Chaos Control and Synchronization of a Complex Rikitake Dynamo Model

A novel chaotic system called complex Rikitake system is proposed. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Further, based on feedback control method, the complex Rikitake system can be controlled to any equilibrium points. Additionally, this paper not only proves the existence of two types of synchronization schemes in the complex Rikitake system but also designs adaptive controllers to realize them. The proposed results are verified by numerical simulations.

Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system

The aim of this paper is to investigate a generalized Rikitake system from the integrability point of view. For the integrable case, we derive a family of integrable deformations of the generalized Rikitake system by altering its constants of motion, and give two classes of Hamilton–Poisson structures which implies these integrable deformations, including the generalized Rikitake system, are bi-Hamiltonian and have infinitely many Hamilton–Poisson realizations. By analyzing properties of the differential Galois groups of normal variational equations (NVEs) along certain particular solution, we show that the generalized Rikitake system is not rationally integrable in an extended Liouville sense for almost all parameter values, which is in accord with the fact that this system admits chaotic behaviors for a large range of its parameters. The non-existence of analytic first integrals are also discussed.

Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System

Integrable deformations of an integrable case of the Rikitake system are constructed by modifying its constants of motions. Hamilton-Poisson realizations of these integrable deformations are given. Considering two concrete deformation functions, a Hamilton-Poisson approach of the obtained system is presented. More precisely, the stability of the equilibrium points and the existence of the periodic orbits are proved. Furthermore, the image of the energy-Casimir mapping is determined and its connections with the dynamical elements of the considered system are pointed out.

Jacobi stability analysis of Rikitake system

We study the Rikitake system through the method of differential geometry, i.e. Kosambi–Cartan–Chern (KCC) theory for Jacobi stability analysis. For applying KCC theory we reformulate the Rikitake system as two second-order nonlinear differential equations. The five KCC invariants are obtained which express the intrinsic properties of nonlinear dynamical system. The deviation curvature tensor and its eigenvalues are obtained which determine the stability of the system. Jacobi stability of the equilibrium points is studied and obtain the conditions for stability. We study the dynamics of Rikitake system which shows the chaotic behaviour near the equilibrium points.

Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System

In this paper, an extended Rikitake system is studied. Several issues, such as Hopf bifurcation, coexistence of stable equilibria and hidden attractor, and dynamics analysis at infinity are investigated either analytically or numerically. Especially, by a simple linear transformation, the wide range of hidden attractors is noticed, and the Lyapunov exponents diagram is given. The obtained results show that the unstable periodic solution generated by Hopf bifurcation leads to the hidden attractor. The existence of hidden attractors that may render the system's behavior unpredictable not only depends on the value of system parameters but also on the value of initial conditions. The phenomena are important and potentially problematic in engineering applications.