| Research Abstract |
Generating function and its formal derivatives for one dimensional discrete dynamical systems are introduced. By taking its expansion parameter as complex, analytic properties of these functions are investigated. Especially it is found that the Lyapunov exponent is given by the radius of convergence of the first derivative function. Functional equations of these functions are also discussed. By introducing another one dimensional map in addition to the original one, a model of the generating function is given. It is found that the function is expressed as the basin boundary.
Extending one of the independent variables into complex, nonlinear interactions between solitons for the integrable equations, such as the Korteweg-de Vries equation, the Boussinesq equation, the Hirota-Ito equation and the Toda lattice, are investigated by observing behavior of poles of soliton solutions.
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