For autonomous systems why can't phase trajectories intersect each other? 
According to what I have learned, in an autonomous system the force acting on the particle does not depend on time. In the simplest case we can suppose that force depends only on position coordinate $q$. However, I can't understand why phase trajectory can't cut itself in such a situation (like in the diagram below). Can't $q(t+\Delta t)=q(t)$ and $v(t)=v(t+\Delta t)$ ?
For example in simple harmonic motion where $F=-kq$ the phase trajectory should cut itself because position and velocity do become equal to what they were at some previous time. Where am I going wrong?
 A: First of all, the coordinates in phase space are the generalized coordinates $q$ and the generalized momenta $p$, not the generalized velocities $\dot{q}$.  For Lagrangians with kinetic terms more complicated than $(1/2) m \dot{q}^2$, the two are not necessarily proportional.
But to answer your main question: phase-space trajectories certainly can intersect themselves.  As you said, a simple harmonic oscillator just wraps around the same closed curve in phase space over and over again.  But a trajectory can't cross itself - that is, you can't have two different curves that meet at a point and then go off in different directions, as you have in your diagram.  That's because for an autonomous system, a system's time evolution is determined entirely by its current location in phase space.  So if you're a given point in phase space, the next place you go is uniquely determined.  Two trajectories that meet at the same point in phase space are completely physically identical at that point, so they can't evolve in two different ways from identical initial conditions.
Another way of seeing this is that for an autonomous system, the time evolution can be visualized by a single-valued vector field on the phase space.  So if you're at a point in the phase space, you can only go in the direction pointed to by the vector field at that point.
