During my PhD studies I was wondering at what point a system becomes complex or even chaotic. I conducted a laboratory study under defined boundary conditions. I conducted my research on fairly homogeneous rock samples. So I could predict pretty well how the samples will behave. Then I read a publication about fractals in geomechanics, the complexity of systems and chaotic behaviour. Obviously complex or chaotic behaviour increases with the number of variables and uncontrollable factors.
But what is the point where a system becomes complex? Is it a matter of the size? Is it a matter of the number of variables? Is it a matter of the view point? Is there any quantification when a system becomes complex or even chaotic?
Doctor Christian Ungewitter , if you have a deterministic system but it is much sensitive to initial conditions your system is in a state of chaos. It is depend of your precision in the initial condition. For example, imagine that the display of a computer shows numbers with 5 digits bur it is really working internally with 8 digits. If you stop your simulation in a specif moment, to write these 5 digits on the screen and to continue your simulation, the results will be very different than the original (8 digits precision). If this happens your system is in a state of chaos. Because it is very sensitive to initial conditions.
So the quantitative description of chaos depends on where your uncertain digit is. If any changes in this digits lead a very different results, your system is chaotic.
Dear Christian Ungewitter
For a simple harmonic system, the frequency of a specific mode is an integer or rational number, therefore you can deterministically determine at what time-periods the signal of oscillation variable repeats itself, e.g. T=2. But if it is irrational as; T =sqrt(2), then you cannot certainly calculate the periodicity interval. Hence, the oscillation signal gets irregular (chaotic) through a computer simulated illustration, due to precision arithmetic logic.
sqrt(2)=1.41421356237309504880168872420969807856967187537694...
π = 3.141592653589793238...
Christian Ungewitter A good question! What the people above are saying is that complexity is independent of size - there are many examples of complexity in a system comprising only two interacting elements.
There are some, well-know, crucial properties that can be used to classify a system as being complex, one of them being chaos (or sensitive dependence on initial conditions, as shown above). For me, another crucial feature is the property of "emergence" - the idea that the output is 'greater than the sum of the parts', and therefore difficult to predict.
PS: I came upon your question as it's related to a question that I posed here: What is the simplest complex system? Any ideas welcome!
I don't think that the system of three ordinary differential equations, called the Lorenz equations, is too complex. And yet one can use it to generate chaotic behavior. Of course, the parameters of the equations and the initial conditions are important here.
Examples are worth of thousands words.
1) Logistic map
X(n+1) = a * X(n) * (1 - X(n))
is expressing chaotic behavior and period doubling. Is this map simple enough?
https://en.m.wikipedia.org/wiki/Logistic_map
2) John Conway's 'Game of Life' (google this)
It is simple cellular automaton that creates emergent structures. Gliders are one such emergent. They can live on a chessboard. Is this simple enough?
From this and many other biological examples we can see that the world is full of simple systems expressing chaos and complexity. We are just unable to observe them directly due to to our insufficient instruments.