Spatial and temporal chaos in some Cellular Neural Networks
博士 === 國立交通大學 === 應用數學系 === 91 === This dissertation investigates the spatial and temporal chaos of some classes of Cellular Neural Networks(CNN). We describe more details as follows. Chapter 1 study the complexity of one-dimensional CNN mosaic...
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ndltd-TW-091NCTU05070272016-06-22T04:14:28Z http://ndltd.ncl.edu.tw/handle/37035056862322991905 Spatial and temporal chaos in some Cellular Neural Networks 在某些CellularNeuralNetworks的空間與時間的混沌 Ting-Hui Yang 楊定揮 博士 國立交通大學 應用數學系 91 This dissertation investigates the spatial and temporal chaos of some classes of Cellular Neural Networks(CNN). We describe more details as follows. Chapter 1 study the complexity of one-dimensional CNN mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates. Chapter 2 investigates bifurcations and chaos in two-cells CNN with periodic inputs. Without the inputs, the time periodic solutions are obtained for template $A=[r,p,s]$ with $p>1$, $r>p-1$ and $-s>p-1$. The number of periodic solutions can be proven to be no more than two in exterior region. The input is $b\sin 2\pi t/T$ with period $T>0$ and amplitude $b>0$. The typical trajectories $\Gamma(b,T,A)$ and their $\omega$-limit set $\omega(b,T,A)$ vary with $b$, $T$ and $A$ are considered. The asymptotic limit cycles $\Lambda_\infty(T,A)$ with period $T$ of $\Gamma(b,T,A)$ are obtained as $b\rightarrow\infty$. When $T_0\leq T_0^*$(given in \ref{t0start} ), $\Lambda_\infty$ and $-\Lambda_\infty$ can be separated. The onset of chaos can be induced by crises of $\omega(b,T,A)$ and $-\omega(b,T,A)$ for suitable $T$ and $b$. The ratio $\mathcal{A}(b)=|a_T(b)|/|a_1(b)|$, of largest amplitude $a_1(b)$ and amplitude of the $T$-mode of the Fast Fourier Transform (FFT) of $\Gamma(b,T,A)$, can be used to compare the strength of sustained periodic cycle $\Lambda_0(A)$ and the inputs. When $\mathcal{A}(b)\ll 1$, $\Lambda_0(A)$ dominates and the attractor $\omega(b,T,A)$ is either a quasi-periodic or a periodic. Moreover, the range $b$ of the window of periodic cycles constitutes a devil's staircase. When $\mathcal{A}(b)\sim 1$, finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large $b$, the input dominates, $\omega(b,T,A)$ becomes simpler, from quasi-periodic to periodic as $b$ increases. Song-Sun Lin 林松山 2003 學位論文 ; thesis 89 en_US |
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博士 === 國立交通大學 === 應用數學系 === 91 === This dissertation investigates the spatial and temporal
chaos of some classes of Cellular Neural Networks(CNN). We
describe more details as follows.
Chapter 1 study the complexity of one-dimensional CNN mosaic
patterns with spatially variant templates on finite and infinite
lattices. Various boundary conditions are considered for finite
lattices and the exact number of mosaic patterns is computed
precisely. The entropy of mosaic patterns with periodic templates
can also be calculated for infinite lattices. Furthermore, we show
the abundance of mosaic patterns with respect to template periods
and, which differ greatly from cases with spatially invariant
templates.
Chapter 2 investigates bifurcations and chaos in two-cells CNN
with periodic inputs. Without the inputs, the time periodic
solutions are obtained for template $A=[r,p,s]$ with $p>1$,
$r>p-1$ and $-s>p-1$. The number of periodic solutions can be
proven to be no more than two in exterior region. The input is
$b\sin 2\pi t/T$ with period $T>0$ and amplitude $b>0$. The
typical trajectories $\Gamma(b,T,A)$ and their $\omega$-limit set
$\omega(b,T,A)$ vary with $b$, $T$ and $A$ are considered. The
asymptotic limit cycles $\Lambda_\infty(T,A)$ with period $T$ of
$\Gamma(b,T,A)$ are obtained as $b\rightarrow\infty$. When
$T_0\leq T_0^*$(given in \ref{t0start} ), $\Lambda_\infty$ and
$-\Lambda_\infty$ can be separated. The onset of chaos can be
induced by crises of $\omega(b,T,A)$ and $-\omega(b,T,A)$ for
suitable $T$ and $b$. The ratio
$\mathcal{A}(b)=|a_T(b)|/|a_1(b)|$, of largest amplitude $a_1(b)$
and amplitude of the $T$-mode of the Fast Fourier Transform (FFT)
of $\Gamma(b,T,A)$, can be used to compare the strength of
sustained periodic cycle $\Lambda_0(A)$ and the inputs. When
$\mathcal{A}(b)\ll 1$, $\Lambda_0(A)$ dominates and the attractor
$\omega(b,T,A)$ is either a quasi-periodic or a periodic.
Moreover, the range $b$ of the window of periodic cycles
constitutes a devil's staircase. When $\mathcal{A}(b)\sim 1$,
finitely many chaotic regions and window regions exist and
interweave with each other. In each window, the basic periodic
cycle can be identified. A sequence of period-doubling is observed
to the left of the basic periodic cycle and a quasi-periodic
region is observed to the right of it. For large $b$, the input
dominates, $\omega(b,T,A)$ becomes simpler, from quasi-periodic to
periodic as $b$ increases.
|
author2 |
Song-Sun Lin |
author_facet |
Song-Sun Lin Ting-Hui Yang 楊定揮 |
author |
Ting-Hui Yang 楊定揮 |
spellingShingle |
Ting-Hui Yang 楊定揮 Spatial and temporal chaos in some Cellular Neural Networks |
author_sort |
Ting-Hui Yang |
title |
Spatial and temporal chaos in some Cellular Neural Networks |
title_short |
Spatial and temporal chaos in some Cellular Neural Networks |
title_full |
Spatial and temporal chaos in some Cellular Neural Networks |
title_fullStr |
Spatial and temporal chaos in some Cellular Neural Networks |
title_full_unstemmed |
Spatial and temporal chaos in some Cellular Neural Networks |
title_sort |
spatial and temporal chaos in some cellular neural networks |
publishDate |
2003 |
url |
http://ndltd.ncl.edu.tw/handle/37035056862322991905 |
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