Do all dynamical systems have attractors? Do all dynamical systems have attractors?
Is there any chance that there are two or more absolutely the same sets of states in one attractor?
 A: Conservative systems are an entire class of dynamical systems that doesn’t have attractors. You might argue that no real system (except perhaps the entire universe) is truly conservative, but then most systems aren’t truly deterministic either. All dynamical systems you can write down are just an approximation of reality that hopefully provides insights.
For example, the frictionless double pendulum is conservative and thus has no attractor. While adding friction makes it more realistic, it is not relevant for many insights on its dynamics.
Non-conservative systems may not have attractors either. However, non-conservative systems without attractor have to have an escalating dynamics, which is somewhat limited in the insights it may provide, as no real dynamics can escalate indefinitely. Therefore they are usually not very interesting from a dynamical-systems point of view.
A: No.
One-dimensional example. As an example, $\dot{x} = x$ has no attractor, being the unstable equilibrium $\overline{x} = 0$ a repulsor (broadly speaking, it becomes an attractor if you reverse time).
Every trajectory starting out of the unstable equilibrium moves infinitely away from it,
$x(t) = x_0 e^t$,
being $x_0$ the initial condition.
"Two-dimensional" example - charges with the same sign. The dynamical system describing the evolution of a positive charge $q_1$ with mass $m_1$, repelled by another positive $q_0$ charge kept fixed in space reads
$m_1 \ddot{\mathbf{r}} = kq_0 q_1 \dfrac{\mathbf{r}}{|\mathbf{r}|^3}$.
Projecting along the radial direction (assuming that the mass has no azimutal initial velocity), we get the second-order scalar equation
$\ddot{r} = \dfrac{c}{r^2}$,
being $c = \frac{k q_0 q_1}{m_1}$. This equation has no equilibrium, and a repulsor coinciding with the fixed charge.
A: A very, very simple example with no attractor or even repulsor: Phase space $\mathbb{R}$, time space $\mathbb{R}$:
$$\Phi^t(x) = x + at$$
where $a > 0$ is some constant. No point is fixed, much less attracting or repulsing.
A: No not all dynamical systems have attractors. An attractor is a subset of the state space of a dynamical system that attracts nearby states and determines their long-term behavior.
