Generalized Fractional Hybrid Hamilton Pontryagin Equations

In this work we present a new approach on the study of dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have investigated the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generalized fractional Liu process and the combination between them, generalized fractional hybrid process. Corresponding generalized fractional stochastic, respectively fuzzy, respectively hybrid dynamical systems were defined. We have applied the theory for generalized fractional hybrid Hamilton-Pontryagin (HP) equation and generalized fractional Hamiltonian equations. We have found fractional Langevin equations from the general fractional hybrid Hamiltonian equations. For these cases and specific parameters, numerical simulations were done.