Ytukyg

I know this might not be an easy question, I've already read the wikipedia page, and there is an interesting view:

Therefore, the main difference between chaotic systems and complex systems is their history. Chaotic systems do not rely on their history as complex ones do. Chaotic behaviour pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'. On the other hand, complex systems evolve far from equilibrium at the edge of chaos.

However, I want to know a little bit more of their differences and intersections. If I'm not (terribly) mistaken, according to how the development of chaotic systems started, the emphasis was put on deterministic systems (should that part be of dynamic systems?), but now, with complex systems, according to what I understood of what Prigogine said, the emphasis should be put in other types of systems more close to reality.

So what can you guys tell me about these two?

I agree with comments by @Did and @Wrzlprmft that the quote is not worthy of inclusion in an encyclopedia. But it is a valid question how the notions of chaotic systems and complex systems relate. (Please update the title!)

I find the Wikipedia page of chaos theory fine and informative. Although chaos in everyday language has a broad meaning, in the theory of dynamical systems, it has a rather specific one. (Similarly to my pet peeve: exponential growth.) It is not as a mysterious phenomenon as many laypeople think it is. Like with the scopes of different areas of mathematics (e.g. number theory, discrete mathematics), there is no single accepted definition because mathematics works without there being one. Here is a possible definition from the Wikipedia page:

to classify a dynamical system as chaotic, it must have these properties:

1. it must be sensitive to initial conditions


2. it must be topologically mixing


3. it must have dense periodic orbits

About what a complex system is, a highly cited review is James Ladyman, James Lambert, Karoline Wiesner. What is a complex system? European Journal for Philosophy of Science, 2013, 3(1):33–67 (link to its text).

Let me hazard an own explanation of complex systems from a dynamical systems viewpoint. It will be `soft' just like how the question itself was.

• There are systems like a simple harmonic oscillator or planetary motion which are described by a system of ordinary differential equations with few variables (because it is one object doing one kind of thing) and which behave in a way that is easy to understand. This is not complex.
  


• There are systems where many objects or agents that follow the same set of rules (with perhaps different parameters) do a similar activity or interact with each other. Examples (not very good ones because they aren't dynamical systems): investors reallocating their portfolios between a risky stock and a safe bond when the interest rate changes. The kinetic theory of gases. As long as averaging over all agents (mean-field theory) gives an accurate and simple model of the whole system, I wouldn't call it a complex system.
  


• There are cases where identical agents following identical rules create emergent patterns. This is called a complex system by most people.


• There are systems where several different types of agents interact with each other, each following their own set of behavioural rules. These are definitely complex systems. Even if one does not exhibit complicated behaviour, it cannot be ruled out a priori. Examples: predator–prey equations (although it is quite tractable, so it is a bit of a grey zone whether people would actually call it complex). Chemical reaction networks modelled by the reaction rate equation. The Belousov–Zhabotinsky (BZ) reaction, which is a spatial problem and it exhibits chaotic behaviour.

Complexity doesn't require stochasticity in the above examples and I don't want to open a discussion of the role of non-determinism in mathematical modelling of physical systems either.

Complexity does not imply chaotic behaviour at all in my opinion. There are controlled systems which are complex but not chaotic precisely because they are under control. Concretely, they do not violate point 1 of the quoted definition (they are not sensitive to initial conditions because the feedback control drives every reasonable initial state to a similar operating regime). (E.g. the dynamical system models of the internal workings of a living cell, or of the aerodynamics of a flying aeroplane on autopilot.) The control might accidentally break down, but this causes unstability (the divergence of some values to zero or to infinity, at least temporarily until a stronger force kicks in), not chaos.

The other way around, chaotic behaviour can arise without a complex system. Think of discrete iterations giving rise to the Julia set. This is called complex dynamics, but in this case, complex refers to complex numbers and not to complex interactions.

An idea that I had to discard, the Lorenz attractor is an ordinary differential equation with only three variables and it is chaotic. It turns out that it is the model of a complex system, a particle ensemble, `a two-dimensional fluid layer uniformly warmed from below and cooled from above'.

A flow network is a dynamic system, but any particular flow network isn't a complex system.

However, the study of the behaviors of flow networks, as a whole, is a complex systems problem. The difference is that you're analyzing from a independent point view -- outside the dynamics in question.