# Chaos Theory Go With The Flow

Trajectories do not intersect. A trajectory in the state space M is the set of points one gets by evolving x ∈ M forwards and backwards in time:

$$C_x = \{y ∈ M : f'(x) = y \text{ for }t ∈ R\}.$$

Show that if two trajectories intersect, then they are the same curve.

Some Definitions:

M = state space (set of all possible values in a dynamic system) f'(x) = the output (or y) ∈ = to be a subset, for example, t∈R is t inside of R, R being the set of all real numbers, t = time.

Note: A dynamic system is a system who's state evolves over a state space using a fixed rule.

Thanks!

• What is $t$? something is missing Oct 4, 2015 at 16:43
• I'm sorry, t stands for time. Oct 4, 2015 at 16:49
• Yes, but you don't use $t$ anywhere, so why is it relevant in your definition? Oct 4, 2015 at 16:52
• @ThomasAndrews The more important is that the definition of dynamical system is missing here :) Oct 4, 2015 at 16:53
• Good point! I added that in as well. Oct 4, 2015 at 16:54