F(x)-x bağlantılı bir kaotik hücresel yapay sinir ağının nümerik olarak incelenmesi

Abstract

Bu tezde,f(x)-x bağlantılı Chua devresi biçimindeki hücrelerin iki boyutlu bir dizisi şeklinde tanımlı olan, 2-boyutlu 1 -katmanlı kaotik Hücresel Yapay Sinir Ağının (HYSA) bilgisayar benzetimi yardımıyla nümerik incelemesi yapılmıştır. Bu amaçla CCNN bilgisayar programı yazılarak, bir kişisel bilgisayar ortamında benzetim düzem elde edilmiştir. Hopfield ağı ve Hücresel Yapay Sinir Ağı (HYSA) gibi bilinen bir çok dinamik YSA modeli, kalıcı rejim dinamik davranışı sabit yörüngeler olan tam kararlı dinamik sistemler olarak çalışırlar. Tam kararlı YSA'lar ile karşılaştırıldığın da kaotik HYSA nın kalıcı rejim dinamik davranışı olarak sabit yörüngeler, kaos ve osilasyonlan içine alan daha zengin bir dinamik davranış repertuarı vardır. Bu tip kaotik YSA'lan araştırma çahşmalannın itici gücü olarak, karmaşık dinamik davranışların karmaşık bilgi ve işaret işlemede sağlayacağı olası olanaklar ile biyolojik sinir ağlarının koku alma bölgesi, görme bölgesi gibi çeşitli altağlannda gözlemlenen osilasyon ve kaotik davranışlar gösterilebilir. Bu tezde kaotik HYSA'nın bilgisayar benzetimi yoluyla nümerik incelenmesi amacıyla pek çok deney yapılmıştır. Bu deney sonuçlarına göre kaotik HYSA nın sınır etkisine çok duyarlı olduğu bu etkiyi gidermek için ağın uzayda bir torus üzerinde tanımlanması gerektiği bulunmuştur.Hücrelerin dış uyan altında ve P-as parametre değişiklikleri altoda dallanma analizleri yapılmış ve ağın Çift Lüle Çekicisi, Spiral Chua Çekicisi, Periodic Çekiciler, Sabit Yörüngeler ve Büyük Limit Çevrimi içine alan geniş bir dinamik davranış kümesindeki çekicilere dallanabildiği gözlemlenmiştir. Hücrelerin farklı başlangıç koşullarından başlatılması durumunda faz senkronizasyonunu sağlayan bir kısıtlama, deneysel yolla bulunmuş ve faz senkronizasyonundan yararlanarak örüntü tanımaya yönelik iki uygulama örneği verilmiştir. Hücrelerin büyük limit çevrim rejiminde çalışmasının, hücrenin bağlantı ağırlıklarını değiştirmek yoluyla diğer dinamik rejimlere dallanmasını engelleyen bir kriz durumuna yol açtığı anlaşılmıştır. Elde edilen benzetim sonuçlan ağın daha sonra yapılacak teorik incelemesine taban olarak kullanılabileceği gibi teorik yolla bulunan sonuçların sağlamasında da kullanılabilir.

