This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech6t differentiable maps in Banach spaces.A criterion of chaos in-duced by a regular nondegenerate homoclinic orbit is established.Chaos of discrete dy-namica0 systems in the n-dimensiona0 rea0 space is also discussed.with two criteria de-rived for chaos Induced by nondegenerate snap-back repellers，one of which is a modified version of Marotto's theorem.In particular.a necessary and sufficient condition is obtained for an expanding fixed point Of a differentiable map in a general Banach space.and in an n-dimensiona0 real space.respectively.It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point In an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.