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 & 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. doi: doi:10.1016/0093-6413(95)00016-K. Google Scholar

[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. doi: doi:10.1142/S0218127400000943. Google Scholar

[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. doi: doi:10.1023/A:1008353609572. Google Scholar

[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. doi: doi:10.1016/0020-7462(91)90086-9. Google Scholar

[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. doi: doi:10.1016/0022-460X(90)90528-8. Google Scholar

[6]

W. G. Choe and J. Guckenheimer, Computing periodic orbits with high accuracy,, Computer Methods in Applied Mechanics and Engineering, 170 (1999), 331. doi: doi:10.1016/S0045-7825(98)00201-1. Google Scholar

[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. doi: doi:10.1016/j.jsv.2006.06.059. Google Scholar

[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. doi: doi:10.1002/nme.1620110708. Google Scholar

[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. doi: doi:10.1137/030600746. Google Scholar

[10]

J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with autonomous differentiation,, SIAM Journal of Scientific Computations, 22 (2000), 950. Google Scholar

[11]

B. D. Hassard, N. D. Hazzarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,", Cambridge University Press, (1981). Google Scholar

[12]

C. S. Hsu, On approximation a general linear periodic system,, Journal of Mathematics Analysis and Application, 45 (1974), 234. doi: doi:10.1016/0022-247X(74)90134-6. Google Scholar

[13]

Grisela R. Itovich and Jorge L. Moiola, On period doubling bifurcations of cycles and the harmonic balance method,, Chaos, 27 (2006), 647. doi: doi:10.1016/j.chaos.2005.04.061. Google Scholar

[14]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Volume \textbf{112} of Applied Mathematics Science, 112 (1998). Google Scholar

[15]

S. L. Lau, The incremental harmonic balance method and its application to nonlinear vibrations,, in, (1995), 50. Google Scholar

[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. doi: doi:10.1115/1.3157762. Google Scholar

[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, 405 (1986), 103. doi: doi:10.1098/rspa.1986.0043. Google Scholar

[18]

G. Moore, Floquet theory as a computational tool,, SIAM Journal of Numerical Analysis, 42 (2005), 2522. doi: doi:10.1137/S0036142903434175. Google Scholar

[19]

A. H. Nayfeh and B. Balachandran, Motion near a Hopf bifurcation of a three dimensional system,, Mechanics Research Communication, 17 (1990), 191. doi: doi:10.1016/0093-6413(90)90078-Q. Google Scholar

[20]

R. H. Rand, An analytical approximation for period-doubling following a Hopf bifurcation,, Mechanics Research Communication, 16 (1989), 117. doi: doi:10.1016/0093-6413(89)90022-0. Google Scholar

[21]

E. Reithmeier, "Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation, Stability Bifurcation and Transition to Chaos,", Lecture Notes of Mathematics \textbf{1483}, 1483 (1991). Google Scholar

[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. doi: doi:10.1142/S0218127404011491. Google Scholar

[23]

R. Seydel, "Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos,", Springer-Verlag, (1994). Google Scholar

[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. doi: doi:10.1007/s11071-007-9289-z. Google Scholar

[25]

J. D. Skufca, Analysis still matters: A surprising instance of failure of Runge-Kutta-Felberg ODE solvers,, SIAM Review, 46 (2004), 729. doi: doi:10.1137/S003614450342911X. Google Scholar

[26]

B. H. Tongue, Characteristics of numerical simulations of chaotic systems,, ASME Journal of Applied Mechanics, 54 (1987), 695. doi: doi:10.1115/1.3173090. Google Scholar

[27]

C. Wulff and A. Schebesch, Numerical continuation of symmetric periodic orbits,, SIAM Journal of Applied Dynamical Systems, 5 (2006), 435. doi: doi:10.1137/050637170. Google Scholar

[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. doi: doi:10.1137/040614207. Google Scholar

[29]

W. X. Zhong, On precise integration method,, Journal of Computational and Applied Mathematics, 163 (2004), 59. doi: doi:10.1016/j.cam.2003.08.053. Google Scholar

show all references

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. doi: doi:10.1016/0093-6413(95)00016-K. Google Scholar

