2011, 1. Aufl., ISBN 9780470977859, xvi, 369

The Duffing Equation: Nonlinear Oscillators and their Behaviourbrings together the results of a wealth of disseminated research literature on the Duffing equation, a key engineering model with a vast number...

Duffing equations | Nonlinear oscillators | Mathematical models

Duffing equations | Nonlinear oscillators | Mathematical models

Book

Nonlinear Dynamics, ISSN 0924-090X, 9/2016, Volume 85, Issue 4, pp. 2449 - 2465

In this paper, we study the application of a version of the method of simplest equation for obtaining exact traveling wave solutions of the Zakharov...

Engineering | Vibration, Dynamical Systems, Control | Modified Zakharov–Kuznetsov equation | Modified method of simplest equation | Exact solutions | Mechanics | Automotive Engineering | Zakharov–Kuznetsov equation | Mechanical Engineering | Zakharov-Kuznetsov equation | 1-SOLITON SOLUTION | DE-VRIES EQUATION | ENGINEERING, MECHANICAL | TRAVELING-WAVE SOLUTIONS | MECHANICS | PARTIAL-DIFFERENTIAL-EQUATIONS | SYMBOLIC COMPUTATION | TIME-DEPENDENT COEFFICIENTS | BOUSSINESQ EQUATIONS | Modified Zakharov-Kuznetsov equation | KDV EQUATION | NONLINEAR EVOLUTION-EQUATIONS | VARIABLE SEPARATION APPROACH | Information science | Methods | Traveling waves | Nonlinear equations | Partial differential equations | Mathematical analysis | Nonlinear differential equations

Engineering | Vibration, Dynamical Systems, Control | Modified Zakharov–Kuznetsov equation | Modified method of simplest equation | Exact solutions | Mechanics | Automotive Engineering | Zakharov–Kuznetsov equation | Mechanical Engineering | Zakharov-Kuznetsov equation | 1-SOLITON SOLUTION | DE-VRIES EQUATION | ENGINEERING, MECHANICAL | TRAVELING-WAVE SOLUTIONS | MECHANICS | PARTIAL-DIFFERENTIAL-EQUATIONS | SYMBOLIC COMPUTATION | TIME-DEPENDENT COEFFICIENTS | BOUSSINESQ EQUATIONS | Modified Zakharov-Kuznetsov equation | KDV EQUATION | NONLINEAR EVOLUTION-EQUATIONS | VARIABLE SEPARATION APPROACH | Information science | Methods | Traveling waves | Nonlinear equations | Partial differential equations | Mathematical analysis | Nonlinear differential equations

Journal Article

1981, ISBN 0471080330, xiv, 519

Book

Journal of Sound and Vibration, ISSN 0022-460X, 03/2014, Volume 333, Issue 7, pp. 2040 - 2053

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation...

ACOUSTICS | VIBRATIONS | MECHANICS | OPTIMIZATION | ENGINEERING, MECHANICAL | Monte Carlo method | Physicians (General practice) | Electric power production | Magnets, Permanent | Electric potential | Vibration | Computer simulation | Mathematical analysis | Voltage | Nonlinearity | Excitation | Filled plastics | Energy harvesting

ACOUSTICS | VIBRATIONS | MECHANICS | OPTIMIZATION | ENGINEERING, MECHANICAL | Monte Carlo method | Physicians (General practice) | Electric power production | Magnets, Permanent | Electric potential | Vibration | Computer simulation | Mathematical analysis | Voltage | Nonlinearity | Excitation | Filled plastics | Energy harvesting

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 12/2015, Volume 39, Issue 23-24, pp. 7420 - 7426

We study a Wick-type stochastic reaction Duffing equation, and obtain some new exact solutions with the help of the white-noise theory and the exact solution of the Riccati equation...

Wick-type stochastic reaction Duffing equation | Exact solutions | White-noise theory | Wick-type stochastic reaction Duffing | Equation | CHAOS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | EXP-FUNCTION METHOD | BIFURCATION

Wick-type stochastic reaction Duffing equation | Exact solutions | White-noise theory | Wick-type stochastic reaction Duffing | Equation | CHAOS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | EXP-FUNCTION METHOD | BIFURCATION

Journal Article

International journal of non-linear mechanics, ISSN 0020-7462, 10/2018, Volume 105, pp. 173 - 178

The nonlinear dynamics of action potentials in the FitzHugh–Nagumo model is addressed using a modified Van der Pol equation with fractional-order derivative and periodic parametric excitation...

