About the intuitive picture of a Hamiltonian for a dissipative system

A Hamiltonian for a one-dimensional (1-D) dissipative system is given which shows that the trajectories in the spaces ($x,\dot{x}$) and ($x,p$) are completely different. The trajectory in the space ($x,p$) has an unexpected contra-intuitive behavior, and a canonical transformation does not solve this behavior.