Global attractor and topological chaos of second-order difference equations in discrete Hamiltonian systems

• Main Authors: Huang, Po-Ying, 黃柏穎
• Other Authors: Li, Ming-Chia
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/44498827261819052017

碩士 === 國立交通大學 === 應用數學系所 === 99 === In this thesis, we discuss two distinct dynamics of the difference equation ∆[p∆x(t-1)]+qx(t)=f(x(t-1)) or f(x(t)), t∈Z, where ∆x(t-1)=ax(t)-bx(t-1). These two dynamics are the behavior of globally attracting and topological chaos. We have several results. Under some conditions of a, b, p and q, every orbit of the equation asymptotically converges to a global attractor. See theorems 2.2 and 2.3. If there exists a function relating to f which has more than one simple zeros or positive topological entropy at an expected parametric value, then the shift map restricted to the set of solutions of this equation has topological chaos. See theorems 2.6, 2.7, 2.8 and 2.9. Finally, we transform this equation into a parameterized continuous function by changing variables. We can also write it as the form of a discrete Hamiltonian system. For the case f(x(t)), theorem 2.10 says that there exists a function relating to f which has positive topological entropy such that the corresponding function has topological chaos. For the case f(x(t-1)), with an additional assumption that the function relating to f is locally trapping, theorem 2.11 says that the corresponding function has also topological chaos.