  Nonlinear systems possessing nonattracting chaotic sets, such as chaotic
saddles, embedded in their state space may oscillate chaotically for a
transient time before eventually transitioning into some stable attractor. We
show that these systems, when networked with nonlocal coupling in a ring, are
capable of forming chimera states, in which one subset of the units oscillates
periodically in a synchronized state forming the coherent domain, while the
complementary subset oscillates chaotically in the neighborhood of the chaotic
saddle constituting the incoherent domain. We find two distinct transient
chimera states distinguished by their abrupt or gradual termination. We analyze
the lifetime of both chimera states, unraveling their dependence on coupling
range and size. We find an optimal value for the coupling range yielding the
longest lifetime for the chimera states. Moreover, we implement transversal
stability analysis to demonstrate that the synchronized state is asymptotically
stable for network configurations studied here.
