  The amplitude-dependent frequency of the oscillations, termed
\emph{nonisochronicity}, is one of the essential characteristics of nonlinear
oscillators. In this paper, the dynamics of the Rossler oscillator in the
presence of nonisochronicity is examined. In particular, we explore the
appearance of a new fixed point and the emergence of a coexisting limit-cycle
and quasiperiodic attractors. We also describe the sequence of bifurcations
leading to synchronized, desynchronized attractors and oscillation death states
in the coupled Rossler oscillators as a function of the strength of
nonisochronicity and coupling parameters. Further, we characterize the
multistability of the coexisting attractors by plotting the basins of
attraction. Our results open up the possibilities of understanding the
emergence of coexisting attractors, and into a qualitative change of the
collective states in coupled nonlinear oscillators in the presence of
nonisochronicity.
