Nonlinear Dyn (2009) 56: 255–268 DOI 10.1007/s11071-008-9397-4 ORIGINAL PAPER A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems S.H. Chen · J.H. Shen · K.Y. Sze Received: 8 January 2008 / Accepted: 9 July 2008 / Published online: 12 August 2008 © Springer Science+Business Media B.V. 2008 Abstract By introducing a new parametric transfor- mation and a suitable nonlinear frequency expansion, the modified Lindstedt–Poincaré method is extended to derive analytical approximations for limit cycles in three-dimensional nonlinear autonomous dynamical systems. By considering two typical examples, it can be seen that the results of the present method are in good agreement with those obtained numerically even if the control parameter is moderately large. More- over, the present prediction is considerably more ac- curate than some published results obtained by the multiple time scales method and the normal form method. Keywords New perturbation procedure · Limit cycle analysis · Modified L–P method · Three-dimensional systems S.H. Chen ( ) · J.H. Shen Department of Applied Mechanics and Engineering, Sun Yat-sen University, 510275, Guangzhou, People’s Republic of China e-mail: stscsh@mail.sysu.edu.cn K.Y. Sze Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, People’s Republic of China 1 Introduction Over the last several decades, many researchers have devoted their effort to extend and improve the clas- sical perturbation methods including the Lindstedt– Poincaré (L–P) method, the averaging method, the Krylov–Bogoliubov–Mitropolsky (KBM) method and the multiple scales method, see Nayfeh [1] and Mick- ens [2], to strongly nonlinear dynamical systems. Un- til now, various improved and novel perturbation tech- niques have been developed such as the modified L–P method [3], the elliptic L–P method [4], the elliptic perturbation method [5], the nonlinear time transfor- mation method [6], the generalized averaging method [7], the nonlinear scales method [8], the modified KBM method [9], etc. In particular, Chen et al. [10] pointed out that the use of conditions of constant phase angles in the perturbation procedures could provide more accurate results for limit cycle analysis, espe- cially for the strongly nonlinear cases. Starting from different viewpoints, He [11, 12] also extended the classical L–P method and developed two modified ver- sions of the L–P methods. All the aforementioned modified and extended per- turbation methods have their own advantages in ob- taining approximate analytical solutions. They are suitable for limit cycles analysis in dynamical systems with weak or strong nonlinearity. However, without exception, all these methods are limited to second- order nonlinear oscillators and none of them are ap- plicable to higher-dimensional nonlinear autonomous