No abstract

This work investigates modified function projective synchronization between two different hyperchaotic dynamical systems, namely, hyperchaotic Lorenz system and hyperchaotic Chen system with fully unknown parameters. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between two diffierent hyperchaotic dynamical systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

PurposeThe purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control.Design/methodology/approachThe authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019’s dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem.FindingsThe occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization.Originality/valueAs the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited.

In this paper, we analyze the dynamical behaviors of Liu system using the complementary-cluster energy-barrier criterion (CCEBC). Moreover, the Hopf bifurcation of this system is investigated using the first Lyapunov coefficient. Also, it is proved that this system has two Hopf bifurcation points, at which these Hopf bifurcations are non degenerate and subcritical.

Projective synchronization and generalized projective synchronization have recently been observed in the coupled hyperchaotic systems. In this paper a generalized projective synchronization technique is applied in the hyperchaotic Lorenz system and the hyperchaotic Lü. The sufficient conditions for achieving projective synchronization of two different hyperchaotic systems are derived. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.