ON THE MARGIN OF COMPLETE STABILITY FOR A CLASS OF CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (05) ◽  
pp. 1343-1361 ◽  
Author(s):  
MAURO DI MARCO ◽  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Neural Networks ◽  
Dynamical Behavior ◽  
Stability Margin ◽  
Cellular Neural Networks ◽  
Third Order ◽  
Dimensional Parameter ◽  
The Family ◽  
A New Technique ◽  
Two Parameters

In this paper, the dynamical behavior of a class of third-order competitive cellular neural networks (CNNs) depending on two parameters, is studied. The class contains a one-parameter family of symmetric CNNs, which are known to be completely stable. The main result is that it is a generic property within the family of symmetric CNNs that complete stability is robust with respect to (small) nonsymmetric perturbations of the neuron interconnections. The paper also gives an exact evaluation of the complete stability margin of each symmetric CNN via the characterization of the whole region in the two-dimensional parameter space where the CNNs turn out to be completely stable. The results are established by means of a new technique to investigate trajectory convergence of the considered class of CNNs in the nonsymmetric case.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

BIFURCATIONS AND OSCILLATORY BEHAVIOR IN A CLASS OF COMPETITIVE CELLULAR NEURAL NETWORKS

2000 ◽  
Vol 10 (06) ◽  
pp. 1267-1293 ◽  
Author(s):  
M. DI MARCO ◽  
A. TESI ◽  
M. FORTI
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Opposite Sign ◽  
Symmetric Matrix ◽  
General Theorem ◽  
Dynamical Behavior ◽  
Cellular Neural Networks ◽  
Oscillatory Behavior ◽  
Third Order ◽  
Small Perturbations

When the neuron interconnection matrix is symmetric, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988a] are known to be completely stable, that is, each trajectory converges towards some stationary state. In this paper it is shown that the interconnection symmetry, though ensuring complete stability, is not in the general case sufficient to guarantee that complete stability is robust with respect to sufficiently small perturbations of the interconnections. To this end, a class of third-order CNN's with competitive (inhibitory) interconnections between distinct neurons is introduced. The analysis of the dynamical behavior shows that such a class contains nonsymmetric CNN's exhibiting persistent oscillations, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This result is of obvious relevance in view of CNN's implementation, since perfect interconnection symmetry in unattainable in hardware (e.g. VLSI) realizations. More insight on the behavior of the CNN's here introduced is gained by discussing the analogies with the dynamics of the May and Leonard model of the voting paradox, a special Volterra–Lotka model of three competing species. Finally, it is shown that the results in this paper can also be viewed as an extension of previous results by Zou and Nossek for a two-cell CNN with opposite-sign interconnections between distinct neurons. Such an extension has a significant interpretation in the framework of a general theorem by Smale for competitive dynamical systems.

ON THE EFFECT OF NEURON ACTIVATION GAIN ON ROBUSTNESS OF COMPLETE STABILITY

2004 ◽  
Vol 14 (05) ◽  
pp. 1807-1811 ◽  
Author(s):  
M. DI MARCO ◽  
M. FORTI ◽  
P. NISTRI ◽  
A. TESI
Keyword(s):  
Neural Networks ◽  
Stability Margin ◽  
Third Order ◽  
Stability Robustness ◽  
Neuron Activation

The paper addresses robustness of complete stability with respect to perturbations of the interconnections of nominal symmetric neural networks. The influence of the maximum neuron activation gain on complete stability robustness is discussed for a class of third-order neural networks. It is shown that high values of the gain lead to an extremely small complete stability margin of all nominal symmetric neural networks, thus allowing to conclude that complete stability robustness cannot be, in general, guaranteed.

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

DYNAMICAL AND COMPLEXITY RESULTS FOR HIGH ORDER NEURAL NETWORKS

1994 ◽  
Vol 05 (03) ◽  
pp. 241-252 ◽  
Author(s):  
ERIC GOLES ◽  
MARTÍN MATAMALA
Keyword(s):  
Neural Networks ◽  
Dynamical Behavior ◽  
Energy Functional ◽  
Iteration Scheme ◽  
High Order ◽  
The Other ◽  
The Family ◽  
Linear Neural Network ◽  
Sequential Iteration ◽  
Complex Dynamical Behavior

We present dynamical results concerning neural networks with high order arguments. More precisely, we study the family of block-sequential iteration of neural networks with polynomial arguments. In this context, we prove that, under a symmetric hypothesis, the sequential iteration is the only one of this family to converge to fixed points. The other iteration modes present a highly complex dynamical behavior: non-bounded cycles and simulation of arbitrary non-symmetric linear neural network.5 We also study a high order memory iteration scheme which accepts an energy functional and bounded cycles in the size of the memory steps.

Sign in / Sign up

Export Citation Format

Share Document