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One of the best, more original examples of chaos theory comes from Ray Bradbury’s A Sound of Thunder. People in the chaos-theory community have always said that the hunters that time safari allowed to go back in time and kill dinosaurs could do so because that dinosaur was going to die a few seconds later anyway.

Chaos theory is all about sensitive dependence on initial conditions. So wouldn’t killing that dinosaur a few seconds before it is “supposed” to die actually alter the future? Is chaos theory that time-dependent? So does forcing an event to happen in the past have greater future implications if you force that event earlier than it would have otherwise happened? For example, if one time-safari hunter killed a dinosaur three seconds before it is supposed to die, and then another hunter on a different time safari kills it seven seconds before it is supposed to die, then does the latter have greater implications on the future?

People in the chaos-theory community have always said that the hunters that time safari allowed to go back in time and kill dinosaurs could do so because that dinosaur was going to die a few seconds later anyway.

Forgive my bluntness, but: Citation needed.
This claim is what you would get from a typical popular misinterpretation of chaos theory.
I have been in said community for quite a while and I have never heard anything remotely resembling that claim (not that chaos theorist talk much about such scenarios).

Here is what you do get from chaos theory:

Reality is clearly sensitive to initial conditions (a.k.a. the butterfly effect).
There are several good arguments for this, but it suffices to note that it contains many clearly chaotic subsystems such as the weather, which in turn affect everything else.

If a system is sensitive to initial conditions, almost every perturbation to the state of the system will on the long run completely change its behaviour.
The only question is how long it takes, which depends on the magnitude of the perturbation and the largest Lyapunov exponent of the system.
More specifically, perturbation propagate exponentially.
The difference of the magnitude of change between killing a dinosaur three or seven seconds early is negligible in this respect.

Note that the almost every here serves to exclude highly special cases such as perturbating the entire system in a way that corresponds to moving it half a second into the future.
For all practical purposes, you can consider it to be every.
In particular, there is no way to predict that a certain perturbation has no effect.

Now, if we consider a perturbation of the scale of an atom, it only depends on the largest Lyapunov exponent of the system how quick it propagates.
This is rather difficult to find out for such a complex system as reality, in particular since it requires finding what subsystem is the weakest link, i.e., it propagates the change most quickly.
Answering the latter is not only tedious but also pointless except for such purely hypothetical discussions.
For example, for earth, the weather may be the weakest link, but it may also be the stock market, some neuron in the brain of some head of state, or something else entirely.

So, to estimate an upper limit, we can resort to estimates that chaos makes weather prediction impossible beyond a time scale of a few weeks. Perturbations on the scale of an atom will take a bit longer to grow to the scale of the entire system, but it won’t be off by more than a few orders of magnitude due to the exponential nature of the system. As a conservative estimate, a change of the scale of an atom affects the weather within a few years.

To come back to your dinosaur scenario, Bradbury’s story does not comply with chaos theory in at least two aspects:

• The system Earth is much more sensitive to initial conditions. Even perturbation on the scale of a single atom a million years ago would affect the present colossally. For example, mankind would probably have never evolved this way. After all evolution is based on mutations, and mutations happen on the molecular level.




• There is no way to orchestrate a perturbation that has no effect (such as killing a dinosaur that would have died anyway).

As a sidenote, mind that the story in question was written before chaos theory was a thing.