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This paper finds and investigates the application of single-exponential local decay pulse (SELDP) to suppress and induce chaos in a family of nonlinear oscillators subject to weak damping and external periodic excitations. Fourier series of SELDP defined on a period is derived and the approximate series is extended to that defined on the set of whole positive real numbers. To show the feasibility of the obtained results, we first give an effective design scheme of electrical circuit related to SELDP signal and this may be helpful in future implementations. We concentrate on this case in which the unforced system possesses two homoclinic orbits. In order to apply Melnikov's approach to make the underlying parameter conditions for suppressing and inducing chaos more clear, a generic numerical algorithm is proposed to compute complicated Melnikov functions. Two propositions, serving as designing the correct parameters in the SELDP function are also given. From our study, we find that chaos can be induced (suppressed) according to the corresponding Melnikov functions have (do not have) simple zeros. Our work can help to understand the underlying mechanisms of suppressing and inducing chaos. The simulation results show the effectiveness of our proposed approach.