On Independent Period-m Evolutions in a Periodically Forced Brusselator
DOI:
https://doi.org/10.5890/JVTSD.2018.12.004Abstract
In this paper, the analytical solutions of independent period-m evolutions (m=3;5;7;9) of chemical concentrations in a periodically forced Brusselator are obtained through the generalized harmonic balance method. Stability and bifurcation of independent periodic evolutions are determined through eigenvalue analysis. The nonlinear frequency-amplitude characteristics of independent periodic evolutions are discussed. To illustrate the analytical solutions, numerical simulations of stable and unstable period-m evolutions (m = 3;5;7;9) are presented herein. The harmonic amplitude spectrums give an approximate estimation of harmonic effects on analytical solutions of periodic motions.References
[1] Prigogine, I. and Lefever, R. (1968), Symmetry breaking instabilities in dissipative systems. II, The Journal of Chemical Physics, 48(4), 1695-1700.
[2] Lefever, R. and Nicolis, G. (1971), Chemical instabilities and sustained oscillations, Journal of theoretical Biology, 30(2), 267-284.
[3] Tyson, J.J. (1973), Some further studies of nonlinear oscillations in chemical systems, The Journal of Chemical Physics, 58(9), 3919-3930.
[4] Tomita, K., Kai, T., and Hikami, F. (1977), Entrainment of a limit cycle by a periodic external excitation, Progress of Theoretical Physics, 57(4), 1159-1177.
[5] Hao, B.L. and Zhang, S.Y. (1982), Hierarchy of chaotic bands, Journal of Statistical Physics, 28(4), 769-792.
[6] Roy, T., Choudhury, R., and Tanriver, U. (2017), Analytical prediction of homoclinic bifurcations following a supercritical Hopf bifurcation, Discontinuity, Nonlinearity, and Complexity, 6(2), 209-222.
[7] Maaita, J.O. (2016), A theorem on the bifurcations of the slow invariant manifold of a system of two linear oscillators coupled to a k-order nonlinear oscillator, Journal of Applied Nonlinear Dynamics, 5(2), 193-197.
[8] Yamgoué, S.B., Nana, B., and Pelap, F.B. (2017), Approximate analytical solutions of a nonlinear oscillator equation modeling a constrained mechanical system, Journal of Applied Nonlinear Dynamics, 6(1), 17-26.
[9] Shayak, B. and Vyas, P. (2017), Krylov Bogoliubov type analysis of variants of the Mathieu equation, Journal of Applied Nonlinear Dynamics, 6(1), 57-77.
[10] Rajamani, S. and Rajasekar, S. (2017), Variation of response amplitude in parametrically driven single Duffing oscillator and unidirectionally coupled Duffing oscillators, Journal of Applied Nonlinear Dynamics, 6(1), 121-129.
[11] Cochelin, B. and Vergez, C. (2009), A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324(1-2), 243-262.
[12] Luo, A.C.J. (2012), Continuous dynamical systems, HEP /L&H, Beijing/Glen Garbon.
[13] Luo, A.C.J. and Huang, J. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. Journal of Vibration and Control, 18(11), 1661-1674.
[14] Luo, A.C.J. and Huang, J. (2013), Analytical period-3 motions to chaos in a hardening Duffing oscillator, Nonlinear Dynamics, 73(3), 1905-1932.
[15] Luo, A.C.J. and Lakeh, A.B. (2014), An approximate solution for period-1 motions in a periodically forced van der Pol oscillator, ASME Journal of Computational and Nonlinear Dynamics, 9(3), Article No:031001 (7 pages).
[16] Luo, A.C.J. and Lakeh, A.B. (2013), Analytical solution for period-m motions in a periodically forced, van der Pol oscillator, International Jounral of Dynamics and Control, 1(2), 99-115.
[17] Luo, A.C.J. and Lakeh, A.B. (2014), Period-m motions and bifurcation in a periodically forced van der Pol-Duffing oscillator, International Journal of Dynamics and Control, 2(4), 474-493.
[18] Wang, Y.F. and Liu, Z.W. (2015), A matrix-based computational scheme of generalized harmonic balance method for periodic solutions of nonlinear vibratory systems, Journal of Applied Nonlinear Dynamics, 4(4), 379-389.
[19] Luo, H. andWang, Y. (2016), Nonlinear dynamics analysis of a continuum rotor through generalized harmonic balance method, Journal of Applied Nonlinear Dynamics, 5(1), 1-31.
[20] Akhmet, M. and Fen, M.O. (2017), Almost periodicity in chaos, Discontinuity, Nonlinearity, and Complexity, 7(1), 15-19.
[21] Luo, A.C.J. (2014), Toward Analytical Chaos in Nonlinear systems, Wiley, New York.
[22] Luo, A.C.J. (2014), Analytical routes to chaos in nonlinear engineering, Wiley, New York.
[23] Luo, A.C.J. and Guo, S. (2018), Analytical solutions of period-1 to period-2 motions in a periodically diffused Brusselator, ASME Journal of Computational and Nonlinear Dynamics,13(9), Article No: 090912 (8 pages).
[24] Luo, A.C.J. and Guo, S. (2018), Period-1 evolutions to chaos in a periodically forced Brusselator, International Journal of Bifurcation and Chaos, in press.
Article Metrics
Usage tracking begins September 1, 2026.