On some properties of nearly conservative dynamics of Ikeda map and its   relation with the conservative case
  The behavior of the well-known Ikeda map with very weak dissipation (so called nearly conservative case) is investigated. The changes in the bifurcation structure of the parameter plane while decreasing the dissipation are revealed. It is shown that when the dissipation is very weak the system demonstrates an "intermediate" type of dynamics combining the peculiarities of conservative and dissipative dynamics. The correspondence between the trajectories in the phase space in conservative case and the transformations of the set of initial conditions in the nearly conservative case is revealed. The dramatic increase of number of coexisting low-period attractors and the extraordinary growth of the transient time while the dissipation decreases have been revealed. The method of plotting a bifurcation trees for the set of initial conditions has been used to classify existing attractors by it's structure. Also it was shown that most of coexisting attractors are destroyed by rather small external noise, and the transient time in noisy driven systems increases still more. The new method of two-parameter analysis of conservative systems was proposed. 
A. P. Kuznetsov, A. V. Savin and D. V. Savin