Nonlinear Hamiltonian equations with fractional damping

Existence and uniqueness are proved for a class of Hamiltonian nonlinear differential equations of second order with a Caputo fractional derivative of order λ∈]0,2[. For noninteger λ the rate of convergence of solutions to a critical point is algebraic. For 0<λ⩽1 a non-negative dissipation function is constructed. For λ=2 the equation is Hamiltonian. The solution is an analytic function of λ for 0<λ<1 and for 1<λ<2. Numerical studies of nonlinear pendulum and the Duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations.