Liouvillian Integrability and the Poincaré Problem for Nonlinear Oscillators with Quadratic Damping and Polynomial Forces

2020 ◽

Vol 150 (6) ◽

pp. 3231-3251 ◽

Cited By ~ 1

Author(s):

Maria V. Demina ◽

Claudia Valls

Keyword(s):

Differential Equations ◽

Algebraic Curves ◽

First Integral ◽

Complete Classification ◽

Invariant Algebraic Curves ◽

Liouvillian Integrability ◽

Poincaré Problem

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

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Dynamics of Coupled Nonlinear Oscillators: The Formation of Long-Lived Vibrational States in the Case of Molecular Systems

Прикладная механика и техническая физика ◽

10.15372/pmtf20160109 ◽

2016 ◽

Vol 57 (1) ◽

Keyword(s):

Nonlinear Oscillators ◽

Vibrational States ◽

Molecular Systems ◽

Coupled Nonlinear Oscillators

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Performance analysis of global local mean square error criterion of stochastic linearization for nonlinear oscillator

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/12015 ◽

2019 ◽

Author(s):

Nguyen Cao Thang ◽

Luu Xuan Hung

Keyword(s):

Performance Analysis ◽

Mean Square Error ◽

Nonlinear Oscillators ◽

Degree Of Freedom ◽

Equivalent Linearization ◽

Mean Square ◽

Error Criterion ◽

Stochastic Linearization ◽

Local Mean ◽

Mean Square Error Criterion

The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of stochastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi degree of freedom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL).

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A new modification of the homotopy perturbation method

Noise & Vibration Worldwide ◽

10.1177/0957456521999875 ◽

2021 ◽

pp. 095745652199987

Author(s):

Magaji Yunbunga Adamu ◽

Peter Ogenyi

Keyword(s):

Comparative Analysis ◽

Error Analysis ◽

Perturbation Method ◽

Homotopy Perturbation Method ◽

Nonlinear Oscillators ◽

Absolute Error ◽

New Method ◽

Optimal Analysis ◽

Homotopy Perturbation ◽

Accuracy And Precision

This study proposes a new modification of the homotopy perturbation method. A new parameter alpha is introduced into the homotopy equation in order to improve the results and accuracy. An optimal analysis identifies the parameter alpha, aimed at improving the solutions. A comparative analysis of the proposed method reveals that the new method presents results with higher degree of accuracy and precision than the classic homotopy perturbation method. Absolute error analysis shows the convenience of the proposed method, providing much smaller errors. Two examples are presented: Duffing and Van der pol’s nonlinear oscillators to demonstrate the efficiency, accuracy, and applicability of the new method.

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Reconstructing the structure of directed and weighted networks of nonlinear oscillators

Physical Review E ◽

10.1103/physreve.95.042302 ◽

2017 ◽

Vol 95 (4) ◽

Cited By ~ 13

Author(s):

Francesco Alderisio ◽

Gianfranco Fiore ◽

Mario di Bernardo

Keyword(s):

Nonlinear Oscillators ◽

Weighted Networks

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Nonlinear Resonance in Neuron Dynamics

Zeitschrift für Naturforschung A ◽

10.1515/zna-1995-0803 ◽

1995 ◽

Vol 50 (8) ◽

pp. 718-726 ◽

Cited By ~ 1

Author(s):

Scott Rader ◽

Diek W. Wheeler ◽

W.C. Schieve ◽

Pranab Das

Keyword(s):

Neural Networks ◽

Energy Transfer ◽

Power Spectrum ◽

Nonlinear Resonance ◽

Nonlinear Oscillators ◽

Hopfield Neural Networks ◽

Neuron Dynamics

Abstract Hübler's technique using aperiodic forces to drive nonlinear oscillators to resonance is analyzed. The oscillators being examined are effective neurons that model Hopfield neural networks. The method is shown to be valid under several different circumstances. It is verified through analysis of the power spectrum, force, resonance, and energy transfer of the system.

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