Nonlinear Dynamics

, Volume 94, Issue 4, pp 3011–3028 | Cite as

A multiple focus-center-cycle bifurcation in 4D discontinuous piecewise linear memristor oscillators

  • Enrique PonceEmail author
  • Andrés Amador
  • Javier Ros
Original Paper


The dynamical richness of 4D memristor oscillators has been recently studied in several works, showing different regimes, from stable oscillations to chaos. Typically, only numerical simulations have been reported and so there is a lack of mathematical results. We focus our analysis in the existence of multiple stable oscillations in the 4D piecewise linear version of the canonical circuit proposed by Itoh and Chua (Int J Bifurc Chaos 18(11):3183–3206, 2008). This oscillator is modeled by a discontinuous piecewise linear dynamical system. By adding one parameter that stratifies the 4D dynamics, it is shown that the dynamics in each stratum is topologically equivalent to a 3D continuous piecewise linear dynamical system. Some previous results on bifurcations in such reduced system allow to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.


Bifurcations Memristor oscillator Piecewise linear systems Periodic orbit Boundary equilibrium bifurcation Extreme multistability Focus-center-limit cycle bifurcation 



Andrés Amador is supported by Pontificia Universidad Javeriana Cali-Colombia. Enrique Ponce and Javier Ros are partially supported by the Spanish Ministerio de Economía y Competitividad, in the frame of project MTM2015-65608-P, and by the Consejería de Economía y Conocimiento de la Junta de Andalucía under grant P12-FQM-1658.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaEscuela Técnica Superior de IngenieríaSevillaSpain
  2. 2.Facultad de Ingeniería Departamento de Ciencias Naturales y MatemáticasPontificia Universidad JaverianaCaliColombia

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