A new asymptotics exploiting the disparity of the orders of magnitude of Damköhler numbers is uncovered in complex kinetics. In contrast to the conventional activation energy asymptotics where the non-dimensional activation energy 0 is algebraically large, the new asymptotics here only requires 0 to be logarithmically large, and is therefore a more appropriate approach for the analysis of problems where the activation energy 0 is not very large. The two-step Zeldovich-Liñán model with a chain-branching and a chain-breaking step is chosen to demonstrate the use of this new asymptotics.