The my first idea is Lorenz system or other dynamical systems have chaotic behavior. There are a lot of other dynamical systems with similar properties. Some of them includes only two first-order equations, but they are usually extended by external force can be considered as third component.

What  about   ONE,  say logistic map ? Is it not enough complex ?

to not be confused with chaotic behavior.

I'd be careful about evaluating "complexity" solely  on number of constituent components.  It's very simple to imagine any number of purely linear, additively decomposable models from the addition of "very, very many random variables." Also, I really enjoy Cariani's treatment (please see attached) in which an important aspect of complexity might be the capacity of components/dimensions to be created or destroyed. 

You may probably know N. Johnson's book: Two's company, Three is Complexity. Please take a look at that.

The important point however are the dynamics and relations among the few components, not the components as such.

Anyway, it' s been established (rather) long ago in the community of complexologists that any complex systems originates with a minimum complexity; meaning, by a synthesis, not as aggregation of items.

That being the case, nonetheless, in quantum physics one single element is already complex. The best example: the electron itself.

I agree with Mohanty and Tsiberkin too. It depends on what are you considering as 'complexity'. Chaotic systems are considered as complex and they do not need more that one component. If you are talking about complex networks, then you need a network, so at least you need two components. But networks with two nodes can be too trivial (even for synchronization problems) so I guess you need 3 nodes minimum. If you  need some structural properties for your network, then I'm afraid that there is no general minimum and it will depend on your domain.

You can read also the 1st chapter of the book that Carlos has recommended here http://www.uvm.edu/rsenr/nr385se/readings/complexity.pdf but I guess it has not a clear answer to your question: that is, a concrete number

These answers are all very helpful. The question was prompted by my attempt to describe complex systems in my own words for a book that I am writing on complex adaptive systems. This is quite challenging as there is considerable variation in complex systems (whatever they are) and the definitions thereof (as the examples above indicate).

I like the phrase "Two's company, three is complexity" and was thinking along the lines of "Two is game theory, three is complexity theory".

And, of course, I am not really looking for a magic number. I just wanted to get a sense how others are describing complex systems. Many thanks for your answers.

Hi

however, thinking about an army... If you have many components like soldiers walking, if you can summarize their behavior as one single set behavior i.e. ordered, then you do not take into account the remaining behaviors (behaviors not exactly ordered, one looking at his girls friend with a little smile per example): that would mean the order is stable and important enough to include all behaviors under observation and does not leave space to chaos.

You might only need two-three free persons or elements to create a complexity (some order with a good deal of chaos), but sometimes unison behaviors and interactions does not leave place to complexity (I.e. It would not be worth to describe the behaviors using complexity, e.g. A good playing orchestra with a maestro). A mechanical watch is therefore more complicated than it is complex, but if we include the fragility of that system into the considerations (same for the orchestra) it's complexity becomes a relevant model of potential behaviors...

I think we almost always have complexity if we open our eyes and ears but that it is not always worth considering it for the purpose we pursue.

Have good writing.

Complexity has nothing to do with complication. It is not a question of differential equation or of  "large number of variables".

Complexity is the attitude of the Observer in which he considers the System as a whole and not composed by parts. Thermodynamics is a complex science statistical mechanics is a complicated model. Complexity is not an attribute of a System but it is (as I said before) an attitude by the Observer.

A system is a set of interacting components so a complex system, being a system, requires at least two components in order to have interaction. In logistic map something is interacting with its previous states, so there are at least two interacting components. Thank you guys for the suggestion to read "Two's company, Three is Complexity." It looks quite interesting.

I thouht I answered yesterday. My point is that "complexity" is one way the Observer observes the System and is not a property of Systems (to be complex or not...).

For instance a gas watched as a thermodynamical System is a complex System while watched as the ensemble of, say, 1024 molecules can be at most a complicated System and is described with the tools of Statistical Mechanics. From the Thermodynamical point of view a gas (or whatever else) could be thought of even as a continuous it has not to be viewed as composed by descrete particles.

Any object can be complex if described as a whole and described by its overall properties, or be considered composed by n sub-systems if we aim at describing its properties as some result of the properties of the sub-systems.

I would challenge the statement that two elements do not add up to a complex system. My understanding is that a complex system is a system that engages in complex behaviour (non-linear, not-quite-unpredictable, chaotic), regardless of the number of its components.

Secondly, a complex system is -in many standard definitions- open, or ambiguously bounded. Therefore, it is influenced by activity 'outside' its boundaries, which means that the number of components 'inside' them is, perhaps, not so important. 

An example of a complex system (i.e., as system that behaves chaotically) with just two components is a double pendulum.

Update 11/10: a revised version of my answer is in the link below.

http://achilleaskostoulas.com/2014/10/10/how-many-components-must-a-complex-system-have/

You may have a complex system with two components if they have many interactions with their environement, but of course complexity increase with the number of elements and the numer of their internal and external interactions.

Tipycally, a family may qui ckly tunr into a turbulent complex system.

The fact that a system has either one or a very large number of components does not mean the system is necessarily complex, therefore the question "what is the minimum number of components in a complex system" is a trivial one. Numerosity of components is a necessasy but not sufficient condition of complexity. This basically means that you will need to consider a bunch of other properties (such as non-linearity, hierarchy, decomposability, etc.) to characterize a system as complex. Finally I would add that the "set of properties" featuring a complex system may differ depending on the scientific discipline, i.e. Physics, economics, biology, psychology...

Dear Francisco, there is this discussion going on around, namely: is there a universal measure of/for complexity? Or, else, each complex system has its own complexity measuring.

The literature about the discussion is broad and deep, and to-date, no consensus has been reached about it. My point is: what you say is true; however, it's just half of the truth, if you allow me the expression.

This guys there at the SFI were seeking for the universal laws of complex systems, which, they claimed, ought to be simple - at the end of the day. Such was the program since the SFI was set up in 1984. In a seminar published in 1994 they recognized that such a search was… well… fruitless. Since then none of them has been talking around about the search for the universal laws of complexity. (Except for a beautiful book by M. Mitchell, where the subject is just mentioned, not treated).

Can we safely claim that each complex science -say, biology, physics, etc., has its own understanding of complexity? Can' t we attack the problem from a different perspective? These are just a few questions I' ve been working along.

Dear Carlos, your questions are quite relevant to understanding complexity and a good starting point for someone breaking ground in disciplines other than natural sciences.

I'm sorry if this is not the scope of the Q&A, but I'd like to answer Carlos' questions saying "YES": to my understanding complexity needs to be adapted to every discipline under investigation. This is not to assert that complexity is not an objective matter or that there is not a "core" understanding of what complexity entails; I agree that complexity is both an objective measure of complex systems, and that there exists a "core" common frame. However, complexity must (and needs) to be measured in different ways depending on the discipline. 

Natural sciences insights into complexity are of course the reference standpoint for any further inquiry into complexity, but other non-natural disciplines, i.e social sciences, need to build upon those insights and come up with its own science of complexity.

Let's realize that complexity science emerged from the understanding of the atom as the basic building block, however, this does not apply to social sciences, where you cannot theorize from this viewpoint and we need different coordinates. Furthermore, properties such as near-decomposability,  homeostasis, spontaneous order-disorder ...(just to mention some) need to be further investigated and approached under a different perspective in non-natural disciplines.

Dear Francisco, thank you for your comments. Let me please start by the end of your post. In fact it is not true that complexity science remerged from the understanding of the atom. The -so to speak- grand-father of complexity science is H. Poincaré. The cradle is the three-body problem - thanks to which chaos was first discovered. (We need wait till Lorenz for a full discovery of chaos).

For the sake of the conversation, I would like to say two things. First, it is not true any longer (at all!) that social sciences are to be built upon natural sciences. Such was the program of the SFI. However, currently no one thinks like that. The reason is that -then I come to my second point- that once you speak of complexity science, the division between natural and social sciences (not to mention the particularly of physics or anthropology) vanishes, so to speak. Complexity sincere is a larger category that -once you stand on it- encompasses and surpasses at the same time the division of sciences that is characteristic of modernity.

Besides, I like your take about complexity adapting to each singular discipline - instead of a universal (formal) one. For the sake of brevity I must stop here. Cheers!

I didn't say social sciences are to be built upon natural sciences, but instead that social sciences need the insights of natural sciences (in fact I emphasize social sciences need new coordinates), otherwise the less advanced understanding of complexity in social sciences is doomed to reinvent the wheel. 

As a matter of fact, both groups of sciences -to stick to the classical expression- have learned from each other, and they keep influencing each other. Such is the marvelous fact when one is working on complexity science.

There are far more small components than large ones in a complex system. This is what we have recently discovered. Based on the discovery, we believe that complex systems are like human brains that are hard to decompose, while simple systems like mechanical watches that are decomposable.

Jiang B. and Ma D. (2015), Defining least community as a homogeneous group in complex networks, Physica A: Statistical Mechanics and its Applications, 428, 154-160,

https://www.researchgate.net/publication/271771290_Defining_Least_Community_as_a_Homogeneous_Group_in_Complex_Networks?ev=prf_pub

Article Defining Least Community as a Homogeneous Group in Complex Networks

Two or three, depending on whether you count the interaction as a component.  Therefore, three. 

http://1.bp.blogspot.com/-gvAF4-jyXWU/T0vHhHr0dJI/AAAAAAAAArM/JMnckNxSxrY/s1600/turing-patterns.png

https://ardabiomedics.files.wordpress.com/2013/12/gm_activatorinhibitor.gif

https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

Also, here are some examples that would appeal to humanist sensibilities for why three and not two:

http://www.iupui.edu/~arisbe/menu/library/bycsp/guess/guess.htm#1

http://www.angelfire.com/md2/timewarp/buber.html

http://www.iupui.edu/~peirce/writings/v1/v1intro.htm

http://www.centertao.org/tao-te-ching/dc-lau/#chapter-42

http://www.meetup.com/The-Chicago-Philosophy-Meetup/events/226011892/

Trick question, isn't it ? Is the content of a glass of still water a complex system (with its myriads of water molecules) ? Is the quantum vacuum not a complex system (with its energy and creation / annihilation of virtual particles) ?