Abstract
As established by the second law of thermodynamics, an isolated system is unable to
support complex phenomena. Conversely, a system, which communicates with the surrounding
environment, may exhibit complex behaviors, provided some of its constitutive components
are capable to amplify infinitesimal fluctuations in energy under suitable polarization [1], a
property known as Local Activity. Back in 1974 the American luminary Stephen Smale [2]
observed a counterintuitive phenomenon, later referred to as Smale Paradox, over the course
of an experiment on a reaction-diffusion system.

Two identical 4th-order reaction cells, sitting on a common quiet state on their own, were found
to undergo sustained limit-cycle oscillations when immersed in a coupling diffusive medium.
An explanation for this unexpected phenomenon may only be found in the Theory of Local
Activity [3], and recurring, particularly, to the Edge of Chaos Theorem, which asserts that a
stable operating point Q of an isolated cell may be destabilized, as the cell is coupled to a
dissipative environment, if and only if the isolated cell is capable to amplify a small-signal
superimposed on Q, i.e. if and only if the isolated cell is both stable and locally-active, i.e. on
the Edge of Chaos, at Q.

In this seminar we shall introduce the simplest ever-reported bio-inspired oscillatory network
[4], consisting of two resistively-coupled and identical 2nd-order memristor cells, which
supports the counterintuitive emergent phenomena, that mesmerized Stephen Smale in the
seventies. The Smale paradox will be resolved here, once and for all, by demonstrating how
static and dynamic patterns may develop in the reaction-diffusion system if and only if the
isolated memristor oscillatory cell is biased on a stable and locally-active operating point.
An in-depth study, based upon linearization analysis and large-signal phase-portrait
investigations, allows to draw a comprehensive picture for the local and global dynamics of
the reaction cell, including a niobium oxide memristor [5], which features a peculiar ð-shaped
DC current-voltage locus, and is fabricated and characterized at NaMLab gGmbH [6]. This
allows to develop a rigorous methodology to tune the design parameters of the two-cell array
so as to induce diffusion-driven instabilities therein. This work sheds light on the precious role
that nonlinear system-theory may assume in the years to come to support circuit designers in
the exploration of the full potential of memristors in bio-inspired electronics.

References:

[1] L.O. Chua, "Local activity is the origin of complexity," Int. J. on Bifurcation and Chaos,
vol. 15, no. 11, pp. 3435-3456, 2005

[2] S. Smale, "A Mathematical Model of Two Cells via Turing's Equation," American
Mathematical Society, Lectures in Applied Mathematics, vol. 6, pp. 15-26, 1974

[3] K. Mainzer, and L.O. Chua, "Local Activity Principle," Imperial College Press, 2013,
ISBN-13: 978-1-908977-09-0

[4] A. Ascoli, A.S. Demirkol, L. Chua, and R. Tetzlaff, "Edge of Chaos Theory Resolves Smale
Paradox in the Simplest Memristor Oscillatory Network," IEEE Trans. on Circuits and
Systems-I: Regular Papers, 2021, in press

Biography

Alon Ascoli (IEEE member) received a Ph.D. Degree in Electronic Engineering from University College Dublin in 2006. He currently holds a tenure faculty position at the Institute of Principles of Electrical and Electronic Engineering of Technische Universität Dresden. He develops system-theoretic methods for the analysis and design of bio-inspired memristive circuits. He was honoured with Best Paper Awards from IJCTA in 2007 and MOCAST in 2020. In April 2017 he was conferred the habilitation title as Associate Professor in Electrical Circuit Theory from the Italian Ministry of Education. He is a member of the Scientific Advisory Board of the Chua Memristor Center and of the IEEE Nanoelectronics and Gigascale Systems Technical Committee (Nano-Giga TC), and the Chair of the IEEE Cellular Nonlinear Networks and Memristive Array Computing (CNN-MAC) TC.