Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

doi:10.1007/s10483-012-1611-6

Applied Mathematics and Mechanics (English Edition) ›› 2012, Vol. 33 ›› Issue (9): 1137-1152.doi: https://doi.org/10.1007/s10483-012-1611-6

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Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

Yang-yang CHEN¹, Le-wei YAN², Kam-yim SZE³, Shu-hui CHEN ⁴

1. Key Laboratory of Vibration Control and Structural Safety of Ministry of Education of China, Guangzhou University, Guangzhou 510405, P. R. China;
2. Department of Engineering Mechanics, Guangzhou University, Guangzhou 510405, P. R. China;
3. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong, P. R. China;
4. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275, P. R. China

Received:2012-05-08 Revised:2012-05-16 Online:2012-09-10 Published:2012-09-10
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 10972240 and 11102045), the Natural Science Foundation of Guangdong Province of China (No. S20110400040), the Foundation of Guangdong Education Department of China (No. LYM10108), the Foundation of Guangzhou Education Bureau of China (No. 10A024), and the Research Grant Council of Hong Kong of China (No.GRF-HKU-7173-09E)

Abstract

Abstract:

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce-dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlin-earity are studied in detail to illustrate the efficiency and accuracy of the present method.

Key words: gas atomization, spray forming, instability of interfacial wave

2010 MSC Number:

O322

Yang-yang CHEN;Le-wei YAN;Kam-yim SZE;Shu-hui CHEN . Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems. Applied Mathematics and Mechanics (English Edition), 2012, 33(9): 1137-1152.

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References

[1] Chen, S. H. Quantitative Analysis Methods for Strongly Nonlinear Vibration (in Chinese), SciencePress, Beijing (2005)
[2] Liu, Z. R. Analytical Methods for Study of Chaos (in Chinese), Science and Technology EducationPress, Shanghai (2002)
[3] Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations ofVector Fields, Springer, New York (1983)
[4] Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York(1990)
[5] Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics, Analytical, Computational, andExperimental Methods, Wiley, New York (1995)
[6] Li, J. B. and Dai, H. H. On the Study of Singular Nonlinear Traveling Wave Equation: DynamicalSystem Approach (in Chinese), Science Press, Beijing (2005)
[7] Chen, Y. S. and Ding, Q. C-L method and its application to engineering nonlinear dynamicalproblems. Applied Mathematics and Mechanics (English Edition), 22(2), 127-134 (2001) DOI10.1007/BF02437879
[8] Chen, L. Q. Chaos in pertrubation planar non-Hamiltonian integrable systems with slowly-varyingangle parameters. Applied Mathematics and Mechanics (English Edition), 22(8), 1301-1305 (2001)DOI 10.1007/BF02437854
[9] Vakakis, A. F. Exponentially small splittings of manifolds in a rapidly forced Duffing system.Journal of Sound and Vibration, 170(1), 119-129 (1994)
[10] Vakakis, A. F. and Azeez, M. F. A. Analytic approximation of the homoclinic orbits of the Lorenzsystem at σ = 10, b = 8/3, and ρ = 13.926 · · · . Nonlinear Dynamics, 15(3), 245-257 (1998)
[11] Xu, Z., Chan, H. S. Y., and Chung, K. W. Separatrices and limit cycles of strongly nonlinearoscillators by the perturbation-incremental method. Nonlinear Dynamics, 11(3), 213-233 (1996)
[12] Chan, H. S. Y., Chung, K. W., and Xu, Z. Stability and bifurcations of limit cycles by theperturbation-incremental method. Journal of Sound and Vibration, 206(4), 589-604 (1997)
[13] Belhaq, M. Predicting homoclinic bifurcations in planar autonomous systems. Nonlinear Dynam-ics, 18(4), 303-310 (1999)
[14] Belhaq, M. and Lakrad, F. Prediction of homoclinic bifurcation: the elliptic averaging method.Chaos, Solitons and Fractals, 11(10), 2251-2258 (2000)
[15] Belhaq, M., Fiedler, B., and Lakrad, F. Homoclinic connections in strongly self-excited nonlinearoscillators: the Melnikov function and the elliptic Lindstedt-Poincaré method. Nonlinear Dynam-ics, 23(1), 67-86 (2000)
[16] Mikhlin, Y. V. Analytical construction of homoclinic orbits of two-and three-dimensional dynam-ical systems. Journal of Sound and Vibration, 230(5), 971-983 (2000)
[17] Mikhlin, Y. V. and Manucharyan, G. V. Construction of homoclinic and heteroclinic trajectoriesin mechanical systems with several equilibrium positions. Chaos, Solitons and Fractals, 16(2),299-309 (2003)
[18] Manucharyan, G. V. and Mikhlin, Y. V. The construction of homoclinic and heteroclinic orbitsin nonlinear systems. Journal of Applied Mathematics and Mechanics, 69(1), 42-52 (2005)
[19] Cao, H. J., Jiang, Y. Z., and Shan, Y. L. Primary resonant optimal control for nested homoclinicand heteroclinic bifurcations in single-dof nonlinear oscillators. Journal of Sound and Vibration,289(1-2), 229-244 (2006)
[20] Zhang, Q. C., Wang, W., and Li, W. Y. Heteroclinic bifurcations of strongly nonlinear oscillator.Chinese Physics Letters, 25(5), 1905-1907 (2008)
[21] Zhang, Y. M. and Lu, Q. S. Homoclinic bifurcation of strongly nonlinear oscillators by frequency-incremental method. Communications in Nonlinear Science and Numerical Simulation, 8(1), 1-7(2003)
[22] Izydorek, M. and Janczewska, J. Homoclinic solutions for a class of the second order Hamiltoniansystems. Journal of Differential Equations, 219(2), 375-389 (2005)
[23] Izydorek, M. and Janczewska, J. Heteroclinic solutions for a class of the second order Hamiltoniansystems. Journal of Differential Equations, 238(2), 381-393 (2007)
[24] Cao, Y. Y., Chung, K.W., and Xu, J. A novel construction of homoclinic and heteroclinic orbits innonlinear oscillators by a perturbation-incremental method. Nonlinear Dynamics, 64(3), 221-236(2011)
[25] Chen, Y. Y. and Chen, S. H. Homoclinic and heteroclinic solutions of cubic strongly nonlinearautonomous oscillators by the hyperbolic perturbation method. Nonlinear Dynamics, 58(1-2),417-429 (2009)
[26] Chen, S. H., Chen, Y. Y., and Sze, K. Y. A hyperbolic perturbation method for determininghomoclinic solution of certain strongly nonlinear autonomous oscillators. Journal of Sound andVibration, 322(1-2), 381-392 (2009)
[27] Chen, Y. Y., Chen, S. H., and Sze, K. Y. A hyperbolic Lindstedt-Poincaré method for homoclinicmotion of a kind of strongly nonlinear autonomous oscillators. Acta Mechanica Sinica, 25(5),721-729 (2009)
[28] Chen, S. H., Chen, Y. Y., and Sze, K. Y. Homoclinic and heteroclinic solutions of cubic stronglynonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method. Science Sincia, Tech-nological Science, 53(3), 1-11 (2010)
[29] Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Dover, New York (1972)
[30] Merkin, J. H. and Needham, D. J. On infinite period bifurcations with an application to rollwaves. Acta Mechanica, 60(1-2), 1-16 (1986)

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

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