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I am struggling to find a proper source or reference where examples of complex systems which are chaotic are given. Based on my understanding, complex systems consist of interacting components, each component has some set of properties whose values can change due to interaction or external force. Following are my question:

  1. Are 1D chaotic maps such as Logistic, Tent chaotic maps, and other maps complex systems? Complex systems show an emergence behavior a pattern. Similarly, chaotic systems also show a pattern via their attractor. Moreover, in low dimensional chaotic systems such as iterative maps, since their current values depend on past values and the variables interact through an iterating equation, I was thinking that through such iterations the variables are interacting with each other and their emergence pattern via attractors make chaotic systems as complex system. Is my understanding correct?

  2. Are all chaotic systems complex systems for example Henon, Lorenz, Rossler, Chua etc?

  3. Any reading material which I could cite that lists chaotic systems as complex systems?

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    $\begingroup$ Hi Sm1. Welcome to Phys.SE. Why do you doubt that the answer is just simply Yes? $\endgroup$
    – Qmechanic
    Apr 14, 2021 at 16:22
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    $\begingroup$ @Qmechanic I think complex usually implies many degrees of freedom. But this is a personal opinion - there is no a universally agreed upon definition. $\endgroup$
    – Roger V.
    Apr 14, 2021 at 16:36
  • $\begingroup$ @Qmechanic: In wikipedia, en.wikipedia.org/wiki/Complex_system the diagram on the right for different components/examples of has nonlinear dynamics as a component of complex system. Within nonlinear dynamics, chaos is mentioned. Hence I thought the answer would be in terms of a yes or no. $\endgroup$
    – Sm1
    Apr 14, 2021 at 16:45
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    $\begingroup$ "Complexity" is more of a marketing term than a sharply defined mathematical term. At this point I've seen basically anything more complex than the harmonic oscillator called a "complex system". It is something you say in order to impress people with all the great implications, like how your carton of eggs says it's "farm fresh". $\endgroup$
    – knzhou
    Apr 14, 2021 at 17:32

3 Answers 3

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Dynamical chaos
We are talking about dynamical chaos here, in the sense that it is a chaotic behavior that appears in systems with a few degrees of freedom, due to lack of rigidity (e.g., in respect to the initial conditions). The surprizing nature of the chaotic behavior in such systems is precisely because they are simple. In this context simple means having few degrees of freedom, whereas complex means many (perhaps even infinite) number of degrees of freedom.

Chaos/stochasticity in systems with many degrees of freedom
Systems with the infinite number of degrees of freedom routinely exhibit chaotic/stochastic behavior, simply because the cannot describe them in necessary details to predict their behavior. These are usually dealt with in the context of statistical physics, and the nature of the stochastic behavior is so obvious that it does not surprize anyone. Indeed, lately the word chaos is usually used to mean dynamical chaos, while stochasticity refers to statistical behavior of many degrees of freedom.

Emergence
Emergence is appearance of new types of behavior in systems composed of small parts, which themselves do not exhibit such a behavior. There is no a 100% agreed upon definition, but the term usually implies a very large number of simple components - i.e., many degrees of freedom.

On the other hand, stable points, separatrices, attractors exist in simple nonlinear systems, even those that do not exhibit chaotic behavior. In this sense these are not emergent properties, but rather basic properties of such systems.

Remark
In the answer above I have described complex systems as those that have many (usually infinite) number of degrees of freedom. In classical physics we usually have two differential equations per degree of freedom (for position and momentum). Thus, the dynamical systems that show chaotic behavior are usually not complex - e.g., Lorentz system has only three equations. This is precisely what is remarkable about these systems - that they exhibit chaotic behavior despite their simplicity.

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  • $\begingroup$ Thank you for answering. So based on your answer chaotic systems that are deterministic and sensitively dependent on initial conditions are complex systems? So the systems mentioned in my question such as logistic, tent maps, Lorenz etc are complex systems? Did I follow you correctly? The wiki link en.wikipedia.org/wiki/Complex_system shows that chaos is a component of complex systems, but I don't know if complex systems include discrete and continuous chaotic systems or whether they mean synchronized chaotic systems. Nothing is clear. $\endgroup$
    – Sm1
    Apr 14, 2021 at 16:48
  • $\begingroup$ I think chaotic systems, such as Lorentz etc., are not complex. Lorentz has only three equations, i.e. one-and-a-half degrees of freedom. But this is a personal opinion - there is no generally agreed upon terminology. $\endgroup$
    – Roger V.
    Apr 14, 2021 at 16:55
  • $\begingroup$ Could you please mention examples of chaotic systems that are complex systems ? Are spatio-temporal chaotic systems such as Kuramoto-Shivansky chaotic system and predator-prey chaotic system a complex system? $\endgroup$
    – Sm1
    Apr 15, 2021 at 0:47
  • $\begingroup$ Systems with a few equations are not complex - they have few degrees of freedom (two ODEs are tipically counted as one degree of freedom). This is precisely why the chaos in these systems is remarkable. $\endgroup$
    – Roger V.
    Apr 15, 2021 at 7:19
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Are chaotic systems examples of complex systems?

