RABINOVICH-FABRIKANT ATTRACTOR

The Rabinovich-Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 1979.Danca and Chen note that the Rabinovich-Fabrikant system is difficult to analyse (due to the presence of quadratic and cubic terms) and that different attractors can be obtained for the same parameters by using different step sizes in the integration, see on the right an example of a solution obtained by two different solvers for the same parameter values and initial conditions. Also, recently, a hidden attractor was discovered in the Rabinovich - Fabrikant system.

and a system of three ordinary differential equations describing the structure as-

$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&y ( z - 1 + x^2 ) + \gamma x\\ \frac{dy}{dt}&=&x ( 3 z + 1 - x^2 ) + \gamma y\\ \frac{dz}{dt}&=& - 2 z ( \alpha + x y ) \end{eqnarray*}\right. $

When $\alpha = 0.14$, $\gamma = 0.10$, a solution curve of this system has the shape of wave like structure and the position of particles will follow a similar path.

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