Temporal Mutability — AKA The Sliding Scale Of How Easy It Is For Time Travelers To Change The Past, And Why.
Time Travel is one of the richest concepts in Speculative Fiction; altering the past is easily one of the richest Time Travel plots.
Apparently, people (or at least SF writers) in general have a deep-seated dissatisfaction with the past—either their own recent past, or with the whole history of the world—because every time the subject of time travel comes up, characters inevitably start wondering whether they can use their Time Machine to change the past. Even if the characters have no intention of changing the past—even if the characters don't actually travel to the past at any point—some smartass will ask about the Grandfather Paradox, which will in turn lead to a discussion on the possibility (and morality) of altering the past:
Could you go back and save your brother from that fatal car crash? Could you punch your boss in the face, then go back and stop yourself? Could you preventWorld War IIby going back to 1930 and killingAdolf Hitler? Nope.And if you could, should you?
Seeing as time travel is currently just a pipe dream, there's really no saying what would be possible when traveling to the past in Real Life. Writers are thus free to invent and follow whatever chronophysics they like, and as long as it's consistent the fans will usually accept it.
These settings tend to fall into one of the following categories (arranged here from least changeable to most changeable):
Timey Wimey Ball: The series says outright that time travel follows no rhyme or reason. Or, it starts off following the rules of one of the above categories, only to later contradict these rules (sometimes justified by stating that the original time travel expert was wrong, or that this new case is some kind of special exception to the general rules of time travel).
As an aside, it's interesting that no one ever seems to be nearly as concerned about time travelers altering the present or the future. No one says "But what if saving that guy somehow causes World War Three — because it didn't 'really' happen or it's not the 'correct' outcome, and humans are not supposed to change history?"
See also this page, for a more in-depth discussion.
It's even more complicated than that — the first of the two times the former method (a Sario Rip) appears, it's short-circuited when Jake dies, his consciousness merges with the other version of himself, and he proceeds to prevent the whole thing from happening — also, he was apparently the only one really there because of that. And apparently there was a risk of both versions of themselves being annihilated when they caught back up to the point when the explosion happened, even though after that, the first iteration of them would be gone, back into the past. As for the Time Matrix, it turned out to be even more powerful than it seemed.
In Dragonriders of Pern, turns out their crisis (Thread's back after 400 years, and they only have one Weyr, with nowhere near enough dragons to protect the whole continent) was caused by Lessa's trip back in time to bring the other five Weyrs forward to solve that crisis.
Harry Harrison's The Technicolor Time Machine has that, although this is not shown until the very end of the book, when the protagonist (a film director who wanted to make a cheap movie about Vikings using a time machine) realizes that the only reason the Vikings settled Vinland (and were the first Europeans to reach the New World) is because of the movie. Their Viking friend Ottar reveals that he is, in fact, Thorfinn Karlsefni, whom history recorded as the leader of the expedition. Furthermore, the director's name is Barney Hendrickson, which is eerily similar to the Real Life historical figure Bjarni Herjólfsson, who was key in ensuring Karlsefni's success.
The first time happens when the ship accidentally ends up in the past, about a year after Hunt got trapped in the black hole. Hunt even asks Rommie if she believes in fate. Rommie does, but Hunt doesn't, claiming "a man makes his own fate". They arrive just before the final battle of the Nietzschean Rebellion, where the Nietzschean fleet obliterates the remains of the High Guard (although that turns into a Pyrrhic Victory for them). Knowing one ship won't make a difference in the battle, Hunt prepares to travel back to the future. Then 3 times as many Nietzschean ships appear as recorded by history. Realizing the Nietzscheans still need to have a Pyrrhic Victory for the galaxies to plunge into chaos, Hunt uses a device that wipes out the "extra" 2/3 of the fleet and leaves. Tyr later reveals that he knew about this all along from old stories but didn't feel like sharing.
Later on, Hunt's fiancée tries to pull the Andromeda Ascendant from the black hole in the past and, actually, looks like she has a chance of doing that. Then a Nietzschean ship shows up and forces her to abandon the efforts. Rommie later tells Hunt that she did manage to nudge the ship a little, which put it in a position to be able to be rescued 300 years later by the Eureka Maru.
Ellone in Final Fantasy VIII claims that changing the past is impossible; this appears to be why, with a side of Stable Time Loop. The influences that Squall and Ultimecia have on their respective pasts are already in effect in their presents - in particular, Squall gives Edea the idea for the founding of SeeD thirteen years in his own past, and Ultimecia's efforts to change the past are at least heavily implied to directly bring about the events she is trying to prevent.
Beast Wars uses this form of time travel, with the eponymous conflict resulting in drastic changes to Planet Earth, allowing the events of the original Transformers to occur.
