Approximate first integrals of the H enon–Heiles system revisited G. € Unal a, * , C.M. Khalique b, * a Faculty of Sciences and Letters, Istanbul Technical University, Maslak, 80626 Istanbul, Turkey b Department of Mathematical Sciences, International Institute for Symmetry, Analysis and Mathematical Modelling, University of North-West, P. Bag X2046, Mmabatho 2735, South Africa Received 22 January 2003; received in revised form 26 May 2003; accepted 29 May 2003 Available online 12 August 2003 Abstract Approximate first integrals (conserved quantities) of H enon–Heiles system have been obtained based on the approximate Noether symmetries for the resonance (x 2 ¼ x 1 ) and off resonance. It has been shown that system undergoes complicated bifurcations near resonances. Analytical results have been verified by nu- merically obtained KAM curves on the Poincar e surface of section. Ó 2003 Published by Elsevier B.V. Keywords: Hamiltonian dynamical systems; Approximate Noether symmetries; Resonances; NoetherÕs theorem 1. Introduction The regular behaviour (order) observed in the numerical studies of the nearly integrable Hamiltonian systems [1,2] led to search for analytical methods on the approximate first integrals. Various perturbative methods have been developed to construct approximate first integrals, e.g., direct method of Contopoulos [1] and Birkhoff–Gustavson normal form method [3]. A compre- hensive study of these methods and others can be found in [4]. However, none of these methods resort to the celebrated NoetherÕs theorem which provides a link between the exact Noether * Corresponding authors. Tel.: +90-212-285-32-88; fax: +90-212-285-63-86 (G. € Unal), tel.: +27-18-389-2009; fax: +27- 18-389-2594 (C.M. Khalique). E-mail addresses: gunal@itu.edu.tr (G. € Unal), khaliquecm@uniwest.ac.za (C.M. Khalique). 1007-5704/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/S1007-5704(03)00063-7 Communications in Nonlinear Science and Numerical Simulation 10 (2005) 73–83 www.elsevier.com/locate/cnsns