We present a new type of strange attractors generated by an odd-symmetric three-dimensional vector field with a saddle-focus having two homoclinic orbits at the origin. This type of attractors is intimately related to the double scroll. We present the mathematical properties which prove rigorously the chaotic nature of this strange attractor to be different from that of a Lorenz-type attractor or a quasiattractor. In particular, we proved that for certain nonempty intervals of parameters, our two-dimensional map has a strange attractor with no stable orbits. Unlike other known attractors, this strange attractor contains not only a Cantor set structure of hyperbolic points typical of horseshoe maps, but also unstable points (i.e., stable in reverse time). This implies that the points from the stable manifolds of the hyperbolic points must necessarily attract the unstable points.
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