Limitation of time-delay induced amplitude death

The present Letter provides the sufficient condition for avoiding amplitude death in two identical nonlinear systems coupled by a delay diffusive connection. This condition is that if a Jacobi matrix at a fixed point in an isolated system has an odd number of real positive eigenvalues, then the time-delay connection never induces amplitude death at the fixed point for any coupling parameters. Furthermore, these results are valid for globally-coupled systems and general types of coupling.