  We explore and experimentally demonstrate the phenomena of amplitude death
(AD) and the corresponding transitions through synchronized states that lead to
AD in coupled {\it intrinsic time-delayed} hyperchaotic oscillators interacting
through mean-field diffusion. We identify a novel synchronization transition
scenario leading to AD, namely transitions among AD, generalized anticipatory
synchronization (GAS), complete synchronization (CS), and generalized lag
synchronization (GLS). This transition is mediated by variation of the
difference of {\it intrinsic time-delays} associated with the individual
systems, and has no analogue in non-delayed systems or coupled oscillators with
{\it coupling time-delay}. We further show that, for equal intrinsic
time-delays, increasing coupling strength results in a transition from the
unsynchronized state to AD state via in-phase (complete) synchronized states.
Using Krasovskii--Lyapunov theory, we derive the stability conditions that
predict the parametric region of occurrence of GAS, GLS, and CS; also, using a
linear stability analysis we derive the condition of occurrence of AD. We use
the error function of proper synchronization manifold and a modified form of
the similarity function to provide the quantitative support to GLS and GAS. We
demonstrate all the scenarios in an electronic circuit experiment; the
experimental time-series, phase-plane plots, and generalized autocorrelation
function computed from the experimental time series data are used to confirm
the occurrence of all the phenomena in the coupled oscillators.
