Complex dynamical behavior in memristor–capacitor systems

  • Lijuan Chen
  • Yuan Zhou
  • Fangyan YangEmail author
  • Shouming Zhong
  • Jianwei Zhang
Original paper


The parallel and series circuits of a Hewlett–Packard memristor and a capacitor are foundational building blocks for realistic memristive circuits. Due to the nonlinearity of the memristor, traditional studies with a single sinusoidal stimulus are limited in their ability to reveal the complex characteristics of memristors. By converting the circuits to an autonomous dimensionless dynamical system, we show that both memristors and capacitors can generate complex dynamical behaviors such as high periodic limit cycles and chaos under a combined periodic stimulus of two sinusoidal signals. To verify the existence of chaos, we present a computer-assisted rigorous proof by a topological horseshoe as well as a circuit implementation. In this way, we uncover a new property of the memristor–capacitor systems: for a typical memristor, e.g., \(R_\mathrm{OFF}/R_\mathrm{ON}=100\), no matter what values other parameters take, there often exists a periodic stimulus to make the circuits chaotic. Furthermore, under combined excitation of multiple periodic stimulus, the chaos in the systems still exists irrespective of a certain pattern that the frequencies of these stimulus have.


Chaos HP memristor Memristor–capacitor circuit Topological horseshoe 


Compliance with ethical standards

Conflict of interest

We declare that we have no financial and personal relationships with other people or organizations that inappropriately influence our work, and there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, Complex Dynamical Behavior in Memristor–Capacitor Systems.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Technical Aspects of Multimodal Systems Group (TAMS), Department of InformaticsUniversity of HamburgHamburgGermany
  3. 3.Chongqing Key Laboratory of Complex Systems and Bionic ControlChongqing University of Posts and TelecommunicationsChongqingChina
  4. 4.School of Optical Electrical and Computer EngineeringUniversity of Shanghai for Science and TechnologyShanghaiChina

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