FitzHugh–Nagumo model | Amplitude–frequency curves | Primary resonance | Van der Pol oscillator | Fractional-order derivatives | FitzHugh-Nagumo model | 2 KINDS | PATTERNS | Amplitude-frequency curves | MECHANICS | ELECTROENCEPHALOGRAM | DUFFING OSCILLATOR | IMPULSES | MATHIEU EQUATION | ROSE NEURAL-NETWORKS | THEORETICAL-MODELS

FitzHugh–Nagumo model | Amplitude–frequency curves | Primary resonance | Van der Pol oscillator | Fractional-order derivatives | FitzHugh-Nagumo model | 2 KINDS | PATTERNS | Amplitude-frequency curves | MECHANICS | ELECTROENCEPHALOGRAM | DUFFING OSCILLATOR | IMPULSES | MATHIEU EQUATION | ROSE NEURAL-NETWORKS | THEORETICAL-MODELS

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 02/2018, Volume 20, Issue 1, p. 1750022

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant...

Duffing equation | Hill equation | nonlinear beam equation | stability | MATHEMATICS | MATHEMATICS, APPLIED | STABILITY REGIONS | Mathematics - Classical Analysis and ODEs

Duffing equation | Hill equation | nonlinear beam equation | stability | MATHEMATICS | MATHEMATICS, APPLIED | STABILITY REGIONS | Mathematics - Classical Analysis and ODEs

Journal Article

Results in physics, ISSN 2211-3797, 03/2020, Volume 16, p. 102816

•Perturbed Shrödinger equation with Kerr law nonlinearity and Kundu-Mukherjee-Naskar equation are considered...

Kundu-Mukherjee-Naskar equation | Cubic Duffing equation | New extended algebraic method | Shrödinger’s equation

Kundu-Mukherjee-Naskar equation | Cubic Duffing equation | New extended algebraic method | Shrödinger’s equation

Journal Article

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

... Liénard equation with a indefinite singularity $$\begin{aligned} x''+f(x)x'+\frac{b(t)}{x}=p(t), \end{aligned}$$ x ′ ′ + f ( x ) x ′ + b ( t ) x = p ( t ) , where $$b...

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

Optical and Quantum Electronics, ISSN 0306-8919, 3/2018, Volume 50, Issue 3, pp. 1 - 13

The generalized projective Riccati equation method is proposed to establish exact solutions for generalized form of the reaction Duffing model in fractional sense namely, Khalil’s derivative...

Generalized projective Riccati equations method | Generalized reaction duffing model | Conformable fractional derivative | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Electrical Engineering | ORDER | 1ST INTEGRAL METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | ENGINEERING, ELECTRICAL & ELECTRONIC

Generalized projective Riccati equations method | Generalized reaction duffing model | Conformable fractional derivative | Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Computer Communication Networks | Physics | Electrical Engineering | ORDER | 1ST INTEGRAL METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | OPTICS | ENGINEERING, ELECTRICAL & ELECTRONIC

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2011, Volume 62, Issue 10, pp. 3894 - 3901

Approximate periodic solutions for the Helmholtz–Duffing oscillator are obtained in this paper. He’s Energy Balance Method (HEBM) and He’s Frequency Amplitude...

Helmholtz–Duffing oscillator | He’s Frequency Amplitude Formulation | He’s Energy Balance Method | He's Energy Balance Method | HelmholtzDuffing oscillator | He's Frequency Amplitude Formulation | ENERGY-BALANCE | MATHEMATICS, APPLIED | HOMOTOPY PERTURBATION | VARIATIONAL APPROACH | Helmholtz-Duffing oscillator | VIBRATIONS | NONLINEAR OSCILLATORS | HES FREQUENCY FORMULATION | U(1/3) FORCE | Energy of formation | Approximation | Error analysis | Mathematical analysis | Resonant frequency | Oscillations | Mathematical models | Oscillators

