Convergence analysis of algorithms for memristive oscillator system
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RESEARCH
Open Access
Convergence analysis of algorithms for memristive oscillator system Ailong Wu1,2,3 and Chao-Jin Fu2* *
Correspondence: [email protected] 2 College of Arts and Science, Hubei Normal University, Huangshi, Hubei, China Full list of author information is available at the end of the article
Abstract Memristive oscillator systems are common models for many problems in physics, engineering, and systems biology. This paper presents a convergence analysis of two types of algorithms for solving a fourth-order memristive oscillator system. For the first algorithm, a parallel algorithm, a limiting state of the iterate sequence generated by a Jacobi iterative scheme and the Euler polygonal method, is a solution of the system under some weaker conditions. With the second algorithm, a partial difference method, which is based on the partial difference concept and exponential convergence, is also presented. The proposed algorithms in this paper can be applied to general nonlinear hybrid systems. Keywords: memristor; hybrid systems; parallel algorithm; partial difference method; convergence
1 Introduction Recently, a large number of applications for memristive oscillator systems have been reported, including nonvolatile memristor memories, digital and analog circuits; see [–]. The models of memristive oscillator systems suggest possible responses of simple intelligences in biomimetics as well as new approaches for bio-inspired reconfigurable circuits. Considerable attention has been devoted to the theoretical properties of memristive oscillator systems, and their relationship to memristor dynamics (see [–, , , ] and references therein). A very lower dimensional memristor equation can appear with complex double-loop behavior. Therefore, the memristive system is a complicated system that has strongly nonlinear behavior, as it includes the switched network cluster and shows a high uncertainty []. Deep studies of the memristive system are important in various applications due to the important guiding role in memristor-based physical commercially available devices. For example, nonlinear dynamics has been shown to play an important role in the understanding of a wide spectrum of memristor-based technological and biological systems [–, , –]. In this paper, we consider a fourth-order memristive oscillator system described by the following differential equations: ⎧ ⎪ ˙ = v (t), ⎪ ⎪ ϕ(t) ⎪ ⎨ v˙ (t) = v (t) – v (t) – W (ϕ(t))v (t), C R C R C ⎪ v˙ (t) = C R v (t) – C R v (t) – C (t), ⎪ ⎪ ⎪ ⎩ (t) ˙ = v (t),
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© 2014 Wu and Fu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wu and Fu Advances in Difference Equations 2014, 2014:160 http://www.advancesindifferenceequations.com/content/2014/1/160
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