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:
- either chaos itself is considered to be enough of an emergent/complex behavior, so that any chaotic system is a complex system;
- 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.
