On discrete Lorenz-like attractors in three-dimensional maps with axial symmetry.

We describe a class of three-dimensional maps with axial symmetry {x→-x,y→-y,z→z} and the constant Jacobian. We study bifurcations and chaotic dynamics in quadratic maps from this class and show that these maps can possess discrete Lorenz-like attractors of various types. We give a description of bifurcation scenarios leading to such attractors and show examples of their implementation in our maps.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane.

Doubling of invariant curves and chaos in three-dimensional diffeomorphisms.

This paper gives a review of doubling bifurcations of closed invariant curves. We also discuss the role of the curve-doubling bifurcations in the formation of chaotic dynamics. In particular, we study scenarios of the emergence of discrete Lorenz and Shilnikov attractors in three-dimensional Hénon maps.

On discrete Lorenz-like attractors.

We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors. We show that such robustly chaotic (pseudohyperbolic) attractors can appear as a result of universal bifurcation scenarios, for which we give a phenomenological description and demonstrate certain examples of their implementation in one-parameter families of three-dimensional Hénon-like maps. We pay special attention to such scenarios that can lead to period-2 Lorenz-like attractors.

Scenarios of hyperchaos occurrence in 4D Rössler system.

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a one-dimensional unstable invariant manifold and Neimark-Sacker bifurcations of stable cycles.

On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps.

We study the 1:4 resonance for the conservative cubic Hénon maps C with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues ±i and for 4-periodic orbits. While for C, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map C has a different type of degeneracy because the resonant term can vanish.

Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits.

Recent results describing non-trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather non-trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown.