According to my understanding, the butterfly effect says, very informally, that even a tiny perturbation in a physical system can lead to significant alterations in future states of the physical system. But, I am looking for a more formal and rigorous definition of the butterfly effect, for two reasons. The first reason is that if a physical system is deterministic, then it does not make sense to talk about perturbations of it, for there is only one way for a deterministic system to be, or so I think. The second, more important reason, is with the informal word "significant" in "significant alterations in...". What does "significant" actually mean? It seems too subjective and human-centered. That is why I am looking for a formalization of the butterfly effect, if there is any.
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9$\begingroup$ en.wikipedia.org/wiki/Chaos_theory $\endgroup$– hftCommented Apr 20 at 1:17
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1$\begingroup$ Chaotic systems have non-periodic attractor, defined by ODE system of equations. $\endgroup$– Agnius VasiliauskasCommented Apr 20 at 10:18
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1$\begingroup$ This isn't an answer because it's not a mathematical formalization, but an analogy: Consider throwing a bowling ball. A difference of an inch or cm in where the ball leaves your hand can result in a big difference in the number of pins knocked down. $\endgroup$– BarmarCommented Apr 20 at 18:22
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4$\begingroup$ "The first reason is that if a physical system is deterministic, then it does not make sense to talk about perturbations of it". It might be better to think of a perturbation as being a measure of our ignorance of the exact position and velocity of a particle in its initial condition. Remember the uncertainty principle means we can never be 100% sure of the exact position and momentum of a particle, no matter how good our measuring devices and techniques are. $\endgroup$– KDPCommented Apr 21 at 3:14
5 Answers
... if a physical system is deterministic, then it does not make sense to talk about perturbations of it ...
The future (and past) behaviour of a deterministic system is fixed once we know its state at some point in time. For convenience, we can label this point in time $t=0$. Although the system is deterministic, what we can perturb is its initial state $\mathbf x(0)$, and we can ask how small changes in $\mathbf x(0)$ affect the subsequent behaviour of the system. If we have a measure of distance in the space of possible states, then we can quantify this question as follows: if we look at all the possible initial states within some small distance $\epsilon$ of $\mathbf x(0)$, and we take some later time $t$, can we establish some limit on how far the states of these perturbed systems can be from $\mathbf x(t)$ - this limit will be a function of $\epsilon$ and $t$.
The significance of this question is that in a real physical system there is a limit to how precisely we can measure its state. So if we imagine the system's initial state as being spread out over some small volume of state space due to this limit on the precision of our measurements, then we are asking how fast this small volume grows as the system evolves over time, and so how quickly does our uncertainty about the future state of the system grow.
What does "significant" actually mean ?
"Significant" means that the size of the limit we described above (the maximum distance between $\mathbf x(t)$ and the state of a slightly perturbed system at time $t$) grows exponentially with time. In this case, our approximate knowledge of the initial state of the system (which may be very precise, but always has some uncertainty) quickly loses its predictive power.
The formalisation of all of this is an area of mathematics called chaos theory.
Usually butterfly effect is mathematized as "sensitivity to initial conditions". This is a bit different to what you are saying. Namely you consider two close initial conditions $x_1(0),x_2(0)$ and see where the dynamical flows bring them, namely you estimate
$$\Vert x_2(t) -x_1(t)\Vert .$$
For "nice behaving" dynamical systems the worst that can happen is an exponential divergence of the above quantity $e^{t\lambda}$. The exponent (or the maximal of such exponents) is called Lyapunov exponent. Essentially a positive Lyapunov exponent can be considered synonym for sensitivity to initial conditions, because nearby points end up very far (exponentially) after some time.
To your first comment, it makes sense to talk about a perturbation of a system. You are correct that if the system is deterministic, it can lead to only one outcome, but we can consider additional initial conditions besides the "true" initial conditions. We may consider a "measured" initial condition, using the best measurements we have. This measured state is never quite the same as the true state. We might ask how much being off by just a little on the measurements might affect our ability to predict where a system evolves to.
As for formalizing this, one very powerful tool we use is a Lyapunov Exponent for the system. The Lyapunov Exponent for a system, $\lambda$, is the value such that:
$$|\delta Z(t)|\approx e^{\lambda t}|\delta Z_0|$$
The $\delta Z$ is an infinitesimal perturbation... as small as can be. In a chaotic system, this perturbation grows by a function of $e^{\lambda t}$ – an exponential increase as time goes on.
