Detecting resonances in conservative maps using evolutionary algorithms

Article history: Received 5 December 2007 Received in revised form 29 September A numerical method is proposed for detecting resonances of conservative maps which reduces this task to an optimization problem. We then solve this problem using evolutionary algorithms, which are methods for global optimization inspired by biological evolution. The proposed methodology is simple and can be easily applied to maps of arbitrary dimensions. In this Letter we apply it to several examples of 2- and 4-dimensional conservative maps, with quite promising results concerning integrability, the location of resonances and the presence of chaotic regions surrounding the island chains that correspond to these resonances.

[1]  Jeffrey Horn,et al.  Handbook of evolutionary computation , 1997 .

[2]  G. Contopoulos,et al.  A fast method for distinguishing between ordered and chaotic orbits. , 1997 .

[3]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[4]  W. Scandale,et al.  Nonlinear Problems in Future Particle Accelerators: Proceedings of the Workshop , 1991 .

[5]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[6]  M. Glasser,et al.  Mel'nikov's function for two-dimensional mappings , 1989 .

[7]  Ch. Skokos,et al.  Application of the SALI chaos detection method to accelerator mappings , 2006 .

[8]  B. M. Fulk.  MATH , 1992 .

[9]  Algebraic Non-Integrability of the Cohen Map , 1998 .

[10]  Michael N. Vrahatis,et al.  Structure and Breakdown of Invariant Tori in a 4-D Mapping Model of Accelerator Dynamics , 1997 .

[11]  Todesco,et al.  Dynamic-aperture estimates and phase-space distortions in nonlinear betatron motion. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Ezio Todesco,et al.  A normal form approach to the theory of nonlinear betatronic motion , 1994 .

[13]  Jacques Laskar,et al.  The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones , 1990 .

[14]  Riccardo Poli,et al.  New ideas in optimization , 1999 .

[15]  Michael N. Vrahatis,et al.  An efficient method for locating and computing periodic orbits of nonlinear mappings , 1995 .

[16]  A. Goriely,et al.  A Mel'nikov vector for N-dimensional mappings , 1995 .

[17]  Dimitris K. Tasoulis,et al.  Clustering in evolutionary algorithms to efficiently compute simultaneously local and global minima , 2005, 2005 IEEE Congress on Evolutionary Computation.

[18]  Jerrold E. Marsden,et al.  Time–frequency analysis of the restricted three-body problem: transport and resonance transitions , 2004 .

[19]  J. Meiss.  Symplectic maps, variational principles, and transport , 1992 .

[20]  Michael G. Epitropakis,et al.  Balancing the exploration and exploitation capabilities of the Differential Evolution Algorithm , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[21]  Thomas Bäck,et al.  Evolutionary algorithms in theory and practice - evolution strategies, evolutionary programming, genetic algorithms , 1996 .

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  Heinz Hanmann,et al.  Hamiltonian Dynamical Systems , 2007 .

[24]  Thomas Bäck,et al.  Evolutionary Algorithms in Theory and Practice , 1996 .