Chaotic Dynamics of a Real Exchange Rate Determination Model and an OLG Model

• Main Authors: Yung-Ju Lin, 林咏儒
• Other Authors: Ming-Chia Li
• Format: Others
• Language: en_US
• Published: 2005
• Online Access: http://ndltd.ncl.edu.tw/handle/11294635170028422446

碩士 === 國立彰化師範大學 === 數學系所 === 93 === In the first part of the thesis, we study some dynamics of the map $$F_{\lambda}(q_{t})=q_{t}+\frac{1}{\lambda}(sq^3_{t}-rq^2_{t}+mq_{t}-n),$$ where $\lambda>0$ is a parameter and $s,r,m,n>0$. It is induced from a model of real exchange rate determination with linear demand functions for imports and exports. If $F_{\lambda}$ has only one fixed point, we prove that the fixed point is globally repelling. If $F_{\lambda}$ has exactly two fixed points, we prove that there exist no periodic point. If $F_{\lambda}$ has three distinct fixed points, we prove that $F_{\lambda}$ is chaotic in sense of Li and Yorke for all sufficiently small $\lambda$. In the second part, we study some dynamics of the map $m_{\alpha,\beta,\sigma,n,A}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ defined by $$m_{\alpha,\beta,\sigma,n,A}(k)= \frac{A(1-\alpha)k^\alpha}{(1+n)\left[1+\beta^{-\sigma}(\alpha A)^{1-\sigma}k^{(\alpha-1)(1-\sigma)}\right]},$$ where $\alpha\in(0,1)$, $\beta>0$, $\sigma>0$, $n\in(-1,\infty)$ and $A>0$ are parameters. It is induced from an OLG model with a Cobb-Douglas production function and a CIES utility function under myopic foresight. For $\beta>\frac{1-\alpha}{\alpha(1+n)}$, we prove that all orbits of $m_{\alpha,\beta,\sigma,n,A}$ are asymptotic to the positive fixed point for all sufficiently large $\sigma$. For $\beta<\frac{1-\alpha}{\alpha(1+n)}$, if $\frac{1-\alpha}{\alpha(1+n)(1+\beta)}<1$ and $(\frac{1-\alpha}{\alpha(1+n)(1+\beta)})^{1+\alpha}<\frac{\beta}{1+\beta}$ we prove that $m_{\alpha,\beta,\sigma,n,A}$ is chaotic in sense of Li and Yorke for all sufficiently large $\sigma$.