# American Institute of Mathematical Sciences

January  2011, 15(1): 231-254. doi: 10.3934/dcdsb.2011.15.231

## Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems

 1 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China 2 Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, China, China

Received  May 2009 Revised  March 2010 Published  October 2010

A semi-analytical procedure for studying stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous dynamical systems is developed. This procedure is based mainly on the incremental harmonic balance (IHB) method. It is composed of three key steps, namely, the determination of limit cycles by IHB method, the calculation of transition matrix by precise integration (PI) algorithm and the discrimination of limit cycle stability by Floquet theory. As an application, the procedure is used to investigate the dynamics of the limit cycle of a three-dimensional nonlinear autonomous system. The symmetry-breaking bifurcation, the first and the second period-doubling bifurcations of the limit cycle are identified. The critical parameter values corresponding to these bifurcations are calculated. The phase portraits and bifurcation points agree well with those of direct numerical integrations by using Runge-Kutta method.
Citation: Jianhe Shen, Shuhui Chen, Kechang Lin. Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 231-254. doi: 10.3934/dcdsb.2011.15.231
##### References:
 [1] M. Belhaq and A. Houssni, Symmetry-breaking and first period-doubling following a Hopf bifurcation in a three dimensional system, Mechanics Research Communication, 22 (1995), 221-231. doi: doi:10.1016/0093-6413(95)00016-K. [2] M. Belhaq, A. Houssni, E. Freire and A. J. Rodriguez-Luis, Analytical prediction of the two first period-doublings in a three dimensional system, International Journal of Bifurcation and Chaos, 10 (2000), 1497-1508. doi: doi:10.1142/S0218127400000943. [3] M. Belhaq, A. Houssni, E. Freire and A. J. Rodriguez-Luis, Asymptotics of homoclinic bifurcation in a three dimensional system, Nonlinear Dynamics, 21 (2000), 135-155. doi: doi:10.1023/A:1008353609572. [4] S. H. Chen, Y. K. Cheung and S. L. Lau, On perturbation procedure for limit cycle analysis, International Journal of Nonlinear Mechanics, 26 (1991), 125-133. doi: doi:10.1016/0020-7462(91)90086-9. [5] Y. K. Cheung, S. H. Chen and S. L. Lau, Application of the incremental harmonic balance method to cubic nonlinearity systems, Journal of Sound and Vibration, 140 (1990), 273-286. doi: doi:10.1016/0022-460X(90)90528-8. [6] W. G. Choe and J. Guckenheimer, Computing periodic orbits with high accuracy, Computer Methods in Applied Mechanics and Engineering, 170 (1999), 331-341. doi: doi:10.1016/S0045-7825(98)00201-1. [7] K. W. Chung, C. L. Chan and B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 299 (2007), 520-539. doi: doi:10.1016/j.jsv.2006.06.059. [8] P. Friedmann and C. E. Hammond, Efficient numerical treatment of periodic systems with application to stability problems, International Journal of Numerical Methods in Engineering, 11 (1977), 1117-1136. doi: doi:10.1002/nme.1620110708. [9] W. Govaerts, Y. A. Kuznetsov and A. Dhooge, Numerical continuation of bifurcation of limit cycles in Matlab, SIAM Journal of Scientific Computations, 27 (2005), 231-252. doi: doi:10.1137/030600746. [10] J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with autonomous differentiation, SIAM Journal of Scientific Computations, 22 (2000), 