Yes, there is a close relationship between symbolic dynamics and periodic orbits. The best results of this type are due to Sharkovskii (1964), although the paper by T.-Y. Li and J. Yorke "Period 3 implies chaos" is an exceptionally accessible introduction to this for interval maps f:[0,1]->[0,1]. Sharkkovskii's results showed that if a periodic orbit of one period exists, periodic orbits of all periods to the right of this period in Sharkovskii's ordering must also exist.

The extension of this to periodic orbits of differential equations in 3 dimensions leads to the intriguing question of whether existence of a periodic orbit that is a knot of one type must force the existence of periodic orbits of certain other knot types. I am not aware of any kind of complete theory regarding conjectures along these lines, although there are related results (for example Philip Boyland, An analog of Sharkovski's theorem for twist maps. in Contemp. Math. 1988). I am not sure just what the state of play is regarding the general question of ODEs in 3-dimensions.

Are you referring to the fact that periodic orbits of certain dynamical systems can be considered as being knots?

Knot Theory is a sub field of Topology. Control Theory and Dynamic Systems are related to Mechanics and to Computer Systems which can bring in Game Theory and Queuing Theory. Chaos Theory is a vague field that includes a lot of different studies in mathematics including Fractals, Biforcation Theory, and Finite Cell Automata.

I would not consider Knot Theory to be directly related to any of these.

Please, look at connections between Legendrian and transverse knots (the usual knots are just special cases of these) and dynamical systems. Floer's theory is a good example of these interconnections. Control theory is part of contact geometry http://www.worldscientific.com/worldscibooks/10.1142/8514

Legendrian and transverse knots are by products of contact geometry and topology. Also, look for the topics related to adiabatic quantum computation

Dan Silver and Susan Williams found connections between knot theory and dynamical systems. The connection is, roughly, this. The commutator subgroup of a classical knot or link is a dynamical system with a presentation given in terms of the action of the meridian.

A nice question.