Study On Some Problems Of Integrable Systems And Chaotic Systems With Symbolic Computation

Based on symbolic computation, some problems of integrable systems and chaotic systems in nonlinear systems are investigated. There are two main parts in this disserta-tion:1. Some integrable properties of nonlinear equations are investigated with help of the prolongation structure method, Riccati type pseudopotentials and Bell polynomials. These properties include Lax pairs, auto-Backlund transformations, conservation laws, singularity manifold equations, bilinear forms and so on. A Maple package to obtain the bilinear forms of nonlinear evolution equations is developed; 2. A fractional-order gener-alized Lorenz canonical form and a new four-dimensional chaotic system are constructed and numerically simulated.In chapter 1, an introduction is devoted to review the background and the current situation related the dissertation, which include integrability of nonlinear system, prolon-gation structure method, symbolic computation and chaotic system.In chapter 2, the prolongation structure technique is improved and applied to Qiao equation. Two potentials and two pseudopotentials are obtained, from which a new spec-tral problem of inverse scattering transformation, Lax equations and infinite number of conserved laws are obtained. The prolongation structure method is generalized to the variable coefficient nonlinear evolution equations and applied to the variable coefficient KdV equation, from which Lax pairs and Pfaffian forms of variable coefficient KdV equa-tion are obtained.In chapter 3, the Riccati-type pseudopotentials of the generalized fifth-order KdV equation are derived, from which Lax pairs and singularity manifold equations can be obtained. Especially, new singularity manifold equations and auto-Backlund transforma-tions can be obtained under three conditions, which include CDG-SK equation, Lax equation and KK equation.In chapter 4, based on the Bell polynomials, a mechanization algorithm is proposed to obtain the bilinear forms of nonlinear evolution equations and the corresponding im-plementation software package in maple is developed. Firstly, the package transforms the equation into a dimensionless equation, which can be expressed in the linear combina-tions of P-polynomials, then the bilinear form of this equation can be obtained directly. The validity and reliability of this algorithm are verified by some examples.In chapter 5, a fractional-order generalized Lorenz canonical form and a new four-dimensional chaotic system are constructed and the dynamical properties are investigated. In addition, the numerical simulations and interesting figures are performed. By choos- ing different coefficient, the fractional-order system of classic Lorenz system, Lii system, Chen system, Shimizu-Morioka system and hyperbolic-type Lorenz system can be ob-tained.In chapter 6, the summary and discussion of this dissertation are given, as well as the outlook of future work is discussed.