Stability of Equilibrium Points and Storage Capacity of Hopfield Neural Networks with Higher Order Nonlinearity

2006 ◽  
Author(s):  
M.R. Rajati ◽  
M.B. Menhaj
Keyword(s):  
Neural Networks ◽  
Storage Capacity ◽  
Equilibrium Points ◽  
Higher Order ◽  
Hopfield Neural Networks ◽  
Stability Of Equilibrium ◽  
And Storage

Storage Capacity of Quaternion-Valued Hopfield Neural Networks with Dual Connections

2021 ◽  
pp. 1-15
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Stochastic Analysis ◽  
Storage Capacity ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Hopfield Neural Networks ◽  
Noise Tolerance ◽  
Complex Valued ◽  
Projection Rule

Abstract A complex-valued Hopfield neural network (CHNN) is a multistate Hopfield model. A quaternion-valued Hopfield neural network (QHNN) with a twin-multistate activation function was proposed to reduce the number of weight parameters of CHNN. Dual connections (DCs) are introduced to the QHNNs to improve the noise tolerance. The DCs take advantage of the noncommutativity of quaternions and consist of two weights between neurons. A QHNN with DCs provides much better noise tolerance than a CHNN. Although a CHNN and a QHNN with DCs have the samenumber of weight parameters, the storage capacity of projection rule for QHNNs with DCs is half of that for CHNNs and equals that of conventional QHNNs. The small storage capacity of QHNNs with DCs is caused by projection rule, not the architecture. In this work, the ebbian rule is introduced and proved by stochastic analysis that the storage capacity of a QHNN with DCs is 0.8 times as many as that of a CHNN.

Nonlinear Dynamics of Three-Neurons-Based Hopfield Neural Networks (HNNs): Remerging Feigenbaum Trees, Coexisting Bifurcations and Multiple Attractors

2019 ◽  
Vol 28 (07) ◽  
pp. 1950121 ◽  
Author(s):  
Z. T. Njitacke ◽  
J. Kengne
Keyword(s):  
Neural Networks ◽  
Synaptic Weight ◽  
Equilibrium Points ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Simplified Model ◽  
Symmetric Pair ◽  
Conditional Stability ◽  
Frequency Spectra ◽  
Dynamical Behaviors

In this work, the dynamics of a simplified model of three-neurons-based Hopfield neural networks (HNNs) is investigated. The simplified model is obtained by removing the synaptic weight connection of the third and second neuron in the original Hopfield networks introduced in Ref. 11 . The investigations have shown that the simplified model possesses three equilibrium points among which origin of the systems coordinates. It is found that the origin is always unstable while the symmetric pair of fixed points with conditional stability has values depending on synaptic weight between the second and the first neuron that is used as bifurcation control parameter. Numerical simulations, carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra are employed to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the modified model of HNNs exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g., coexistence of two, four and six disconnected periodic and chaotic attractors) which have not been reported in previous works focused on the dynamics of HNNs. Finally, PSpice simulations verify the results of theoretical analyses of the simplified model of three-neurons-based HNNs.

New estimate on the domains of attraction of equilibrium points in continuous Hopfield neural networks

2006 ◽  
Vol 351 (3) ◽  
pp. 161-166 ◽  
Author(s):  
Xiaofan Yang ◽  
Xiaofeng Liao ◽  
Chuandong Li ◽  
David J. Evans
Keyword(s):  
Neural Networks ◽  
Equilibrium Points ◽  
Hopfield Neural Networks ◽  
Domains Of Attraction

Multistability Analysis of State-Dependent Switched Hopfield Neural Networks With The Gaussian-Wavelet-Type Activation Function*

2021 ◽  
Author(s):  
Yang Liu ◽  
Zhen Wang
Keyword(s):  
Neural Networks ◽  
Multiple Equilibria ◽  
Storage Capacity ◽  
Activation Function ◽  
Stability Theorem ◽  
Hopfield Neural Networks ◽  
Lyapunov Stability Theorem ◽  
Activation Functions ◽  
State Dependent ◽  
Theoretical Results

Abstract This paper studies the multistability of state-dependent switched Hopfield neural networks (SSHNNs) with the Gaussian-wavelet-type activation function. The coexistence and stability of multiple equilibria of SSHNNs are proved. By using Brouwer's fixed point theorem, it is obtained that the SSHNNs can have at least 7n or 6n equilibria under a specified set of conditions. By using the strictly diagonally dominance matrix (SDDM) theorem and Lyapunov stability theorem, 4n or 5n locally stable (LS) equilibria are obtained, respectively. Compared with the conventional Hopfield neural networks (HNNs) without state-dependent switching or SSHNNs with other kinds of activation functions, SSHNNs with this type of activation functions can have more LS equilibria, which implies that SSHNNs with Gaussian-wavelet-type activation functions can have even larger storage capacity and would be more dominant in associative memory application. Last, some simulation results are given to verify the correctness of the theoretical results.

Fault prognostics using dynamic wavelet neural networks

2001 ◽  
Vol 15 (4) ◽  
pp. 349-365 ◽  
Author(s):  
PENG WANG ◽  
GEORGE VACHTSEVANOS
Keyword(s):  
Neural Networks ◽  
Storage Capacity ◽  
Structure Learning ◽  
Condition Based Maintenance ◽  
Wavelet Neural Networks ◽  
And Performance ◽  
Fault Prognostics ◽  
And Storage ◽  
Useful Lifetime ◽  
Inner Race

Modern industry is concerned about extending the lifetime of its critical processes and maintaining them only when required. Significant aspects of these trends include the ability to diagnose impending failures, prognosticate the remaining useful lifetime of the process and schedule maintenance operations so that uptime is maximized. Prognosis is probably the most difficult of the three issues leading to condition-based maintenance (CBM). This paper attempts to address this challenging problem with intelligence-oriented techniques, specifically dynamic wavelet neural networks (DWNNs). DWNNs incorporate temporal information and storage capacity into their functionality so that they can predict into the future, carrying out fault prognostic tasks. Such fundamental issues as the network structure, learning algorithms, stability analysis, uncertainty management, and performance assessment are studied in a theoretical framework. An example is presented in which a trained DWNN successfully prognoses a defective bearing with a crack in its inner race.

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