Can $x(t) = A\cos(at^2+b)$ be interpreted as a generic chaotic trajectory of some dynamical system?
We have just been introduced to the background theory behind chaotic systems but haven't worked through any practical examples such as this. The three criteria which I am going by to prove a system is chaotic:
1)It must be sensitive to initial conditions.
2)It must be topologically transitive.
3)It must have dense periodic orbits.
I believe that it fulfills criteria 2 as an arbitrarily small subset of the range will eventually cycle through every value. For criteria 1, I am tempted to say it doesn't fulfill as it is bounded by A and I believe this is the same reason that the more simple $A\cos(t)$ fails to be chaotic. Would I be correct in this assumption?
And finally, how would we then analyze the criteria for a hyperbolic function such as $x(t) = A\tanh\left(\frac{at}{1+t}\right)$?
I believe this function fulfills criteria 1 and 2 but I can't be sure about criteria 3 (for either function) and would appreciate any advice regarding this.