Helmholtz–Duffing oscillator | He’s Frequency Amplitude Formulation | He’s Energy Balance Method | He's Energy Balance Method | HelmholtzDuffing oscillator | He's Frequency Amplitude Formulation | ENERGY-BALANCE | MATHEMATICS, APPLIED | HOMOTOPY PERTURBATION | VARIATIONAL APPROACH | Helmholtz-Duffing oscillator | VIBRATIONS | NONLINEAR OSCILLATORS | HES FREQUENCY FORMULATION | U(1/3) FORCE | Energy of formation | Approximation | Error analysis | Mathematical analysis | Resonant frequency | Oscillations | Mathematical models | Oscillators

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 05/2019, Volume 266, Issue 11, pp. 6924 - 6962

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials...

Quasi-periodic solution | Impulsive Duffing equation | Moser's twist theorem | Lagrange stability | MATHEMATICS

Quasi-periodic solution | Impulsive Duffing equation | Moser's twist theorem | Lagrange stability | MATHEMATICS

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 01/2015, Volume 251, pp. 669 - 674

We study the periodic solutions of the following kind of non-autonomous Duffing differential equation y¨+ay-εy3=εh(t,y,ẏ), with a>0,ε...

Duffing differential equation | Periodic solution | Averaging theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | STABILITY | BOUNDARY-VALUE-PROBLEMS | Mathematical models | Duffing equation | Computation | Mathematical analysis

Duffing differential equation | Periodic solution | Averaging theory | PERIODIC-SOLUTIONS | MATHEMATICS, APPLIED | STABILITY | BOUNDARY-VALUE-PROBLEMS | Mathematical models | Duffing equation | Computation | Mathematical analysis

Journal Article

Indian Journal of Physics, ISSN 0973-1458, 2/2014, Volume 88, Issue 2, pp. 177 - 184

... reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity...

Physics, general | Astrophysics and Astroparticles | Fractional generalized reaction duffing model | First integral method | Physics | Solitons | PHYSICS, MULTIDISCIPLINARY | SOLITON PERTURBATION-THEORY | KDV EQUATION | DIFFUSION | DE-VRIES EQUATION | EXPLICIT | Partial differential equations | Fractions

Physics, general | Astrophysics and Astroparticles | Fractional generalized reaction duffing model | First integral method | Physics | Solitons | PHYSICS, MULTIDISCIPLINARY | SOLITON PERTURBATION-THEORY | KDV EQUATION | DIFFUSION | DE-VRIES EQUATION | EXPLICIT | Partial differential equations | Fractions

Journal Article

Optical and Quantum Electronics, ISSN 0306-8919, 10/2017, Volume 49, Issue 10, pp. 1 - 11

.... Nonlinear Schrödinger equation is an important model for optical communication. Our theoretical analysis and numerical simulation show that when the...

Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Secure communication | Nonlinear Schrödinger equation | Chaos synchronization | Feedback control | Computer Communication Networks | Physics | Electrical Engineering | QUANTUM SCIENCE & TECHNOLOGY | Nonlinear Schrodinger equation | DUFFING OSCILLATORS | OPTICS | ENGINEERING, ELECTRICAL & ELECTRONIC | Analysis | Models | Numerical analysis

Optics, Lasers, Photonics, Optical Devices | Characterization and Evaluation of Materials | Secure communication | Nonlinear Schrödinger equation | Chaos synchronization | Feedback control | Computer Communication Networks | Physics | Electrical Engineering | QUANTUM SCIENCE & TECHNOLOGY | Nonlinear Schrodinger equation | DUFFING OSCILLATORS | OPTICS | ENGINEERING, ELECTRICAL & ELECTRONIC | Analysis | Models | Numerical analysis

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 01/2013, Volume 23, Issue 1, pp. 1350014 - 1350015

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation...

