Can synchronized chaotic systems be periodic or stochastic? I am confused regarding the terms chaotic, periodic, and stochastic in the context of coupled chaotic systems:


In general, if chaotic systems are coupled together and synchronized then does the dynamics of the synchronized system to a layman be referred to as still chaotic or periodic?
If the synchronized system becomes desynchronized by removing the chaotic oscillators, then can we say that the dynamics of the desynchronized system is stochastic and there is no chaos? What is the opposite of chaos? Periodicity of randomness?

From the application of chaotic circuits like Chua's chaotic circuit, if the synchronized system is composed of Chua chaotic oscillator and if one of the Chua's oscillator circuit becomes faulty then would it show random dynamics?
What is the opposite of chaos? Periodicity of randomness?


First of all, chaos, periodicity, deterministicity and stochasticity are best used as properties of stand-alone mathematical models¹.
A simple characterisation of these different properties is this:


In a stochastic dynamics, identical initial conditions lead to different outcomes.
A system that is not stochastic is called deterministic.
In modelling, you achieve stochasticity by employing random-number generators.²
This is something you have to explicitly do.
Systems do not become stochastic just like that.
In a chaotic dynamics, identical initial conditions lead to identical outcomes, but similar initial conditions lead to dissimilar outcomes (in general).
In a regular dynamics (i.e., periodic, quasiperiodic or fixed-point), similar initial conditions lead to similar outcomes and identical initial conditions lead to identical outcomes.


These three are clearly distinct.
Chaotic dynamics and regular dynamics are deterministic.
Thinking in terms of opposites is not helpful here.




¹ As to whether reality is any of those: We don’t know, it’s inherently impossible to find out, and even if we knew, this wouldn’t help the least with practical modelling applications.
² You can (and probably will) use pseudorandom number generators here, which are in turn deterministic, but if that affects your model, you are doing something wrong. For all intents and purposes, you should be able to replace a pseudorandom generator with a true random number generator.



  In general if chaotic systems are coupled together and synchronized then does the dynamics of the synchronized system to a layman be referred to as still chaotic or periodic?


If you couple two systems together, you create a new system.
If and only if the original dynamics (and the coupling) are deterministic, the coupled system will be deterministic³.
Thus, if you couple two chaotic systems, the result will be deterministic.
However, in general the resulting dynamics can be chaotic or regular.
For example:


It can exhibit some generalised synchrony, that in turn can either be regular or chaotic.
The systems may not synchronise at all, but exhibit distinct chaotic dynamics.


You can only make statements like this:
If two identical chaotic systems are diffusively coupled, the coupled system will for some initial conditions exhibit a completely synchronised chaotic dynamics, in which the individual systems exhibit their uncoupled dynamics.
However, the same system may already exhibit a different dynamics for different initial conditions.


  If the synchronized system becomes desynchronized by removing the chaotic oscillators, then can we say that the dynamics of the desynchronized system is stochastic and there is no chaos?


If you start with a deterministic system, removing one component cannot possibly make it stochastic.
As to what dynamics you will get, this depends on how exactly your removal looks like, but in general it can be anything.


  From the application of chaotic circuits like Chua's chaotic circuit, if the synchronized system is composed of Chua chaotic oscillator and if one of the Chua's oscillator circuit becomes faulty then would it show random dynamics?


As already mentioned, chaos, periodicity, deterministicity and stochasticity are properties of stand-alone mathematical models.
To translate these to real systems, the question is which kind of model adequately describes the system.
A real system always has a “stochastic component”, i.e., one that we can model only by stochasticity and that is beyond the scope of what we can access, e.g. thermal noise – if such components are negligible for whatever question we want to answer, we can use a deterministic model; otherwise, we cannot.

So, here it depends on what exactly you mean by faulty and how you can adequately model it.
It may be that the original system can be adequately modelled by a deterministic dynamics, while the new system cannot.
If faulty means that the circuit is exhibiting random fluctuations (that you cannot model deterministically), you get a stochastic system.
It doesn't matter wether you are a layman or an academic. In both cases one should use the proper notation and/or define the terms orderly and accurately. Remember a chaotic system can be periodic, if it has pseudorandom properties inside a periodic boundary condition. The question doesn't make much sense that its one or the other.

The opposite of chaos is also non-randomness, but it is a very complicated field to go into. One can say that non-randomness (or non-chaotic systems) have regular patterns that one can easily see or deduce, but sometimes chaotic systems have patterns combined with eachother, and it behaves chaotically that way. There is another thing called Complexity that can be intertwined in these concepts. Simple rules (methods) can produce complex behavior and pseudorandomness. Chua oscillators seems to be real hardware random generators and might be more than pseudorandom. But nobody knows if the real world randomness is pseudorandom. Periodicity is when something repeats, like a pendulum, but if it has a damping factor the state of the pendulum is never equal to a previous state.