Does Fibonacci number sequence represent a chaotic system?

Question

If we define a chaotic system as a non-linear deterministic aperiodic system that is highly sensitive to its initial condition, then we should call Fibonacci number series a chaotic system.

But is it really chaotic? considering that it's almost predictable?

In other words, do we only consider a system chaotic, if we cannot predict it? If this is the case, can we claim that defining chaotic/non-chaotic depends on our current state of knowledge and not the attributes of the system itself?

Answer 1

Fibonacci sequences are not chaotic for several reasons:

1. They are linear and as a result there is a closed form. They are perfectly predictable.
2. They are not bounded, which is a central aspect of any definition of chaos I know. (Often this is implied by requiring recurrence or a property called topological transitivity.) Without this condition, things like exponential growth would be chaotic.
3. If you are only allowing for integer initial conditions (which is up to you), the possible states are discrete and you cannot have initial conditions that are infinitesimally different. Hence there cannot be sensitivity to initial conditions in any way that is comparable to chaos. In general, there is no chaos in discrete-state systems.

In other words, do we only consider a system chaotic, if we cannot predict it?

That’s not a defining property of chaos. While there are many competing definitions of chaos (which mostly only differ for pathological examples), they are all pretty clear and do not feature unpredictability. Rather, a certain lack of predictability is a consequence of the definitions.