950-985. [11] B. D. Hassard, N. D. Hazzarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, New York, 1981. [12] C. S. Hsu, On approximation a general linear periodic system, Journal of Mathematics Analysis and Application, 45 (1974), 234-251. doi: doi:10.1016/0022-247X(74)90134-6. [13] Grisela R. Itovich and Jorge L. Moiola, On period doubling bifurcations of cycles and the harmonic balance method, Chaos, Solitons and Fractals, 27 (2006), 647-665. doi: doi:10.1016/j.chaos.2005.04.061. [14] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Volume 112 of Applied Mathematics Science, 2nd edition, Springer-Verlag, New York, 1998. [15] S. L. Lau, The incremental harmonic balance method and its application to nonlinear vibrations, in "Proceeding of International Conference on Structure Dynamics, Vibration, Noise and Control," Hong Kong, (1995), 50-57. [16] S. L. Lau and Y. K. Cheung, Amplitude incremental variational principle for nonlinear vibration of elastic systems, ASME Journal of Applied Mechanics, 48 (1981), 959-964. doi: doi:10.1115/1.3157762. [17] J. H. Merkin and D. J. Needhamu, An infinite period bifurcation arising in roll wave down an open inclined channel, Proceedings of Royal Society of London, Series A, Mathematical and Physical Sciences, 405 (1986), 103-116. doi: doi:10.1098/rspa.1986.0043. [18] G. Moore, Floquet theory as a computational tool, SIAM Journal of Numerical Analysis, 42 (2005), 2522-2568. doi: doi:10.1137/S0036142903434175. [19] A. H. Nayfeh and B. Balachandran, Motion near a Hopf bifurcation of a three dimensional system, Mechanics Research Communication, 17 (1990), 191-198. doi: doi:10.1016/0093-6413(90)90078-Q. [20] R. H. Rand, An analytical approximation for period-doubling following a Hopf bifurcation, Mechanics Research Communication, 16 (1989), 117-123. doi: doi:10.1016/0093-6413(89)90022-0. [21] E. Reithmeier, "Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation, Stability Bifurcation and Transition to Chaos," Lecture Notes of Mathematics 1483, Springer-Verlag, New York, 1991. [22] F. I. Robbio, D. M. Alonso and J. L. Moiola, Detection of limit cycle bifurcations using harmonic balance methods, International Journal of Bifurcation and Chaos, 14 (2000), 3647-3654. doi: doi:10.1142/S0218127404011491. [23] R. Seydel, "Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos," Springer-Verlag, New-York, 1994. [24] J. H. Shen. K. C. Lin, S. H. Chen and K. Y. Sze, Bifurcation and route-to-chaos analyses for Mathieu-Duffing oscillator by the incremental harmonic balance method, Nonlinear Dynamics, 52 (2008), 403-414. doi: doi:10.1007/s11071-007-9289-z. [25] J. D. Skufca, Analysis still matters: A surprising instance of failure of Runge-Kutta-Felberg ODE solvers, SIAM Review, 46 (2004), 729-737. doi: doi:10.1137/S003614450342911X. [26] B. H. Tongue, Characteristics of numerical simulations of chaotic systems, ASME Journal of Applied Mechanics, 54 (1987), 695-699. doi: doi:10.1115/1.3173090. [27] C. Wulff and A. Schebesch, Numerical continuation of symmetric periodic orbits, SIAM Journal of Applied Dynamical Systems, 5 (2006), 435-475. doi: doi:10.1137/050637170. [28] J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying the weak resonance double Hopf bifurcation in nonlinear systems with delayed feedbacks, SIAM Journal of Applied Dynamical Systems, 6 (2007), 29-60. doi: doi:10.1137/040614207. [29] W. X. Zhong, On precise integration method, Journal of Computational and Applied Mathematics, 163 (2004), 59-78. doi: doi:10.1016/j.cam.2003.08.053.

