On hyperbolic attractors in a modified complex Shimizu-Morioka system.

We present a modified complex-valued Shimizu-Morioka system with a uniformly hyperbolic attractor. We show that the numerically observed attractor in the Poincaré cross section expands three times in the angular direction and strongly contracts in the transversal directions, similar in structure to the Smale-Williams solenoid. This is the first example of a modification of a system with a genuine Lorenz attractor, but manifesting a uniformly hyperbolic attractor instead.

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators.

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration.

A "saddle-node" bifurcation scenario for birth or destruction of a Smale-Williams solenoid.

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale-Williams solenoid in stroboscopic Poincaré map of two alternately excited non-autonomous van der Pol oscillators. The transition occupies a narrow but finite parameter interval and progresses in such way that periodic orbits constituting a "skeleton" of the attractor undergo saddle-node bifurcation events involving partner orbits from the attractor and from a non-attracting invariant set, which forms together with its stable manifold a basin boundary of the attractor.