2011, 2011(Special): 1423-1431. doi: 10.3934/proc.2011.2011.1423

Application of He's method to the modified Rayleigh equation


Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000, Kursk, Russian Federation, Russian Federation

Received  July 2010 Revised  August 2011 Published  October 2011

In this work we analyze the application of He’s variational method for an estimation of limit cycles and oscillation periods for the class of self-sustained oscillations described by the modified Rayleigh equation. The main goal of the research is to find suitable trial functions which allow to reproduce the period of limit-cycle motion with a high degree of accuracy. There is an especial consideration of the Selkov model in the modified Rayleigh form having only one extremum that does not allow to apply the classical method of slow and fast motions. In this case He’s method allows to find the period of a limit-cycle motion with a high accuracy and to predict its value for various parameters of the concerned equations. Thus, it is possible to assert that at a correct choice of trial function the considered method gives exact results not only in the case of harmonic oscillations but also in the case of relaxation ones.
Citation: Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2173-2199. doi: 10.3934/dcdsb.2021027


Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631


Ting-Hao Hsu, Gail S. K. Wolkowicz. A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1257-1277. doi: 10.3934/dcdsb.2019219


Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic and Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008


Jaume Llibre, Arefeh Nabavi. Phase portraits of the Selkov model in the Poincaré disc. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022056


Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229


Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035


Andy M. Yip, Wei Zhu. A fast modified Newton's method for curvature based denoising of 1D signals. Inverse Problems and Imaging, 2013, 7 (3) : 1075-1097. doi: 10.3934/ipi.2013.7.1075


Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066


Marissa Condon, Arieh Iserles, Karolina Kropielnicka, Pranav Singh. Solving the wave equation with multifrequency oscillations. Journal of Computational Dynamics, 2019, 6 (2) : 239-249. doi: 10.3934/jcd.2019012


Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032


Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205


Nakao Hayashi, Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the modified witham equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1407-1448. doi: 10.3934/cpaa.2018069


Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939


José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models. Networks and Heterogeneous Media, 2020, 15 (3) : 369-387. doi: 10.3934/nhm.2020023


M. Berezhnyi, L. Berlyand, Evgen Khruslov. The homogenized model of small oscillations of complex fluids. Networks and Heterogeneous Media, 2008, 3 (4) : 831-862. doi: 10.3934/nhm.2008.3.831


Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893


Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193


Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2017-2032. doi: 10.3934/jimo.2021054


Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365

 Impact Factor: 


  • PDF downloads (111)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]