I am confused about this question.
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!
The question essentially asks that given a solution to a differential equation and an initial condition, show that the trajectory in phase space is unique. That is, if you follow the curve (solution), you will never be confronted with the curve splitting into two curves. This should also be true when following the trajectory backwards.