaos theory | Mathematical analysis | Bifurcations | Mathematical models | Estimates

Hausdorff dimension | nonautonomous equation | pullback attractor | uniform attractor | Liénard equation | LOCALIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lienard equation | PULLBACK ATTRACTORS | DYNAMICAL-SYSTEMS | DIMENSION | MULTIDISCIPLINARY SCIENCES | Periodic functions | Computer simulation | Chaos theory | Mathematical analysis | Bifurcations | Mathematical models | Estimates

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 12/2015, Volume 82, Issue 4, pp. 1953 - 1968

In this paper we develop a systematic and self-consistent procedure based on a set of compatibility conditions for identifying all maximal (eight parameter)...

Liénard-type equation | Engineering | Vibration, Dynamical Systems, Control | Lie point symmetries | Mechanics | Ordinary differential equations | Automotive Engineering | Mechanical Engineering | Parameter identification | Hamiltonian functions | Mathematical analysis | Integral equations | Symmetry

Liénard-type equation | Engineering | Vibration, Dynamical Systems, Control | Lie point symmetries | Mechanics | Ordinary differential equations | Automotive Engineering | Mechanical Engineering | Parameter identification | Hamiltonian functions | Mathematical analysis | Integral equations | Symmetry

Journal Article

Discrete and Continuous Dynamical Systems - Series B, ISSN 1531-3492, 08/2017, Volume 22, Issue 6, pp. 2465 - 2478

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Lienard equation. By...

Iscochronous system | Metriplectic structure | Complex Hamiltonization | Chiellini integrabilty condition | Liénard equation | MATHEMATICS, APPLIED | iscochronous system | Lienard equation | metriplectic structure | LAST MULTIPLIER | complex Hamiltonization | FORMULATION

Iscochronous system | Metriplectic structure | Complex Hamiltonization | Chiellini integrabilty condition | Liénard equation | MATHEMATICS, APPLIED | iscochronous system | Lienard equation | metriplectic structure | LAST MULTIPLIER | complex Hamiltonization | FORMULATION

Journal Article

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