Testing for Local Activity and Edge of Chaos

Abstract

In this paper we study the local activity, local passivity and edge of chaos of continuous-time reaction-diffusion cellular nonlinear networks (CNN) with one-port first-order, one-port secondorder, two-port second-order, two-port third-order and three-port third-order cells. We prove that the local passive regions determined by cell impedance Z Q (s) and cell admittance Y Q (s) for first-and second-order cells are equivalent to each other. We also present an efficient procedure to determine the edge-of-chaos parameter region by combining the local active regions derived from Y Q (s) and the pole locations of Z Q (s). In order to characterize the fundamental limitations of local passivity on the emergence of complexity we study the local active property from a parameter space spanned by both the cell parameters and the external excitations called the cell's port currents (in view of its interpretations from classical circuit theory). Analytical results of locally passive, restricted locally passive, edge-of-chaos and locally active parameter regions for CNN cells modeled by cubic nonlinearities are presented. We illustrate our results by analyzing CNN cells modeled by Chua's circuits with a cubic nonlinearity. We find that the morphology of the edge-of-chaos and the local active parameter regions have a close connection to the pattern formation behaviors of CNNs. Simulation results are presented to verify our theoretical results.

A Gallery of Attractors From Smooth Chua's Equation

“…One of the smooth functions similar to the original three-segment piecewise-linear nonlinearity is a cubic polynomial. Several papers chose Chua's circuit with a cubic nonlinearity [Altman, 1993a[Altman, , 1993bHuang et al, 1996;Hartley, 1993;Khibnik et al, 1993;Zhong, 1994;Pivka et al, 1996;Bykov, 1998;Tang & Man, 1998;Liao & Chen, 1998;Yang et al, 1998;Algaba et al, 1999Algaba et al, , 2000Algaba et al, , 2001Eltawil & Elwakil, 1999;Collantes & Suarez, 2000;Ge & Wang, 2000;Yang & Chua, 2001;Jang et al, 2002;O'Donoghue et al, 2005]. A cubic nonlinearity can also be realized by simple practical circuits [O'Donoghue et al, 2005].…”

“…There were several other publications on Chua's circuit with a cubic (or smooth) nonlinearity, some of which are related to the analyses of chaos and bifurcation phenomena [Altman, 1993a[Altman, , 1993bKhibnik et al, 1993;Huang et al, 1996;Pivka et al, 1996;Bykov, 1998;Algaba et al, 1999Algaba et al, , 2000Algaba et al, , 2001Collantes & Suarez, 2000], control of chaotic systems [Hartley & Mossayeb, 1993;Liao & Chen, 1998;Yang et al, 1998;Ge & Wang 2000;Jang et al, 2002], circuit implementations [Zhong, 1994;Tang & Man, 1998;Eltawil & Elwakil, 1999;O'Donoghue et al, 2005], and CNN (cellular neural network) [Yang & Chua, 2001]. As in [Khibnik et al, 1993;Zhong, 1994;Huang et al, 1996], most phenomena from the piecewise-linear Chua's equation carry over to a cubic Chua's equation.…”

Bio‐Inspired Nano‐Sensor‐Enhanced CNN Visual Computer

“…In practical devices, the locally active region often occupies a very large region of the parameter space, and, even if a device is designed to operate inside the local‐activity region, the architecture where the device is embedded may still not exhibit any computing capability, let alone emergent phenomena. To enhance the likelihood of computing, the cell parameters must be chosen within a much smaller subset of the locally active region called the “edge of Chaos,” another fundamental concept first introduced by Chua (Chua, 1998; Yang and Chua, 2001; Dogaru and Chua, 1988, 2000). Explicit “mathematical criteria for identifying this prime real estate” areas have been derived for CNN cells modeled by various nonlinear differential equations.…”

E∧3: A Universal emulator for complex systems

“…These concepts are now key points of the theory of complexity. Moreover, recent developments have lead to new analyses based on quantitative tests [2,8]. In all these studies the importance of numerical simulations to study complex systems is evident.…”