On the dynamics of a hyperjek memristive system
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)

Date:	2024
Abstract:	From the pioneer work of Chua and Kang many researches have worked proposing different memristive systems having different applications in distinct areas depending on their properties and now it is a very active research subject mainly due to their applications Here we study the dynamics of the hyperjerk memristive system given by the fourth order ordinary differential equation . . . . x = -¨x - a . . . x - bx˙ 2 . . . x - (1 + x)x˙, previously studied by several authors showing that this system exhibits chaos for some values of its parameters a and b, as usual every dot denotes one derivative with respect to the time t. This system has a line filled with equilibria and it has a polynomial first integral H. Until now there are no analytical results on the periodic orbits of this differential system, and in that paper we fill that hole. Writing this differential equation as a first order differential system in R4, first we prove that this differential system has a zero-Hopf equilibrium, i. e. an equilibrium point such that the Jacobian matrix of the differential system evaluated at such equilibrium has a zero with multiplicity two, and one pair of conjugated purely imaginary eigenvalues. Second, we show that from this zero-Hopf equilibrium bifurcate two cylinders filled with periodic orbits parameterized by the levels of the first integral H. Moreover, the three-dimensional system obtained restricting the differential system in R4 to the invariant hypersurface H = h for h ˃-1/2, exhibits two Hopf bifurcations producing periodic orbits in the center manifold of that restriction.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: Reial Acadèmia de Ciències i Arts de Barcelona
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Hopf bifurcation ; Zero-Hopf bifurcation
Published in:	Applied physics. A, Materials science and processing, Vol. 130, Issue 12 (December 2024) , art. 903, ISSN 1432-0630

DOI: 10.1007/s00339-024-08073-7

Available from: 2025-12-31 Postprint

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles