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The modern nonlinear theory, bifurcation and chaos theory are used in
this paper to analyze the dynamics of the Rikitake two-disk dynamo system. The
mathematical model of the Rikitake system consists of three nonlinear
differential equations, which found to be the same as the mathematical model of
the well-known Lorenz system. The study showed that under certain value of
control parameter, the system experiences a chaotic behaviour. The experienced
chaotic oscillation may simulate the reversal of the Earth’s magnetic field.
The main objective of this paper is to control the chaotic behaviour in
Rikitake system. So, a nonlinear controller based on the slide mode control
theory is designed. The study showed that the designed controller was so
effective in controlling the unstable chaotic oscillations.
We introduce nil 3-Armendariz rings, which are generalization of 3-Armendariz rings and nil Armendaiz rings and investigate their properties. We show that a ring R is nil 3-Armendariz ring if and only if for any , Tn(R) is nil 3-Armendariz ring. Also we prove that a right Ore ring R is nil 3-Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result, we can show that a commutative ring R is nil 3-Armendariz if and only if the total quotient ring of R is nil 3-Armendariz.