在某些Cellular Neural Networks的空間與時間的混沌

Abstract

這篇論文主要探討細胞神經網路(CNN)的空間與時間的混沌現象

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.