Rigorous investigation of the Ikeda map by means of interval arithmetic

In this work we present various tools for studying dynamical systems implemented in interval arithmetic. The methods include an algorithm for finding all low-period cycles enclosed in a specified region, finding an upper bound of the invariant and nonwandering part of a given set, finding a lower bound of the basin of attraction of a stable periodic orbit, and proving the existence of symbolic dynamics of a given type embedded in the map. Using these techniques a detailed study of the behaviour of the Ikeda map for different parameter values is performed.

[1]  R. Baker Kearfott,et al.  Algorithm 681: INTBIS, a portable interval Newton/bisection package , 1990, TOMS.

[2]  A. Szymczak.  A combinatorial procedure for finding isolating neighbourhoods and index pairs , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  O. Junge,et al.  Exploring invariant sets and invariant measures. , 1997, Chaos.

[4]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[5]  Ying-Cheng Lai,et al.  Towards complete detection of unstable periodic orbits in chaotic systems , 2001 .

[6]  M. Dellnitz,et al.  A subdivision algorithm for the computation of unstable manifolds and global attractors , 1997 .

[7]  P. Zgliczynski.  Computer assisted proof of chaos in the Rössler equations and in the Hénon map , 1997 .

[8]  R. Bowen.  Periodic points and measures for Axiom $A$ diffeomorphisms , 1971 .

[9]  C. Robinson.  Dynamical Systems: Stability, Symbolic Dynamics, and Chaos , 1994 .

[10]  Gary Froyland,et al.  Rigorous computation of topological entropy with respect to a finite partition , 2001 .

[11]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[12]  C. S. Hsu,et al.  GLOBAL ANALYSIS BY CELL MAPPING , 1992 .

[13]  Zbigniew Galias,et al.  Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map , 2001 .

[14]  Christopher K. R. T. Jones,et al.  Global dynamical behavior of the optical field in a ring cavity , 1985 .

[15]  M. Dellnitz,et al.  Finding zeros by multilevel subdivision techniques , 2002 .

[16]  J. Molenaar,et al.  Mappings of grazing-impact oscillators , 2001 .

[17]  A. Gibbons.  Algorithmic Graph Theory , 1985 .

[18]  S. Newhouse,et al.  On the estimation of topological entropy , 1993 .

[19]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.