Shilnikov attractors in three-dimensional orientation-reversing maps

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing and orientation-preserving cases. We consider the three-dimensional Mirá map $x¯=y,y¯=z, and z¯=Bx+Cy+Az−y2$ with the negative Jacobian (⁠$B<0$⁠) as a basic model demonstrating various types of Shilnikov attractors. We show that depending on values of parameters $A,B$⁠, and $C$⁠, such attractors can be of three possible types: hyperchaotic (with two positive and one negative Lyapunov exponent), flow-like (with one positive, one very close to zero, and one negative Lyapunov exponent), and strongly dissipative (with one positive and two negative Lyapunov exponents). We study scenarios of the formation of such attractors in one-parameter families.