The infinitesimal perturbations are used for many reasons, but one of the powerful reasons is that working infinitesimals linearizes the system. While you may be able to come up with a more practical measure for any given system, a Lyapunov exponent can be used to compare any two chaotic systems.
An equivalent tool which is more intuitive is Lyapunov Time. Lyapunov time is the timescale $t$ on which the perturbation grows by some amount. A scaling of $e$ is common, as is $2$ and $10$. ($e$ is the most common, and is used here.)
Wikipedia gives a table of Lyapunov Times for various chaotic systems:
| System | Lyapunov Time |
|---|---|
| Pluto's orbit | 20 million years |
| Solar System | 5 million years |
| Axial tilt of Mars | 1-5 million years |
| Orbit of 36 Atalante | 4,000 years |
| Rotation of Hyperion | 36 days |
| Chemical chaotic oscillations | 5.4 minutes |
| Hydrodynamic chaotic oscillations | 2 seconds |
| 1 cm3 of argon at room temperature | $3.7×10^{−11}$ seconds |
| 1 cm3 of argon at triple point (84 K, 69 kPa) | $3.7×10^{−16}$ seconds |
Yes, the orbit of planets in the solar system is chaotic. On the scale of millions of years, it is not possible to predict where the planets will be.
In general, the Butterfly Effect is characterized by very small changes in initial conditions having great effect on a larger system.
The literal Butterfly Effect itself (as in the effect of a butterfly's wings on global weather) is more of an allegory than an actual physical/mathematical theory. It is cool for didactic reasons as everybody immediately understands what it means, intuitively.
The closest you would probably come to it, in actual research, are papers like Dependence on initial conditions versus model formulations for medium-range forecast error variations which, although not using butterflies as input, still look at the effects of changes in initial conditions.
There are several common formalizations for other systems than the weather, as well.
For example:
- Bifurcation is a well-known formalization which arises for exceedingly simple systems. The formalization consists of an easily creatable diagram of the chaotic behaviour of the system, especially of the visible effect of starting out simple, but then going into a completely different regime very quickly. You will find plenty of resources on this topic if you look for the term.
- More generally, Chaos Theory works on this kind of formalization, i.e. by giving bounds for variables of systems where the system as a whole will change behaviour between "normal" and "chaotic", and expounding on why and how it is hard-to-impossible to predict the behaviour, once the chaotic regime is reached.
- The double pendulum is a more "classical" example with a, to quote Wikipedia, "rich dynamic behaviour". The part it shares with the Butterfly Effect is that it is very sensitive to changes in the initial parameters, which wildly differentiates it from the single pendulum, which is one of the simplest physical systems imaginable.
Today people typically refer to Liapunov exponents when referring to the butterfly effect, as most answers already pointed out. It is about the exponential growth of infinitesimally close neighbouring trajectories. Historically, this was first formalised proven rigorously for billiards in negatively curved surfaces (related to geodesic deviation if you've seen this in GR).
However, originally, the butterfly effect rather alluded to another possibility that you cannot observe for finite dimensional ODE's. In his 1969 paper, Lorenz suggests that instead, you could have a discontinuity of the system with respect to the initial condition at finite time. Formally, say you have a deterministic system governed by the ODE: $$ \dot x = v(x) $$ with $x$ lying in an abstract high dimensional phase space. By analogy with fluid mechanics, you can define a flow $\phi(x,t)$ which satisfies: $$ \partial_t\phi = v(\phi) $$ so that $t\to\phi(x,t)$ represents a trajectory with initial condition $x$. Lorenz asks the question whether $\phi$ could be discontinuous with respect to $x$ after a certain time. This would mean that if you want to predict the future with an arbitrarily small but finite error, you would need to have perfect certainty in the initial condition. A practical consequence would be the existence of a mathematically well defined finite time predictibility window, no matter the ressources spent in accurately computing the trajectories and specifying the initial conditions. You can check out Lorenz's original paper "The predictability of a flow which possesses many scales of motion", or a more modern discussion by Tim Palmer "The real butterfly effect".
Hope this helps.