You decide — i.e., it depends on the definition you pick.

In 1992, Waldrop wrote in its Complexity: The Emerging Science at the Edge of Order and Chaos:

complexity - a subject that's still so new and so wide-ranging that nobody knows quite how to define it, or even where its boundaries lie.

Well, it's not so new any more, but, like with with many wide areas of research, its precise definition remains open to discussion, as attested by the 30 definitions of complex adaptive systems provided in Table 1 of Turner and Baker's paper, by the answers to the question What is the definition of “Complexity” in physics? Is it quantifiable? and by Trabesinger's Nature editorial:

A formal definition of what constitutes a complex system is not easy to devise; equally difficult is the delineation of which fields of study fall within the bounds of 'complexity'. An appealing approach — but only one of several possibilities — is to play on the 'more is different' theme, declaring that the properties of a complex system as a whole cannot be understood from the study of its individual constituents.

So this is a somewhat subjective issue, that boils down to two options:

  1. either chaos itself is considered to be enough of an emergent/complex behavior, so that any chaotic system is a complex system;
  2. or one demands a characteristic such as multiple components, adaptation, self-organization, or others that exclude 1-D maps, the double pendulum, etc.

Even the linked Wikipedia entry is seemingly contradictory in this regard, supporting both options at the same time: defining a complex system as "a system composed of many components" and that "relationships between a system's parts give rise to its collective behaviors" — which excludes simple classical dynamical systems' models — while at the same time stating (here and here) that "Systems can be complex if, for instance, they have chaotic behavior [...] In a sense chaotic systems can be regarded as a subset of complex systems" — which appears to include any chaotic system.

Perhaps worth mentioning is the description of the journal Complex Systems, which lists first in its areas of focus: "Dynamic, topological and algebraic aspects of cellular automata and discrete dynamical systems" - which certainly include even the simplest dynamical systems; as well as the Aims & Scope of the journal Advances in Complex Systems, that mentions "Population dynamics" and "Fluctuation phenomena" - which also can include low-dimensional chaotic systems. But then, here they're describing their editorial preferences, not trying to define the field they're named after.

So, to the OP's specific questions:

1. Are 1D chaotic maps such as Logistic, Tent chaotic maps, and other maps complex systems?

By most definitions, which emphasize things such as a larger number of "agents" or adaptiveness, no; by a more relaxed "more is different" approach, yes.

1. (part) their current values depend on past values

This is not correct for the examples mentioned - unless we're considering a system with delay or some other type of history dependence, the current state depends only on the previous state.

1. (part) their emergence pattern via attractors make chaotic systems as complex system. Is my understanding correct?
2. Are all chaotic systems complex systems for example Henon, Lorenz, Rossler, Chua etc?

Using a broad definition, yes.

3. Any reading material which I could cite that lists chaotic systems as complex systems?

I couldn't find anything decent. The Nature editorial cited above supports the "more is different" approach, which I understand can include simple chaotic systems, but doesn't mention them explicitly.

The difficulty to find such a reference probably indicates that the consensus tend to the option 2 above, i.e., a 1-D map or even the double pendulum are not generally seen as examples of complex systems. If you're writing about it, you can play it safe and say that "chaotic systems may be considered complex systems", but, unless it's an opinion piece, an even better option might be to avoid the discussion altogether.

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  • $\begingroup$ Thank you, your answer makes sense to me. So if a complex system needs to have many components, in the form of say several variables then maybe 1 D chaotic maps are not complex. However, chaotic systems such as Kuramoto Shivansky, 3 species and 4 species predator prey (Lotka Volterra) systems could be complex systems since they have many variables that are interacting with each other. Similarly coupled map lattices and synchronized systems could be complex systems. To summarize, is it correct to say that chaotic Lotka Volterra, Lorenz, Rossler & spatio-temporal systems are complex systems? K $\endgroup$
    – Sm1
    Apr 16, 2021 at 15:59
  • $\begingroup$ Basically, I am looking for specific names of chaotic systems that are complex systems...this information is not present in many books/papers. What the papers say are just broad definitions followed by research work on popular chaotic systems such as Lotka Volterra,KS, hyper chaos etc. $\endgroup$
    – Sm1
    Apr 16, 2021 at 16:01
  • $\begingroup$ @Sm1 I'd say you're safe with extended systems, such as lattices and spatio-temporal ones; Lorenz, Lotka-Volterra and Co. would probably rather fall into the "only chaos" category. An on-line text book gives some concrete examples it considers complex, such as vortices in a fluid or the Red Spot in Jupiter, but I don't know how good a source this book is. $\endgroup$
    – stafusa
    Apr 16, 2021 at 16:13
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Complex systems are systems where order arises out of chaos or higher levels of order arises from more low level order. This is called emergence.

Chaos by itself is not complex, it is merely chaotic. There are some simple systems that are deterministic but quickly fall into a chaotic movement. For example, the double pendulum.

The Mandelbrot set is a paradigmatic of order arising out of chaos at different scales.

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