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In the catastrophe theory of nonlinear science,the intensity of water-flow and the coefficient of non-uniformsediment m are regarded as two bound variables, and the in-tensity of bed-load transport Φ as the state variable in the mo-tion of non-uniform sediment in cusp-catastrophe model.Based on the standard equation of the cusp-catastrophe theo-ry, the relation equation between the intensity of bed-loadtransport Φ and the intensity of water-flow has been derivedby used coordinate transform and topology transform. The e-quation of bed load transport rate was built on the cusp-catas-trophe theory of nonlinear science. The others are applied toverify this equation, that the results calculated by the cusp-ca-tastrophe equation agree well with the other equations. Thisindicates that the cusp-catastrophe equation is reasonable, and the results fully reflect the characteristics of threshold motionand transport of non-uniform sediment. The purpose of thispaper is to explore the incipient motion and transport laws ofnon-uniform sediment from the viewpoint of nonlinear science.