Two occurrences of fractional actions in nonlinear dynamics

Abstract

Fractional theories have gained recently an increasing interest in dynamical systems since they offer some solutions to a number of puzzles in particular nonconservative and dissipative issues. Most of fractional dynamical theories are formulated by means of one occurrence of action that group kinetic energy and potential energy in one single package. In this work, we introduce a modified fractional dynamics based on the occurrence of two independent actions where fractional and nonfractional Euler-Lagrange equations are mixed together. We show that their combination divulge some properties that offer new insights in nonlinear dynamics. In particular, it was observed that a large family of solutions that could be used to model dissipative systems may be derived from the action with two occurrences of integrals. Moreover, damping systems may be obtained by means of simple Lagrangian functionals.