Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

Abstract

Based on some geometrical properties (symmetries and global analytic first integrals) of the Rabinovich system the closed-form solutions of the equations have been established. The chaotic behaviors are excepted. Moreover, the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). The advantage of this method is to obtain accurate solutions for special cases, with only an analytic first integral. An important output is the existence of complex eigenvalues, depending on the initial conditions and physical parameters of the system. This approach was not still analytically emphasized from our knowledge. A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied to more dynamic systems with these geometrical properties.

Keywords

• optimal auxiliary functions method
• Rabinovich system
• symmetries
• Hamilton–Poisson realization
• periodical orbits