Complex analysis of a piece of Toda lattice

We study a small piece of two-dimensional Toda lattice as a complex dynamical system. In particular, it is shown analytically how the Julia set, which appears when the piece is deformed, disappears as the system approaches the integrable limit.

[1]  S. Saito.  A dual resonance model solves the Yang - Baxter equation , 1997, solv-int/9704008.

[2]  I. Krichever,et al.  Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations , 1996, hep-th/9604080.

[3]  Takahashi,et al.  From soliton equations to integrable cellular automata through a limiting procedure. , 1996, Physical review letters.

[4]  Satoru Saito,et al.  An analysis of a family of rational maps containing integrable and non-integrable difference analogue of the logistic equation , 1996 .

[5]  R. Hirota,et al.  Pfaffian and determinant solutions to a discretized Toda equation for , and , 1995, hep-th/9509039.

[6]  J. Suzuki,et al.  FUNCTIONAL RELATIONS IN SOLVABLE LATTICE MODELS II: APPLICATIONS , 1993, hep-th/9310060.

[7]  J. Suzuki,et al.  FUNCTIONAL RELATIONS IN SOLVABLE LATTICE MODELS I: FUNCTIONAL RELATIONS AND REPRESENTATION THEORY , 1993, hep-th/9309137.

[8]  M. Bernhard.  Introduction to Chaotic Dynamical Systems , 1992 .

[9]  P. Pearce,et al.  Conformal weights of RSOS lattice models and their fusion hierarchies , 1992 .

[10]  R. Carroll.  Topics in soliton theory , 1991 .

[11]  N. Saitoh,et al.  Integrable lattice gauge model based on soliton theory , 1990 .

[12]  Saito.  String vertex on the Grassmann manifold. , 1988, Physical review. D, Particles and fields.

[13]  R. Hirota.  Discrete Two-Dimensional Toda Molecule Equation , 1987 .

[14]  Saito.  String theories and Hirota's bilinear difference equation. , 1987, Physical review letters.

[15]  Saito.  String amplitudes as solutions to soliton equations. , 1987, Physical review. D, Particles and fields.

[16]  K. Sogo.  A way from string to soliton ―introduction of KP coordinate to string amplitudes― , 1987 .

[17]  R. Hirota.  Solutions of the Classical Boussinesq Equation and the Spherical Boussinesq Equation: The Wronskian Technique , 1986 .

[18]  J. Nimmo,et al.  Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique , 1983 .

[19]  T. Miwa.  On Hirota's difference equations , 1982 .

[20]  Junkichi Satsuma,et al.  A Wronskian Representation of N-Soliton Solutions of Nonlinear Evolution Equations , 1979 .