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.

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

We consider reversible non-conservative perturbations of the conservative cubic Hénon maps H ± 3 : x = y, ȳ = −x + M 1 + M 2 y ± y 3 and study their influence on the 1:3 resonance, i.e. bifurcations of fixed points with eigenvalues e ±i2π/3 . It follows from [DM00], this resonance is degenerate for M 1 = 0, M 2 = −1 when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map H + 3 and elliptic orbits in the case of map H − 3 ), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map H + 3 and saddles with the Jacobians less than 1 and greater than 1 in the case of map H − 3 ). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric non-transversal homo-and heteroclinic cycles. We also generalize the results of [DM00] to the case of the p : q resonances with odd q and show that all of them are also degenerate for the maps H ± 3 with M 1 = 0.