It starts out that way, since the "alien planet" turned into Earth All Along thanks to time travel. But then it turns into Type 3 when Megatron nearly kills Optimus Prime, with that action "breaking" the rubber band.
He also has Rampage blow up the top of a mountain and watches as it changes in a future recording.
And then back to the Stable Time Loop in the finale, when an Autobot shuttle wasn't on the records of the Ark because the protagonists used it to save the day and get home.
Type 2: Enforced Immutability
In his preface for The Great Divorce, C. S. Lewis cited as an inspiration a short story (whose author he could no longer remember) from an American SF magazine about a man who traveled into the past "and there, very properly, found raindrops that would pierce him like bullets and sandwiches that no strength could bite—because, of course, nothing in the past can be altered."
Time Scout presents Enforced Immutability that approaches Rubber Band. You can change anything so long as it doesn't matter in any way. If it does matter, you can't change it. Something will happen. Usually to you. Things that can be changed are enforced by uptime laws. And taxed accordingly.
Doctor Who has aspects of almost all of them, ocassionally it depending on who's at the helm. Previously Enforced Immutability was implied via the Timelords who're all dead now. These days the term Fixed Point is used to describe events which are basically historical lynch pins: you can technically change them, but you shouldn't seeing as the universe will fall apart. They also had the Reapers (beyond creepy monsters from outside time who came to "sterilise the wounds" if things got too messed up, but you had to really screw history up to get them. Mostly it's Rubber-Band History in effect, with the occasional burst of You Already Changed The Past for variety's sakes (It's even possible to avoid Paradoxes if you're clever enough to create a Paradox Machine which holds the current time in place regardless of what you do to the past (hence why the third season finale had a bunch of transformed humans killing off their ancestors and suffering no ill effects despite essentially erasing their history.) It helps that the main character can see all of time and space in his head at his whim and pick and choose whether he wants to/can afford to be a conscientious Temporal Balancing Actor or just screw with everything.
LOST seems to be following this rule, in fact continually referring to "the rules" and expounding that history cannot be changed. Desmond's stories drift slightly into Rubber-Band History, in that he can prevent individual instances of Charlie's death, but it will still happen as soon as he fails to intervene.
Star Trek had a Temporal Prime Directive. However, this is poorly enforced; the rules of the universe amount to Type 4.
Continuum is like this. Theoretically, you can change the past, but when the lives of the countless sentients of the post-Aquarian future depend on the past not being changed, it's going to stay unchanged.
In Homestuck, any deviation from the original timeline creates a Doomed Timeline, which is as bad as it sounds. On the other hand, timelines have a limited ability to interact with one another, and the creation of a Doomed Timeline may have consequences reflected in the Alpha Timeline (Davesprite being the most visible example of this phenomenon). While most of these cross-timeline shenanigans have been necessary to the proper continuation of the Alpha Timeline, it's been hinted that the right combination of Doomed Timelines could throw the Alpha Timeline permanently Off the Rails, or at least off of Lord English's rails.
In fact, Stable Time Loops are Dave's preferred method of time travel, because the Alpha Timeline's method of enforcing immutability is to kill off any time travelers who aren't supposed to be there (or to put it another way, doomed timeline = doomed time traveler).
In Feng Shui, trying to change history without capturing any Feng Shui sites will inevitably result in this. Big world events happen with the perpetrators having different names and everything eventually comes around to something resembling the present day. But once you start capturing Feng Shui sites, you can start making changes stick, and can even bring about a Critical Shift if enough sites are in your power.
Looper seems to adhere to this, mostly, with some bits from type 4 put in.
Time travel in Cirque Du Freak seems to work this way. Anything that happens will still happen, even if the "actors" for any given role are different people. For example, the books explain that if someone went back in time to kill Adolf Hitler, you could hypothetically succeed. However, someone else would rise to lead Nazi Germany, and World War II would still happen. It's just that the man known as Furher would not be Adolf Hitler. The basic principle for this is the key to Darren's salvation.
In Primer, Abe and Aaron are able to "preserve causality" by insulating themselves from the outside world prior to their trip back in time, but when they want to, they're able to manipulate past events to their own advantage. It's never specified whether they're Rewriting the Past or causing Branching Timelines.
Presumably, they're rewriting the past. Remember the watch experiment? They were able to tell time was looping inside the box because when the box was started, and then later a watch was placed inside and the box closed and reopened immediately, the time elapsed on the watch was an even multiple of the external time past. The watch was circulating through a time loop a huge number of times, each internal loop presenting a finite probability that it would be the one that was reopened, and by building a bigger box and putting people in it (who can decide to exit at the external beginning of the loop), time travel happens.
Primer may actually fall under Temporal Chaos Theory, as the characters never travel back further than a week—not enough time for wild divergences to manifest.
