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.


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

1 Answer 1


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.

  • $\begingroup$ Nor will you ever have two trajectories that cross each other. $\endgroup$
    – GEdgar
    Jan 28, 2016 at 14:41

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