One such model that allows for travel into the past has recently come to my mind. I haven't found any implementations of it in literature and am so far pretending to authorship. However, it's likely that someone had this idea before me; if it reminds you of some literary work, please comment.
On some planet there happened to arise life. In absence of competition, primitive organisms multiplied, adapted to various environment conditions, evolved into better developed species and finally gave rise to an intelligent being, for unambiguity let's say to human. Little by little, intelligent organisms built a civilization; empires grew, matured and collapsed; ages passed of stone, iron, gunpowder, electricity and atom. Sadly, the technological progress hadn't changed the man's essence, and the science was a slave of warfare in the stone age as well as in the time of atom. After a hundred generations of people who held the nuclear buttons, the hundred and first gave in to the primitive cruel chieftain sitting inside and unleashed the ultimate fatal war that destroys everything living on the planet, from human to amoeba.
The planet became unsuitable for life for billions of years, but numerous times the half-lives of the radioactive nuclei had passed, and the conditions on the surface became favorable for conception of life. Once again from a primitive organism to a humanoid, once again from a savage to a nuclear physicist, once again a fatal button, a world war and death of everything living. This happened several times, with inconceivable gaps of many billions of years between. Every evidence of a former civilization vanished in the abyss of time, and each time the mankind was unaware that there had been intelligent life on the planet before.
Of course, not only nuclear weapons could human minds produce. In its prosperity, each civilization gave many different fruits: developed medicine, computers, great works of art. Among many scientists, there always were those who invented something similar to the achievements of the former civilization because great minds think alike. Each time, by trial and error humans found electricity, radio, aircraft, microprocessor and… time machine. This machine worked according to the Novikov self-consistency principle: it was physically impossible to use it for such a travel that would generate a causality paradox. Because a travel into the past would allow an individual to affect oneself or one's ancestors and thus contradict the present, the machine couldn't perform such a travel. You couldn't go back a year, a hundred years, or a hundred centuries because of the physical limitations of the Universe, not on the designer's whim.
However, those who experimented with the time machine could easily find that it was quite possible to go into the very remote past, billions of years ago. This way, every civilization discovered the others that lived before. In a similar manner, it was possible to travel into the age of a future civilization and come back (but not before the time of departure) without generating a paradox. The possiblity of travel between the ages was determined by the fact that each civilization was doomed to collapse, and by changing something in a previous age, the traveller wouldn't change anything in one's own. Time travel gradually became usual, just like normal travel, and governments even had to pass laws to limit immigration from the past and the future. Cultural exchange between the former and later ages only further increased the similarity of historical processes and secured the same inevitable end.
The stability of this model is based on the postulate of inevitable convergence of macroscopic history to a predetermined outcome. If you like, the gloomy plot can be given a romantic development: in some age, a man is born whom the time machine cannot send into the past, and the reader realizes that this man is “the One”. He is predestined to bring ideas that will finally avert the imminent death. This is why the time machine cannot send him into a former age: otherwise the previous civilization would have survived until the present and generate a paradox. After numerous reincarnations, the new mankind finally finds a spark of hope for a better future.
По-русски: Культурный обмен между обречёнными цивилизациями