Feng Shui revolves around the battle for Places of Power that generate powerful chi. Whoever controls enough Feng Shui sites can change the course of history. The only major catch is that there are only four major "junctures" of time available in the Netherworld, and whatever happens in one juncture, you cannot go back and try to stop what already happened from occurring because time flows normally in each of the junctures.
Chrono Trigger allows major and deliberate changes to the timeline, and one rather spectacular instance of Tricked Out Time. Sometimes, your changes will come back to bite you in the ass, but for the most part that's fixable too — however, the sequel Chrono Cross gives this a Cerebus Retcon and takes it to level five.
In the Sam & Max episode, "Chariots of the Dogs", Bosco accidentally goes back in time and changes his history. When Sam and Max go back in time to stabilize the past they in addition cause Bosco to never been born. They fix it all by the end.
Type 5: Temporal Chaos Theory
The original Marvel UK comics run of The Transformers had this approach, with time-hopping characters from the future causing (both directly and indirectly) horrific effects to the timestream that result in the catastrophic "Time Wars" and numerous other paradoxes. The situation is eventually contained - barely - but the 'future' is no longer set, and indeed takes a wildly different path.
Orson Scott Card's Pastwatch: The Redemption of Christopher Columbus uses the Overwriting the Timeline version of this in order to sabotage Columbus's discovery of the New World and prevent the massacre of Native Americans and the Atlantic slave trade. Temporal paradoxes don't result, because causality and chronology are regarded as completely separate things.
The author includes an explanation of the mathematicians and physicists to the historians about why it's possible for a time traveler to exist when his own timeline has been erased. Essentially, causality is claimed not to be real. Changes in the past cause the points in time from that point on to be overwritten in the same way as a VHS tape is overwritten by a new recording. There can be no alternate timeline because there is no "space" for it to be written. The time traveler's memories are not altered, as the physical state of the brain remains unaffected by the overwriting.
Anime and Manga
This is how time travel works in the Dragon Ball universe, and the reason killing larval Cell in the main timeline didn't do anything to stop the fully-grown one from another timeline. Trunks isn't able to prevent his Bad Future from happening, but he can use the power he gained in the past to stop it from getting any worse.
This is how it used to work in Marvel Comics, particularly around the Stan Lee editorial era, flowing into the Wolfman era. It kind of stopped somewhere in the 90's.
The Abramsverse and the Mirrorverse in Star Trek explicitly do this, presumably to preserve the main continuity.
The non-canonic Countdown comic shows events still occuring after Spock and Nero travel to the past, meaning the old timeline still exists.
In The Legend of Zelda: Ocarina of Time, time-travel shennanigans (or, rather, mental time travel shennanigans) cause the timeline to split into three branches, with one point of divergence known *
the ending, where Zelda sends Link's mind back in time to prevent his removing the Master Sword from its pedestal
most probably the point during the game when Link sent his own mind back in time to retroactively obtain the Silver Gauntlets and Lens of Truth, if it is a known point at all
The time travel of Narbonic was revealed to be this variant in the "Director's Cut" version currently running. Dave Davenport becomes unstuck in time, changes his own past, and as a result... becomes someone who never smoked and never had a nicotine addiction.
Rip Hunter in Booster Gold states that the future is in flux but major events in the past (called "solidified time") are immutable. This strange time scheme hinging on the present is particularly odd when you consider Booster is from the future himself making the present his past, and Rip spent most of his time in the future (and is also Booster's future son).
Since the comic takes place in a Shared Universe, the rule is, in practice, "Booster can only change stuff that wouldn't screw up the storylines of other DC Comics." So, he can go back in time and save some random little girl's puppy from being killed, since the girl and the dog are just background characters, but he can't save Bruce Wayne's parents, since that would kind of negate the entire Batman series.
This is driven home via a recurring subplot where Booster keeps trying to go back and save Barbara Gordon from being crippled. Every time he tries, no matter how Crazy-Prepared he is, he fails, because the event is solidified time.
Andromeda turns into this from You Already Changed The Past after the introduction of tesseracts. Trance somehow manages to swap places with her future self. Then they add the Route of Ages and multiple realities.
Perry Rhodan falls into this category mostly due to being a Long Runner with changing authors. It's generally a mix of #1 and #3, but the past has on occasion been changed (including one old issue featuring a trip into the past that ended up affecting the outcome of a battle in the present, including dead soldiers spooking their superiors by suddenly being alive again). The series also once featured a "time police" for one arc, though its role was not so much to actually 'police' time as to simply mercilessly attack any civilizations discovered to be experimenting with time travel in the here-and-now.
Times Like This: A comic with time travel as its focus, but the environment is mostly satirical humor with a little drama thrown in. Changing the past can have reverberations on the present, but there's no paradoxes that come with, say, giving yourself a massage.