On the dynamics of a hyperjek memristive system On the dynamics of a hyperjek memristive system

• Published in: Applied physics. A, Materials science & processing Vol. 130; no. 12
• Main Authors: Llibre, Jaume, Valls, Claudia
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2024; Springer Nature B.V
• Subjects: Characterization and Evaluation of Materials; Condensed Matter Physics; Derivatives; Differential equations; Eigenvalues; Equilibrium; Hopf bifurcation; Hyperspaces; Integrals; Jacobi matrix method; Jacobian matrix; Machines; Manufacturing; Memory devices; Nanotechnology; Optical and Electronic Materials; Orbits; Ordinary differential equations; Physics; Physics and Astronomy; Polynomials; Processes; Surfaces and Interfaces; Thin Films; 34C05; Zero-Hopf bifurcation; Hopf bifurcation; Hyperjek memristive system
• ISSN: 0947-8396; 1432-0630
• DOI: 10.1007/s00339-024-08073-7

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 ⃛ - b x ˙ 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 R 4 , 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 R 4 to the invariant hypersurface H = h for h > - 1 / 2 , exhibits two Hopf bifurcations producing periodic orbits in the center manifold of that restriction.