On the Periodic Solution for a Class of Perturbed a Hyper Jerk Memristive System

Nonlinear dynamical systems with hidden attractors represent a recent and highly active area of research. In this work, the following novel class of four dimensional dynamical memristive vector field is considered []. χ=y∂/∂x+z∂/∂y+w∂/∂z-(z+(1+x)y+a1w+a2z2w)∂/∂w. Such a vector field exhibits a line of equilibrium points Ex(x,0,0,0) passing through the origin. In a specific region within the parameter space, this system displays hyper-jerk dynamics, resembling a dynamical system with concealed attractors. The behavior of the proposed vector field is analyzed by identifying periodic solutions. First, an explicit condition is determined to ensure that the origin, which is a non-isolated singular point has a pair of purely conjugate eigenvalues. In a particular scenario, the equilibrium point(-1,0,0,0)∈Ex undergoes a zero-Hopf bifurcation. The first-order averaging method is used to find periodic solutions; however, it does provide any information about periodic solutions that could emerge from the zero-Hopf bifurcation point. Furthermore, we show that the Jacobian matrix of the system, when evaluated at the origin, exhibits one zero eigenvalue, one negative eigenvalue, and one pair of conjugated purely imaginary eigenvalues. A periodic solution is then found by introducing a small perturbation to the parameter a1 in the system, expressed as a1=1+ϵ, where ϵ is a sufficintly small parameter. This perturbation is analyzed using the bifurcation theory, leading to the emergence of a periodic solution in the center manifold. The periodic solution bifurcates from a non-isolated singular point located at the origin, as demonstrated through the application of first-order averaging method. This method confirms that the required conditions for the existence of a periodic solution are satisfied.