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##### References:
 [1] M. Belhaq and A. Houssni, Symmetry-breaking and first period-doubling following a Hopf bifurcation in a three dimensional system, Mechanics Research Communication, 22 (1995), 221-231. doi: doi:10.1016/0093-6413(95)00016-K. [2] M. Belhaq, A. Houssni, E. Freire and A. J. Rodriguez-Luis, Analytical prediction of the two first period-doublings in a three dimensional system, International Journal of Bifurcation and Chaos, 10 (2000), 1497-1508. doi: doi:10.1142/S0218127400000943. [3] M. Belhaq, A. Houssni, E. Freire and A. J. Rodriguez-Luis, Asymptotics of homoclinic bifurcation in a three dimensional system, Nonlinear Dynamics, 21 (2000), 135-155. doi: doi:10.1023/A:1008353609572. [4] S. H. Chen, Y. K. Cheung and S. L. Lau, On perturbation procedure for limit cycle analysis, International Journal of Nonlinear Mechanics, 26 (1991), 125-133. doi: doi:10.1016/0020-7462(91)90086-9. [5] Y. K. Cheung, S. H. Chen and S. L. Lau, Application of the incremental harmonic balance method to cubic nonlinearity systems, Journal of Sound and Vibration, 140 (1990), 273-286. doi: doi:10.1016/0022-460X(90)90528-8. [6] W. G. Choe and J. Guckenheimer, Computing periodic orbits with high accuracy, Computer Methods in Applied Mechanics and Engineering, 170 (1999), 331-341. doi: doi:10.1016/S0045-7825(98)00201-1. [7] K. W. Chung, C. L. Chan and B. H. K. Lee, Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 299 (2007), 520-539. doi: doi:10.1016/j.jsv.2006.06.059. [8] P. Friedmann and C. E. Hammond, Efficient numerical treatment of periodic systems with application to stability problems, International Journal of Numerical Methods in Engineering, 11 (1977), 1117-1136. doi: doi:10.1002/nme.1620110708. [9] W. Govaerts, Y. A. Kuznetsov and A. Dhooge, Numerical continuation of bifurcation of limit cycles in Matlab, SIAM Journal of Scientific Computations, 27 (2005), 231-252. doi: doi:10.1137/030600746. [10] J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with autonomous differentiation, SIAM Journal of Scientific Computations, 22 (2000), 950-985. [11] B. D. Hassard, N. D. Hazzarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," Cambridge University Press, New York, 1981. [12] C. S. Hsu, On approximation a general linear periodic system, Journal of Mathematics Analysis and Application, 45 (1974), 234-251. doi: doi:10.1016/0022-247X(74)90134-6. [13] Grisela R. Itovich and Jorge L. Moiola, On period doubling bifurcations of cycles and the harmonic balance method, Chaos, Solitons and Fractals, 27 (2006), 647-665. doi: doi:10.1016/j.chaos.2005.04.061. [14] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Volume 112 of Applied Mathematics Science, 2nd edition, Springer-Verlag, New York, 1998. [15] S. L. Lau, The incremental harmonic balance method and its application to nonlinear vibrations, in "Proceeding of International Conference on Structure Dynamics, Vibration, Noise and Control," Hong Kong, (1995), 50-57. [16] S. L. Lau and Y. K. Cheung, Amplitude incremental variational principle for nonlinear vibration of elastic systems, ASME Journal of Applied Mechanics, 48 (1981), 959-964. doi: doi:10.1115/1.3157762. [17] J. H. Merkin and D. J. Needhamu, An infinite period bifurcation arising in roll wave down an open inclined channel, Proceedings of Royal Society of London, Series A, Mathematical and Physical Sciences, 405 (1986), 103-116. doi: doi:10.1098/rspa.1986.0043. [18] G. Moore, Floquet theory as a computational tool, SIAM Journal of Numerical Analysis, 42 (2005), 2522-2568. doi: doi:10.1137/S0036142903434175. [19] A. H. Nayfeh and B. Balachandran, Motion near a Hopf bifurcation of a three dimensional system, Mechanics Research Communication, 17 (1990), 191-198. doi: doi:10.1016/0093-6413(90)90078-Q. [20] R. H. Rand, An analytical approximation for period-doubling following a Hopf bifurcation, Mechanics Research Communication, 16 (1989), 117-123. doi: doi:10.1016/0093-6413(89)90022-0. [21] E. Reithmeier, "Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation, Stability Bifurcation and Transition to Chaos," Lecture Notes of Mathematics 1483, Springer-Verlag, New York, 1991. [22] F. I. Robbio, D. M. Alonso and J. L. Moiola, Detection of limit cycle bifurcations using harmonic balance methods, International Journal of Bifurcation and Chaos, 14 (2000), 3647-3654. doi: doi:10.1142/S0218127404011491. [23] R. Seydel, "Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos," Springer-Verlag, New-York, 1994. [24] J. H. Shen. K. C. Lin, S. H. Chen and K. Y. Sze, Bifurcation and route-to-chaos analyses for Mathieu-Duffing oscillator by the incremental harmonic balance method, Nonlinear Dynamics, 52 (2008), 403-414. doi: doi:10.1007/s11071-007-9289-z. [25] J. D. Skufca, Analysis still matters: A surprising instance of failure of Runge-Kutta-Felberg ODE solvers, SIAM Review, 46 (2004), 729-737. doi: doi:10.1137/S003614450342911X. [26] B. H. Tongue, Characteristics of numerical simulations of chaotic systems, ASME Journal of Applied Mechanics, 54 (1987), 695-699. doi: doi:10.1115/1.3173090. [27] C. Wulff and A. Schebesch, Numerical continuation of symmetric periodic orbits, SIAM Journal of Applied Dynamical Systems, 5 (2006), 435-475. doi: doi:10.1137/050637170. [28] J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying the weak resonance double Hopf bifurcation in nonlinear systems with delayed feedbacks, SIAM Journal of Applied Dynamical Systems, 6 (2007), 29-60. doi: doi:10.1137/040614207. [29] W. X. Zhong, On precise integration method, Journal of Computational and Applied Mathematics, 163 (2004), 59-78. doi: doi:10.1016/j.cam.2003.08.053.
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