NDLI: Manifolds of Equilibria and Bifurcations without Parameters in Memristive Circuits

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Riaza, Ricardo
Copyright Year	2012
Abstract	The memory-resistor or memristor is a new electrical device governed by a nonlinear flux-charge relation. Its existence was predicted by Leon Chua in 1971, and the report in 2008 of a physical device with such a constitutive relation has driven a lot of attention to this circuit element. The memristor and related devices are expected to play a very relevant role in electronics in the near future, specially at the nanometer scale. The special form of the voltage-current characteristic, which reads as either $v=M(q)i$ or $i=W(\varphi)v$, implies that any equilibrium point is embedded into a center manifold of equilibria whose dimension is defined by the total number of memristors in the circuit. We characterize the normal hyperbolicity of these manifolds of equilibria in graph-theoretic terms. Moreover, when the assumptions supporting the normal hyperbolicity of such manifolds fail, the differential-algebraic nature of circuit models is shown to lead to certain bifurcations without parameters not exhibited by explicit ODEs. The results are illustrated by several circuit examples, some of which arise in the design of superconducting quantum bits based on the Josephson junction.
Starting Page	877
Ending Page	896
Page Count	20
File Format	PDF
ISSN	00361399
DOI	10.1137/100816559
e-ISSN	1095712X
Issue Number	3
Volume Number	72
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-06-21
Access Restriction	Subscribed
Subject Keyword	Implicit equations, differential-algebraic equations Analytic circuit theory Josephson junction qubit Invariant manifolds differential-algebraic equation normal hyperbolicity memristor Bifurcations of singular points Applications of graph theory manifold of equilibria bifurcation without parameters Stability of manifolds of solutions nonlinear circuit
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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