  Reaction-diffusion processes on networked systems have received mounting
attention in the past two decades, and the corresponding network dynamics
theories have been continuously enriched with the advancement of network
science. Recently, time-varying feature and many-body interactions have been
discovered on various and numerous networks in real world like biological and
social systems, and the study of contemporary network science has gradually
moved away from the historically static network frameworks that are based on
pairwise interactions. We are aimed to propose a general and rudimentary
framework for Turing instability of reaction-diffusion processes on
higher-order temporal networks. Firstly, we define a brand Laplacian to depict
higher-order temporal diffusion behaviors on networks. Furthermore, the general
form of higher-order temporal reaction-diffusion systems with frequency of
oscillation is defined, and a time-independent and concise form is obtained by
equivalent substitution and method of averaging. Next, we discuss the two cases
of homogeneous and heterogeneous network systems and give the equivalent
conditions of Turing instability through linear stability analysis. Finally, in
the numerical simulation part, we verify and discuss the validity of the above
theoretical framework and study the effect of the frequency of oscillation of
higher-order temporal network on reaction-diffusion processes. Our study has
revealed that higher-order temporal reaction-diffusion, which takes into
account both time-varying feature and many-body interactions, can formulate
innovative and diversified patterns. Moreover, these are significantly
differences from patterns in continuous space and patterns on traditional
networks.
