Fractional Dynamical Systems

Abstract

The dynamical system is a notion for any fixed map, which describes the time dependence of a position in its space of states. At any given time a dynamical system has a state given by a vector x, which can be represented by a point in an appropriate state space. Infinitesimal changes in the state of the system correspond to small changes in the vectors. The evolution map of the dynamical system is a fixed rule that describes what future states follow from the current state. The notions of gradient and Hamiltonian systems arise in dynamical systems theory (Hirsh and Smale, 1974; Dubrovin et al., 1992; Vilasi, 2001). In this chapter, generalizations of gradient and Hamiltonian systems are suggested. We use differential forms and exterior derivatives of fractional orders. Its allow us to define Hamiltonian and gradient dynamical systems (Gilmor, 1981; Dubrovin et al., 1992; Vilasi, 2001; Godbillon, 1969) of non-integer (fractional) orders. In the general case, the fractional Hamiltonian (or gradient) systems cannot be considered as Hamiltonian (gradient) systems. The suggested class of fractional gradient and Hamiltonian systems is wider (Tarasov, 2005a,b) than the usual class of gradient and Hamiltonian dynamical systems. The systems of gradient and Hamiltonian type can be considered as a special case of fractional gradient and Hamiltonian systems.