| Summary: | 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 $\sigma$.
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