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The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied.The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion.It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the inteal parameters of trichotomous noise,thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise.The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.