  We study the dynamics of circular active particles (AP) on a two dimensional
periodic undulated surface. Each particle has an internal energy mechanism
which is modeled by an active friction force and it is controlled by an
activity parameter $v_0$. It acts as negative friction if the speed of the
particle is smaller than $v_0$ and normal friction otherwise. Surface
undulation is modeled by the periodic undulation of fixed amplitude and
wavelength and is measured in terms of a dimensionless ratio of amplitude and
wavelength, $\bar{h}$. The dynamics of the particle is studied for different
activities, $v_0$ and surface undulations (SU), $\bar{h}$. Three types of
particle dynamics are observed on varying activity and SU. For small $v_0
\lesssim 0.1$ and $\bar{h}$ $\gtrsim 0.8$, particles remain confined in a
surface minimum, for moderate $v_0 \lesssim \bar{h}$, dynamics of particle
shows an intermediate subdiffusion to late time diffusion and for large $v_0
\gtrsim \bar{h}$, it shows initial superdiffusion to late time diffusion. For
all $v_0$'s and $\bar{h} \lesssim 0.2$, the dynamics of particle, satisfies the
Green-Kubo relation between the effective diffusivity and velocity
auto-correlation function. Systematic deviation is found on increasing
$\bar{h}$. Hence, an effective equilibrium can be established for a range of
system parameters in this nonequilibirum system.
