We consider a type of system that has a smooth invariant sub-manifold N in which a chaotic attractor A exists and for which more than one attractor exists in the full phase space. In this case, basins of all or some of attractors distinct from A can be locally intertwined in the neighborhood of A’s basin, which is restricted to N, not only for the region L⊥ 0 of parameters, where L⊥ is the normal Lyapunov exponent on the attractor A. An example of this case in 3-dimensional maps is considered. Further, a new type of crisis — intertwining crisis — of a chaotic attractor whose basin is locally intertwined is discussed and illustrated. In the last decade, two striking types of dynamical behavior — on-off intermittency 1) and riddled basins 2) — have received much attention and have been extensively studied.These two phenomena can appear in systems that possess chaotic dynamics in a smooth invariant manifold N of lower dimension than that of the full phase space.These two types of dynamics have been understood as a common type of local bifurcation, which is called a blowout bifurcation. 3) On-off intermittency is a phenomenon in which an orbit starting in the full phase space spends a long time very near the invariant manifold N and experiences intermittent bursting.In such a situation, typical orbits on A possess a small positive normal Lyapunov exponent L⊥, which is calculated for perturbations transverse to N .It is known that this intermittency differs from the type of Manneville and Pomeau 4) in some statistical properties. 5) On the other hand, basin riddling can occur only for parameter regions in which L⊥ is negative.In this case the basin of attraction for the chaotic attractor A [β(A)] has a strictly positive Lebesgue measure set in the full phase space, and any disk of any given point in β(A) includes pieces of another attractor’s basins.In this way, the basin β(A) is riddled with holes of another attractor’s basins. In this work, we discuss that dynamical behavior which differs from the two types described above can occur, and we consider an example of a 3-dimensional map.The new dynamics, which we refer to as locally intertwinedbasins, are analogous in some respects to riddled basins.But this new basin structure can occur in both regions L⊥ > 0 and L⊥ < 0 of parameters. In order to discuss the phenomenon in question, let us briefly review riddled basins.In the region L⊥ < 0 of parameters, in general, a nonzero measure set of points in the full phase space is attracted to A.When the basin β(A) is riddled, the system must have at least one attractor, B, apart from N .In this case, the heteroclinic orbit, U , from some periodic orbit in A to the attractor B exists, and the basin of B [β(B)] clings to U .When N is a one-dimensional smooth manifold, the dynamics in N are non-invertible, and pre-images of any periodic orbit are dense on the A’s basin [βN (A)], which is restricted in N .This structure leads β(A )t o be