  It is generally believed that a cosmological arrow of time must be associated
with entropy production. Indeed, in his seminal work on cyclic cosmology,
Tolman introduced a viscous fluid in order to make successive
expansion/contraction cycles larger than previous ones, thereby generating an
arrow of time. However, as we demonstrate in this letter, the production of
entropy is not the only means by which a cosmological arrow of time may emerge.
Remarkably, systems which are dissipationless may nevertheless demonstrate a
preferred direction of time provided they possess attractors. An example of a
system with well defined attractors is scalar-field driven cosmology. In this
case, for a wide class of potentials (especially those responsible for
inflation), the attractor equation of state during expansion can have the form
$p \simeq -\rho$, and during contraction $p \simeq \rho$. If the resulting
cosmology is cyclic, then the presence of cosmological hysteresis, $\oint p~dV
\neq 0$ during successive cycles, causes an arrow of time to emerge in a system
which is formally dissipationless. An important analogy is drawn between the
arrow of time in cyclic cosmology and an arrow of time in an $N$-body system of
gravitationally interacting particles. We find that, like the $N$-body system,
a cyclic universe can evolve from a single past into two futures with
oppositely directed arrows of time.
