  We present a model which displays Griffiths phase i.e. algebraic decay of
density with continuously varying exponent in the absorbing phase. In active
phase, the memory of initial conditions is lost with continuously varying
complex exponent in this model. This is 1-D model where fraction r of sites
obey rules leading to directed percolation (DP) class and the rest evolve
according to rules leading to compact directed percolation (CDP) class. For
infection probability $p < p_c$, the fraction of active sites $\rho(t) = 0$
asymptotically. For $p > p_c$, $\rho(infty) > 0$. At $p = p_c$, $\rho(t)$, the
survival probability from single seed and the average number of active sites
starting from single seed decay logarithmically. The local persistence
$P_l(\infty) > 0$ for $p < p_c$ and $P_l(\infty) = 0$ for $p > p_c$. For $p >
p_s$, local persistence $P_l(t)$ decays as a power law with continuously
varying exponents. The persistence exponent is clearly complex as $p\rightarrow
1$. The complex exponent implies logarithmic periodic oscillations in
persistence. The wavelength and the amplitude of the logarithmic periodic
oscillations increases with p. We note that underlying lattice or disorder does
not have self-similar structure.
