scholarly journals Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Xian Yu ◽  
Rong Yuan ◽  
Cheng-Hsiung Hsu ◽  
Ming-Shu Peng
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Signal Transmission ◽  
Distributed Delay ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Switching Speed ◽  
One Dimensional ◽  
The One ◽  
Nondecreasing Functions

This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer latticeZ. The dynamics of each given cell depends on itself and its nearestmleft orlright neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (12) ◽  
pp. 3515-3550 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
CHUN-HSIEN LI ◽  
SUH-YUH YANG
Keyword(s):  
Neural Networks ◽  
Traveling Wave ◽  
Wave Speed ◽  
Distributed Delay ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Monotonicity Condition ◽  
Switching Speed ◽  
Wave Solutions ◽  
Special Cases

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

2001 ◽  
Vol 11 (08) ◽  
pp. 2085-2095 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
KAI-PING CHIEN ◽  
SONG-SUN LIN ◽  
CHENG-HSIUNG HSU
Keyword(s):  
Neural Networks ◽  
Steady State ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
Output Function ◽  
One Dimensional ◽  
Spatial Entropy ◽  
The One ◽  
Steady State Solutions

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

SNAP-BACK REPELLERS AND CHAOTIC TRAVELING WAVES IN ONE-DIMENSIONAL CELLULAR NEURAL NETWORKS

2007 ◽  
Vol 17 (06) ◽  
pp. 1969-1983 ◽  
Author(s):  
YA-WEN CHANG ◽  
JONQ JUANG ◽  
CHIN-LUNG LI
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Discrete Time ◽  
Traveling Waves ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
One Dimensional ◽  
Time Analogy

In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br( z ), the ball of radius r with center at z does not necessarily imply that F is expanding in Br( z ). Subsequent efforts (see e.g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in fixing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give sufficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given.

EXACT NUMBER OF MOSAIC PATTERNS IN CELLULAR NEURAL NETWORKS

2001 ◽  
Vol 11 (06) ◽  
pp. 1645-1653 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
SONG-SUN LIN ◽  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Boundary Conditions ◽  
Cellular Neural Networks ◽  
Transition Matrices ◽  
Lattice Systems ◽  
Range Interaction ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Exact Number ◽  
The One

This work investigates mosaic patterns for the one-dimensional cellular neural networks with various boundary conditions. These patterns can be formed by combining the basic patterns. The parameter space is partitioned so that the existence of basic patterns can be determined for each parameter region. The mosaic patterns can then be completely characterized through formulating suitable transition matrices and boundary-pattern matrices. These matrices generate the patterns for the interior cells from the basic patterns and indicate the feasible patterns for the boundary cells. As an illustration, we elaborate on the cellular neural networks with a general 1 × 3 template. The exact number of mosaic patterns will be computed for the system with the Dirichlet, Neumann and periodic boundary conditions respectively. The idea in this study can be extended to other one-dimensional lattice systems with finite-range interaction.

TRAVELING WAVES IN CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (07) ◽  
pp. 1307-1319 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN ◽  
WENXIAN SHEN
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Shooting Method ◽  
Wave Train ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Cellular Neural Networks ◽  
Periodic Wave Solutions ◽  
Wave Solutions

In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

DEFORMATION OF TRAVELING WAVES IN DELAYED CELLULAR NEURAL NETWORKS

2003 ◽  
Vol 13 (04) ◽  
pp. 797-813 ◽  
Author(s):  
PEIXUAN WENG ◽  
JIANHONG WU
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Delay Differential Equations ◽  
Wave Speed ◽  
Signal Transmission ◽  
Cellular Neural Networks ◽  
Profile Equation ◽  
Damped Oscillation ◽  
Wave Profiles ◽  
Delay Differential

In this paper, we establish the existence and describe the global structure of traveling waves for a class of lattice delay differential equations describing cellular neural networks with distributed delayed signal transmission. We describe the transition of wave profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. We also describe an interval of the wave speed where the structure of the wave solution is unknown since the corresponding profile equation involves distributed argument of both advanced and retarded types, and we present some preliminary numerical simulation to illustrate the complexity.

scholarly journals Traveling wave solutions for the Richards equation with hysteresis

2019 ◽  
Vol 84 (4) ◽  
pp. 797-812 ◽  
Author(s):  
E El Behi-Gornostaeva ◽  
K Mitra ◽  
B Schweizer
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Richards Equation ◽  
One Dimensional ◽  
Homogeneous Solutions ◽  
Wave Solutions ◽  
Relevant Effect ◽  
Saturation Overshoot ◽  
The One

Abstract We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau $-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.

Traveling wave solutions of the one-dimensional Boussinesq paradigm equation

2013 ◽  
Author(s):  
V. M. Vassilev ◽  
P. A. Djondjorov ◽  
M. Ts. Hadzhilazova ◽  
I. M. Mladenov
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
One Dimensional ◽  
Wave Solutions ◽  
The One

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

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