## 1 Introduction  
Nonlinear optics induced by strong field has become a study hotspot, such as second or high harmonic [1; 2; 3; 4; 5], four-wave mixing [6], optical solitons [7; 8; 9; 10], and optical bistability (OB) [11; 12; 13; 14]. OB is one of the significant nonlinear phenomena, in which the transmission characteristics of an optical system depend on the input light intensity [11]. Of which, an input intensity corresponds to two stable resonant outputs. OB has many great application values in optical transistors [12], optical switches [13], and photomemories [14]. How to design and find appropriate structures or materials to reduce the thresholds of OB and the interval of bistability thresholds are the key problems that are to be considered at present [15; 16; 17]. With reference to the lattice structure of solid, people put forward the concept of photonic crystals (PCs), which provides a new approach for solving the above two problems. At the bandgap edge of PCs, the optical field localization at the defect in PCs can effectively reduce the bistability threshold [18; 19]. Otherwise, graphene, which is viewed as a developing two-dimensional material, has many unique optical, mechanical, and electrical properties [20; 21; 22; 23]. In the terahertz band, graphene not only possesses fast speed optical response and tunable conductivity, but it also owns significant nonlinear effects [24; 25], which also affords a golden opportunity for the exploration of OB. In recent studies, synthetic PCs have been widely utilized in manipulating photons to realize the miniaturization and integration of photonic devices [26; 27; 28; 29; 30]. By modulating the refractive indices of materials to form periodic distribution, it can achieve specific optical properties, such as the localization of light field [30; 31; 32], unidirectional transmission [33; 34], directional cloaking [35], and solitons [36; 37]. In the past, it mainly focused on the control of the real part for the index of refraction in periodic PCs, while, in recent years, the regulation of the imaginary part and aperiodic PCs have triggered the exploring enthusiasm of researchers [38; 39; 40]. A system that has loss or gain is non-Hermitian based on quantum mechanics. The eigenvalues of Hamiltonian for the Schrodinger equation generally have complex spectra. However, as the Hamiltonian satisfies parity-time (PT) symmetry, its eigenvalues are real [41]. The concept of non-Hermite can be extended to the regime of optics [42; 43], and then the refractive index of PT-symmetric system obeys \(n(\mathbf{r})=n\times(-\mathbf{r})\)[44; 45; 46; 47; 48]. PT-symmetric systems could induce many unique optical properties, including unidirectional invisibility [49], optical transparency [43; 50], power oscillations [51], and the lateral shift of reflected light beam [52]. Otherwise, anti-PT-symmetric systems that satisfy \(n(\mathbf{r})=-n\times(-\mathbf{r})\) can induce coherent perfect absorption laser points and unidirectional invisibility [53; 54]. In particular, the aperiodic photonic lattices with PT symmetry support defect cavity with a high Q-factor and the energy of light field can be greatly localized at the defect [8; 29]. The nonlinear refractive index of graphene is proportional to the local optical field intensity. The composite system of aperiodic PT-symmetric lattice dielectrics and graphene could therefore be used to realize low-threshold OB. In this study, we simulate the OB of graphene that is embedded in the PT-symmetric system composed of aperiodic photonic lattice dielectrics at terahertz frequencies. Two aperiodic Fibonacci sequence dielectrics constitute a resonant cavity to form a strong local electric field. Subsequently, we show the transverse distribution of the optical field in the resonant cavity. Next, the influences of the gain-loss factor on the Q-factor of resonant cavity and OB are explored. We furthermore demonstrate that the momentum relaxation time of electrons in graphene can regulate the bistability threshold.