[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. doi: doi:10.1142/S0218127400000943. Google Scholar

[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. doi: doi:10.1023/A:1008353609572. Google Scholar

[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. doi: doi:10.1016/0020-7462(91)90086-9. Google Scholar

[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. doi: doi:10.1016/0022-460X(90)90528-8. Google Scholar

[6]

W. G. Choe and J. Guckenheimer, Computing periodic orbits with high accuracy,, Computer Methods in Applied Mechanics and Engineering, 170 (1999), 331. doi: doi:10.1016/S0045-7825(98)00201-1. Google Scholar

[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. doi: doi:10.1016/j.jsv.2006.06.059. Google Scholar

[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. doi: doi:10.1002/nme.1620110708. Google Scholar

[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. doi: doi:10.1137/030600746. Google Scholar

[10]

J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with autonomous differentiation,, SIAM Journal of Scientific Computations, 22 (2000), 950. Google Scholar

[11]

B. D. Hassard, N. D. Hazzarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,", Cambridge University Press, (1981). Google Scholar

[12]

C. S. Hsu, On approximation a general linear periodic system,, Journal of Mathematics Analysis and Application, 45 (1974), 234. doi: doi:10.1016/0022-247X(74)90134-6. Google Scholar

[13]

Grisela R. Itovich and Jorge L. Moiola, On period doubling bifurcations of cycles and the harmonic balance method,, Chaos, 27 (2006), 647. doi: doi:10.1016/j.chaos.2005.04.061. Google Scholar

[14]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Volume \textbf{112} of Applied Mathematics Science, 112 (1998). Google Scholar

[15]

S. L. Lau, The incremental harmonic balance method and its application to nonlinear vibrations,, in, (1995), 50. Google Scholar

[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. doi: doi:10.1115/1.3157762. Google Scholar

[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, 405 (1986), 103. doi: doi:10.1098/rspa.1986.0043. Google Scholar

[18]

G. Moore, Floquet theory as a computational tool,, SIAM Journal of Numerical Analysis, 42 (2005), 2522. doi: doi:10.1137/S0036142903434175. Google Scholar

[19]

A. H. Nayfeh and B. Balachandran, Motion near a Hopf bifurcation of a three dimensional system,, Mechanics Research Communication, 17 (1990), 191. doi: doi:10.1016/0093-6413(90)90078-Q. Google Scholar

[20]

R. H. Rand, An analytical approximation for period-doubling following a Hopf bifurcation,, Mechanics Research Communication, 16 (1989), 117. doi: doi:10.1016/0093-6413(89)90022-0. Google Scholar

[21]

E. Reithmeier, "Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation, Stability Bifurcation and Transition to Chaos,", Lecture Notes of Mathematics \textbf{1483}, 1483 (1991). Google Scholar

[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. doi: doi:10.1142/S0218127404011491. Google Scholar

[23]

R. Seydel, "Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos,", Springer-Verlag, (1994). Google Scholar

[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. doi: doi:10.1007/s11071-007-9289-z. Google Scholar

[25]

J. D. Skufca, Analysis still matters: A surprising instance of failure of Runge-Kutta-Felberg ODE solvers,, SIAM Review, 46 (2004), 729. doi: doi:10.1137/S003614450342911X. Google Scholar

[26]

B. H. Tongue, Characteristics of numerical simulations of chaotic systems,, ASME Journal of Applied Mechanics, 54 (1987), 695. doi: doi:10.1115/1.3173090. Google Scholar

[27]

C. Wulff and A. Schebesch, Numerical continuation of symmetric periodic orbits,, SIAM Journal of Applied Dynamical Systems, 5 (2006), 435. doi: doi:10.1137/050637170. Google Scholar

[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. doi: doi:10.1137/040614207. Google Scholar

[29]

W. X. Zhong, On precise integration method,, Journal of Computational and Applied Mathematics, 163 (2004), 59. doi: doi:10.1016/j.cam.2003.08.053. Google Scholar

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