In this thesis, the simulation results on a 2-D array of f(x)-x coupled Chua's circuits, called 2-D chaotic Cellular Neural Network (CNN) [3] are presented. The chaotic CNN is a special case of the generalized CNN introduced by Güzeliş and Chua [4]. It is observed via computer simulations that depending on the internal parameters of the cells and the connection weights, the chaotic CNN has several attractors such as spiral Chua, double scroll, large limit cycle, period- 1 limit cycle, period-2 limit cycle and stable equilibria. In the past three decades a number of neural network architectures have been developed. The architectures have been inspired both by the principles governing biological neural systems and the well-established theories of engineering and fundamental sciences. Most of the widely applied neural networks fall into two main classes: 1) memoryless neural networks and 2) dynamical neural networks. From a circuit theoretical point of view, the memoryless neural networks are nonlinear resistive circuits, while the dynamical neural networks are non-linear R-L-C circuits. A memoryless neural network defines a nonlinear transformation from the space of input signals into the space of output signals. Such networks have been successfully used in pattern recognition and several problems which can be defined as a nonlinear transformation between two spaces. As in the Hopfield network and Cellular Neural Network, dynamical neural networks have usually been designed as dynamical systems where the inputs are set of some constant values and each trajectory approaches one of the stable equilibrium points depending upon the initial state and therefore called compeletely stable neural networks. Some useful application of these networks includes image processing, pattern recognition and optimization. In the last five years,neural networks which exhibit more complicated nonlinear dynamics including oscilation and chaos, have become an important research field. The observation of chaotic dynamics in biological neural networks and VI the rich repertoire of dynamical behaviours of chaotic systems are two main reasons for investigating such chaotic neural networks. The chaotic CNN model studied in this thesis has slightly more complicated nonlinear dynamics than completely stable neural networks.Each cell in the chaotic CNN becomes a Chua's circuit when it is isolated from the external inputs and from the neighbouring cells.Therefore, the isolated cells exhibit a variety of nonlinear dynamical phenomena including constant trajectories, oscillation and chaos [13], [16], [19], [20]. The whole network is then an interconnection of potentially chaotic subcircuits and it may possesses a collection of constant, oscillatory and chaotic dynamics bifurcating from one to another depending on the connection weights, external inputs and thresholds. The extensive investigations [16], [19], [20] on the bifurcations and chaos in Chua's circuit constitute a basis for the analysis of the chaotic CNN. As depicted in Figure 1, each cell of chaotic CNN is made up of 3 units: i) a linear resistive circuit summing up.the external inputs and the outputs of the neighbouring cells. ii) a 3 rd order linear dynamical circuit consisting of two capacitors, an inductor and two resistors. iii) a nonlinear controlled source having symmetrical, 7-region, piecewise linear characteristic. - 0 I.(u.) yy*) 1 -x I. i VO.i=0i UMTI UNTTH TJNTTIH Figure 1. Circuit diagram of a cell in chaotic CNN. vn The cells in a chaotic CNN are connected only to the cells in their nearest neighborhood defined by the following formula: Nr{i,j) = { C{k,l)\ max{\ k-i\,\ I- j\} <r,l< k="" <="" m;\<="" i="" n="" }="" where="" (i="" j)="" is="" the="" vector="" of="" integers="" indexing="" cell="" c(i="" in="" th="" row="" j="" column="" 2-dimensional="" array="" and="" n,="" m="" are="" numbers="" rows="" columns="" 2="" d="" respectively.="" system="" equations="" describing="" a="" chaotic="" cnn="" with="" neighborhood="" size="" r="l" normalized="" dimensionless="" form="" given="" (l)-(2).="" -y="" 0="" 1="" -1="" -="" 3="" 0.="" +="" a.k,l="" s="" if,="" k,l="" 6="" nr="" (1)="" yk)="7(»»0-OTl)(" x9="" l="" x9-l="" ^="" (mi-m2)="" \="" xy="" e2="" ~="" -m2="" v="" ej="" e^="" (2)="" it="" known="" [3]="" that="" has="" always="" bounded="" output="" under="" excitation.="" values="" connection="" weights="" does="" not="" affect="" boundedness="" cnn.="" therefore="" neccesary="" to="" apply="" constraint="" weights.="" however,="" throughout="" thesis="" feedback="" aid="" input="" ba="" chosen="" symmetrical="" space="" invariant="" order="" reduce="" computational="" costs="" def="" a-l,~\="a\,l" =="" ava-\,q="" 3-1,0="a2" ?="" style="margin: 0px; padding: 0px; outline: 0px;">a-\,l ~ ^1,-1 = ^3>"o,-l ~ a0,l = ^4 ' ^0,0 = **S> def def def dtf def £-1,-1 = Ki = bl > b-W = b1.0 =b2 > K,l = K-\ ~ b3 > K-l = Ki = K > \o = b5. Chaotic CNN considered in this thesis differs from the other known Chua's circuit arrays in the following respects : i) The models in [7], [8], [9] are 1-D arrays of Chua's circuits. On the contrary, the chaotic CNN is a 2-D array of Chua's circuits. vm ii) In the 2-D array of Chua's circuits used in [10], the individual cells are designed not to be operated in the chaotic regime even when they isolated in contrast to chaotic CNN. iii) As a direct consequence of unbounded 3 segment piecewise linear nonlinearity used in models [7], [8], [9], only resistive couplings i.e. excitatory connections are allowed in these models. The chaotic CNN can inhibitory connections as well as excitatory connections, iv) The coupling in the chaotic CNN is nonlinear as opposed to the other known Chua's circuit arrays [7], [8], [9], [10], the output f(x) of a cell in chaotic CNN is fed to the state variable x of other cells. Therefore it is called f(x)-x coupling, v) In the model in [9], the coupling between the cells is unilateral. In order to calculate the trajectories of the chaotic CNN a simulator software which called CCNN has been developed. CCNN employes the forward Euler algorithm with a constant iteration step to integrate the system of differential equations given in (1). It has been observed via computer simulations that chaotic CNN is very sensitive to the boundary effectBecause of this effect the cells at the boundaries of the 2-D chaotic CNN cause global distortion of the phase synchronization. To overcome this problem a cyclic boundary has been chosen. In order to analyze the bifurcations under the absence of any stimulation, a set of simulation experiments has been done. The simulation results show that changing the internal cell parameters and the feedback connection weights cause bifurcations like happened in P-as space as depicted in Figure 2, among double scroll attractor, spiral Chua attractor, period-n limit cycles, stable oscilations, large limit cycle, equilibrium points. Consequently, if the internal cell parameters of each cell are selected to operate in chaotic regime, it has been shown that excitation of chaotic CNN with negative and positive constant sources lead to bifurcations between spiral Chua attractors around the P. and P+ equilibrium points respectively. DC 13 Convergence to the P+ and P. equilibrium points Large chaotic attractor with 2 wings Elyptic limit cycle around P+andP. Q9 + 08 + Spiral Chua attractor Period n > 2 limit cycle around P+ and P. Q7- period-1 limit cycle around P+ and P. OB ÜS - Convergence to the 0 equilibrium point t) tl 12 13 14 15 17 « V Figure 2. P- as bifurcations It has been experimentaly found that the costraint given in (3) lead to completely phase synchronization for the chaotic CNN. a5<0.72, IX = 1, i e {1,2,3,4,5,} (3) Using the phase synchronization property of chaotic CNN, two pattern recognition application examples are given. It has been observed that the large limit cycle lead to a krisis efect Because the cells operating in the large limit cycle regime cannot bifurcate to the other dynamics by varying the connection weights. The structure of the thesis is as follows. Chapter 1 gives a general introduction to chaotic cellular neural networks. The circuit structure of the cells and the connection topology of the whole chaotic CNN is described in Chapter 2. The Chua circuit which corresponds to the isolated cells in chaotic CNN, is presented in Chapter 3. The Computer simulation results according to the chaotic CNN are presented in Chapter 4.Conclusion is given in Chapter 5. Appendix A is a guide for the simulation tool which has been used to simulate chaotic CNN throughout the thesis.</r,l<>