Butterflies, Snowballs and Dominoes
Small changes in circumstances leading to large changes in outcomes is an oft-repeated phrase. This is known by several names, particularly, the butterfly effect, the snowball effect, and the domino effect. Although these are used interchangeably, I’d like to use them for referring to distinct changes in the same system for good reasons.
These terms are going to be used in the context of differential equation models for physical phenomena, where small changes, whether introduced on purpose or due to an error, would lead to vastly different final solutions.
These perturbations could be introduced either in the equation, the initial condition or a boundary condition. And it is here that the analogies with the title of this post fit best.
1. Butterfly Effect
Small perturbations to initial conditions are akin to a butterfly flapping its wings and setting off a tornado. This is observed in the famous Lorenz system, which can give widely different answers for very similar initial conditions.
For example, using σ = 10, ρ = 28, and β = 8/3 and at t = 20 with the below initial conditions, we have:
(1, 1, 1) → (9.03, 7.91, 24.42)
(1.001, 1.001, 1.001) → (2.72, 1.66, 10.66)
Thus, small perturbations in the initial condition of an appropriate system can give rise to widely different solutions. In fact, it can be rigorously shown [1] that a Lyapunov exponent, which characterises the separation rate for infinitesimally close trajectories, for the above system is positive.
This implies that small perturbations can even lead to unbounded differences! It is a fact that has to be kept in mind while modeling realistic scenarios, where initial conditions cannot be captured to the dot.
Note that ‘chaos’ only refers to unbounded trajectories for close initial conditions. There are other perturbations to mathematical models that need to be inspected as well.
2. Snowball Effect
Small perturbations to the mathematical model itself have an interesting analogy. They are similar to a miniscule slope inducing a gravitational source term, which eventually can lead to amassing a large snowball.
Consider a snowball of radius ‘r’, initially at rest, rolling down a slope with a small angle of ‘ϵ’ as its slope. For snow accumulation, let’s assume it is a continuous process and proportional to the distance travelled. Thus, in a small time instance ‘dt’, the snowball travels ‘v dt’ and the radius increase is given by ‘dr = k A v dt’. This equation when written as a differential equation is:
The sum of potential and kinetic energy before and after this infinitesimal roll is the same. Considering the final position as the reference for potential energy, we have:
The system is easier to solve when written in terms of m and v. This simplifies to
Here α is a positive constant that captures the density and accumulation rate k. One can see that m grows with time as the source term is always positive.
The code to generate the above graph can be found here. This captures the overall trend although the program has been written for a specific value, that is, velocity tends to a constant and the snowball mass shoots up with time. Ideally, one must show this trend irrespective of parameters, but that is for another post.
Even for a small inclination, we have an unbounded increase in mass of a snowball, in contrast to when it would have been of constant size on a flat plane. Here, the small inclination is like an infinitesimal perturbation to a source term in the mathematical model and that leads to an unbounded growth in the variable of interest.
2.1 Connecting Snowball Effect to Artificial Viscosity
This very much connects to the notion of artificial viscosity in computational fluid dynamics (CFD), which essentially is a numerical dissipation term that mimics the effects of physical viscosity in the flow. It is added to the governing equations to smooth out gradients and prevent numerical instabilities.
However, adding artificial viscosity to a CFD simulation can also be seen as a small perturbation to the source term in the mathematical model and that can lead to excessive or even infinite damping of the flow over time, analogous to the unbounded growth of the snowball.
Unless one can rigorously show that source term perturbations cannot change the solution by much, using an artificial viscosity approach cannot guarantee accuracy of the solution irrespective of the magnitude of perturbation. Doing this without having to solve the equation, as that might not always be feasible, can also be challenging.
3. Domino Effect
Small perturbations to the force boundary condition on a single domino can lead to a cascading effect of several dominoes toppling. Thus, an infinitesimal perturbation to the boundary condition of an appropriate system can set off an unbounded change to its state.
Consider a simple one-dimensional harmonic oscillator given by the following system:
When a small oscillation is constantly applied to the left boundary condition instead:
This system can be solved analytically by substituting a particular solution :
Thus, it can be seen that when the perturbation matches the natural frequency, the solution can grow in an unbounded fashion. This phenomenon is also commonly known as resonance.
A similar analogy can be drawn to modeling situations, where it is hard to pinpoint the boundary conditions. And it was shown that small variations can also lead to unbounded growth of variables.
4. Summary
In the previous sections, perturbations to various parts of the mathematical model were shown to create unbounded growth in the variable of interest.
Thus, introducing small perturbations as part of regularization for numerical stability can shift it arbitrarily far from the solution of the unperturbed model. Any success such as similarity with experiments should then not be ascribed to the model.
The best conclusion one can draw in such situations is to confirm that the error for canonical problems is indeed negligible and hope that it holds for many more situations.
References
[1] Viswanath, Divakar. Lyapunov exponents from random Fibonacci sequences to the Lorenz equations. Doctoral dissertation. Cornell University, 1998.
