  A triangulated category $\mathcal{C}$ with a canonical Bott's isomorphism
$[2]\xrightarrow{\sim}id$ is called a cyclic category in this paper. We give a
new notion of stability conditions on a $k$-linear Krull-Schmidt cyclic
category. Given such a stability condition $\sigma$, we can assign a Maslov
index to each basic loop in such a category. If all Maslov indexes vanish, we
get $\mathcal{C}',\sigma'$ as the $\mathbb{Z}$-lifts of $\mathcal{C},\sigma$
respectively such that $\mathcal{C}'$ is a $\mathbb{Z}$-graded triangulated
category and $\sigma'$ is a Bridgeland stability condition on $\mathcal{C}'$.
Moreover, we showed that there is an isomorphism
$$Stab^{0,e}(\mathcal{C})\xrightarrow{\simeq} BStab(\mathcal{C}')$$ where
$Stab^{0,e}(\mathcal{C})$ denotes the equivalence classes of stability
conditions which are deformation equivalent to $\sigma$, and
$BStab(\mathcal{C}')$ denotes the space of Bridgeland stability conditions on
$\mathcal{C}'$.
  We provide examples of stability conditions on a simple cyclic category. We
also discuss some interesting phenomena in these examples, such as the
chirality symmetry breaking phenomenon and nontrivial monodromy. The chirality
symmetry breaking phenomenon involves stability conditions which can not be
lifted to Bridgeland stability conditions.
