EXTENDING LOCAL PASSIVITY THEORY AND HOPF BIFURCATION AT THE EDGE OF CHAOS IN OREGONATOR CNN

2012 ◽  
Vol 22 (11) ◽  
pp. 1250285
Author(s):  
CHANGBING TANG ◽  
FANGYUE CHEN ◽  
JIANBO WANG ◽  
XIANG LI
Keyword(s):  
Diffusion Coefficient ◽  
Hopf Bifurcation ◽  
Wide Spectrum ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Diffusion Term ◽  
Cell Parameters ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Coupled Cells

The fundamental local passivity theory asserts that a wide spectrum of complex behaviors may exist if the cells in the reaction–diffusion are not locally passive. This local passivity principle has provided a powerful tool for studying the complexity in a homogeneous lattice formed by coupled cells. In this paper, the complexity matrix YQ(s), which is the tool for testing the local passivity theory, is modified based on the characteristic polynomial AQ(λ). Then, the local passivity theory is applied to the study of the Oregonator CNN to judge if the cell parameters of a CNN are chosen at the edge of chaos. Analysis of the bifurcation and the numerical simulations show that nonzero diffusion term in Oregonator CNN may cause a reaction–diffusion equation oscillating under the appropriate choice of diffusion coefficient if the local passivity theory is not satisfied. That is, if the cell parameters of a CNN are chosen at the edge of chaos, the system is potentially unstable.

ANALYTICAL CRITERIA FOR LOCAL ACTIVITY AND APPLICATIONS TO THE OREGONATOR CNN

2000 ◽  
Vol 10 (01) ◽  
pp. 25-71 ◽  
Author(s):  
LEQUAN MIN ◽  
KENNETH R. CROUNSE ◽  
LEON O. CHUA
Keyword(s):  
Cell Parameter ◽  
High Energy ◽  
Reaction Model ◽  
Cell Parameters ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Coupled Cells ◽  
Emergence Of Complexity

This paper presents analytic criteria for local activity in one-port Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999], and gives the applications to the Oregonator CNN defined by the kinetic chemical reaction model of morphogenesis first introduced in [Field & Noyes, 1974]. Locally active domains, locally passive domains, and the edge of chaos are identified in the cell parameter space. Computer simulations of the dynamics of several Oregonator CNN's with specific selected cell parameters in the above-mentioned domains show genesis and the emergence of complexity. Furthermore, a novel phenomena is observed that "extremely high energy" is concentrated only on a few cells in the dynamic patterns of some Oregonator CNN's whose cell parameters are located in active domains; furthermore, relaxation oscillations and "transient oscillations" can exist if the cell parameters of the Oregonator CNN are located nearby or on the edge of chaos. This research illustrates once again the effectiveness of the local activity theory in choosing the system parameters for the emergence of complex patterns (static and dynamic) in a homogeneous lattice formed by coupled cells.

Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation

1998 ◽  
Vol 08 (02) ◽  
pp. 211-257 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Wide Spectrum ◽  
Reaction Diffusion ◽  
Fundamental Result ◽  
Activity Parameter ◽  
Cell Parameters ◽  
Parameter Domain ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Nagumo Equation

The local activity theory [Chua, 97] offers a constructive analytical tool for predicting whether a nonlinear system composed of coupled cells, such as reaction-diffusion and lattice dynamical systems, can exhibit complexity. The fundamental result of the local activity theory asserts that a system cannot exhibit emergence and complexity unless its cells are locally active. This paper gives the first in-depth application of this new theory to a specific Cellular Nonlinear Network (CNN) with cells described by the FitzHugh–Nagumo Equation. Explicit inequalities which define uniquely the local activity parameter domain for the FitzHugh–Nagumo Equation are presented. It is shown that when the cell parameters are chosen within a subset of the local activity parameter domain, where at least one of the equilibrium state of the decoupled cells is stable, the probability of the emergence of complex nonhomogenous static as well as dynamic patterns is greatly enhanced regardless of the coupling parameters. This precisely-defined parameter domain is called the "edge of chaos", a terminology previously used loosely in the literature to define a related but much more ambiguous concept. Numerical simulations of the CNN dynamics corresponding to a large variety of cell parameters chosen on, or nearby, the "edge of chaos" confirmed the existence of a wide spectrum of complex behaviors, many of them with computational potentials in image processing and other applications. Several examples are presented to demonstrate the potential of the local activity theory as a novel tool in nonlinear dynamics not only from the perspective of understanding the genesis and emergence of complexity, but also as an efficient tool for choosing cell parameters in such a way that the resulting CNN is endowed with a brain-like information processing capability.

scholarly journals Theorems and Application of Local Activity of CNN with Five State Variables and One Port

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Gang Xiong ◽  
Xisong Dong ◽  
Li Xie ◽  
Thomas Yang
Keyword(s):  
Bifurcation Diagram ◽  
Complex Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Local Activity ◽  
Homogeneous Lattice ◽  
Coupled Cells

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

Time-delay-induced instabilities and Hopf bifurcation analysis in 2-neuron network model with reaction–diffusion term

2018 ◽  
Vol 313 ◽  
pp. 306-315 ◽  
Author(s):  
Swati Tyagi ◽  
Subit K Jain ◽  
Syed Abbas ◽  
Shahlar Meherrem ◽  
Rajendra K Ray
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Network Model ◽  
Bifurcation Analysis ◽  
Reaction Diffusion ◽  
Neuron Network ◽  
Diffusion Term

Edge of Chaos and Local Activity Domain of the Gierer–Meinhardt CNN

1998 ◽  
Vol 08 (12) ◽  
pp. 2321-2340 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Parameter Space ◽  
Cell Parameter ◽  
Reaction Diffusion ◽  
Specific Reaction ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Activity Domain

This paper present an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by the model of morphogenesis first proposed in [Gierer & Meinhardt, 1972]. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

scholarly journals Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen–Cahn equation

2010 ◽  
Vol 140 (4) ◽  
pp. 673-706 ◽  
Author(s):  
Matthieu Alfaro ◽  
Harald Garcke ◽  
Danielle Hilhorst ◽  
Hiroshi Matano ◽  
Reiner Schätzle
Keyword(s):  
Transition Layer ◽  
Time Scale ◽  
Mean Curvature ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Finsler Metric ◽  
Diffusion Term ◽  
Almost Everywhere ◽  
Anisotropic Reaction Diffusion ◽  
Anisotropic Mean Curvature

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

Edge of Chaos and Local Activity Domain of the Brusselator CNN

1998 ◽  
Vol 08 (06) ◽  
pp. 1107-1130 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Parameter Space ◽  
Cell Parameter ◽  
Reaction Diffusion ◽  
Specific Reaction ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Activity Domain

This paper presents an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by a trimolecular model, called the Brusselator. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

2010 ◽  
Vol 20 (09) ◽  
pp. 2955-2963
Author(s):  
J. M. ARRIETA ◽  
N. CONSUL ◽  
S. M. OLIVA
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Nonlinear Boundary ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Critical Value ◽  
Reaction Diffusion Equation ◽  
Asymptotically Stable ◽  
One Dimensional ◽  
Stable Periodic

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

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