On the dynamics of a hyperjek memristive system

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 \(\ddddot{x} =-\ddot{x} - a \dddot{x} -b \dot{x}^2 \dddot{x}-(1+x)\dot{x}\), 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 \(\mathbb {R}^4\), 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 \(\mathbb {R}^4\) to the invariant hypersurface \(H=h\) for \(h>-1/2\), exhibits two Hopf bifurcations producing periodic orbits in the center manifold of that restriction.