multiple time scales | Van der Pol-Duffing equation | double Hopf bifurcation | normal form | center manifold reduction | MULTIDISCIPLINARY SCIENCES | PERTURBATION TECHNIQUE | OSCILLATOR | FUNCTIONAL-DIFFERENTIAL EQUATIONS | MULTIPLE SCALES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NONLINEAR-SYSTEMS | BOGDANOV-TAKENS SINGULARITY | NORMAL FORMS | COMPUTATION | FEEDBACK | Dynamic tests | Computer simulation | Mathematical analysis | Bifurcations | Mathematical models | Critical point | Hopf bifurcation | Two dimensional

multiple time scales | Van der Pol-Duffing equation | double Hopf bifurcation | normal form | center manifold reduction | MULTIDISCIPLINARY SCIENCES | PERTURBATION TECHNIQUE | OSCILLATOR | FUNCTIONAL-DIFFERENTIAL EQUATIONS | MULTIPLE SCALES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | NONLINEAR-SYSTEMS | BOGDANOV-TAKENS SINGULARITY | NORMAL FORMS | COMPUTATION | FEEDBACK | Dynamic tests | Computer simulation | Mathematical analysis | Bifurcations | Mathematical models | Critical point | Hopf bifurcation | Two dimensional

Journal Article

CMES - Computer Modeling in Engineering and Sciences, ISSN 1526-1492, 2018, Volume 115, Issue 2, pp. 149 - 215

Exact solutions of the cubic Duffing equation with the initial conditions are presented...

Nonlinear equations | Duffing equation | Leaf functions | Ordinary differential equation | HARMONIC-BALANCE METHOD | nonlinear equations | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | ordinary differential equation | leaf functions | OSCILLATOR | Initial conditions | Free vibration | Exact solutions | Differential equations | Ordinary differential equations | Nonlinear programming | Trigonometric functions | Mathematics - General Mathematics

Nonlinear equations | Duffing equation | Leaf functions | Ordinary differential equation | HARMONIC-BALANCE METHOD | nonlinear equations | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | ordinary differential equation | leaf functions | OSCILLATOR | Initial conditions | Free vibration | Exact solutions | Differential equations | Ordinary differential equations | Nonlinear programming | Trigonometric functions | Mathematics - General Mathematics

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 12/2012, Volume 25, Issue 12, pp. 2349 - 2353

In this paper, we derive a class of analytical solution of the damped Helmholtz–Duffing oscillator that is based on a recently developed exact solution for the...

Damped Helmholtz–Duffing oscillator | Mixed-parity nonlinear oscillator | Asymmetric behavior | Quadratic nonlinearities | Damped Helmholtz-Duffing oscillator | MATHEMATICS, APPLIED | INTEGRABILITY | OSCILLATOR | Mathematical analysis | Oscillators | Exact solutions

Damped Helmholtz–Duffing oscillator | Mixed-parity nonlinear oscillator | Asymmetric behavior | Quadratic nonlinearities | Damped Helmholtz-Duffing oscillator | MATHEMATICS, APPLIED | INTEGRABILITY | OSCILLATOR | Mathematical analysis | Oscillators | Exact solutions

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 08/2015, Volume 39, Issue 16, pp. 4607 - 4616

Exact analytical solutions for a family of autonomous ordinary differential equations arising in connection with many important applied problems such as a...

Jacobian elliptic functions | Nonlinear Schrödinger equation | Exact analytical non-singular solutions | Duffing oscillatory equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Nonlinear Schrodinger equation

Jacobian elliptic functions | Nonlinear Schrödinger equation | Exact analytical non-singular solutions | Duffing oscillatory equation | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | Nonlinear Schrodinger equation

Journal Article

Meccanica, ISSN 0025-6455, 7/2015, Volume 50, Issue 7, pp. 1841 - 1853

.... The mechanical response in deterministic situation is described by the Duffing equation, whose numerical solution is obtained with the Runge–Kutta...

Microsystems | Symbolic computing | Nonlinear vibration | Civil Engineering | Stochastic perturbation technique | Mechanics | Automotive Engineering | MEMS | Microresonators | Mechanical Engineering | Duffing equation | Physics | MECHANICS | SYSTEMS | Algebra | Vibration | Algorithms | College teachers

Microsystems | Symbolic computing | Nonlinear vibration | Civil Engineering | Stochastic perturbation technique | Mechanics | Automotive Engineering | MEMS | Microresonators | Mechanical Engineering | Duffing equation | Physics | MECHANICS | SYSTEMS | Algebra | Vibration | Algorithms | College teachers

Journal Article