Pavel Kuptsov

A spin-transfer oscillator is a nanoscale device demonstrating self-sustained precession of its magnetization vector whose length is preserved. Thus, the phase space of this dynamical system is limited by a three-dimensional sphere. A generic oscillator is described by the Landau – Lifshitz – Gilbert – Slonczewski equation, and we consider a particular case of uniaxial symmetry when the equation yet experimentally relevant is reduced to a dramatically simple form. The established regime of a single oscillator is a purely sinusoidal limit cycle coinciding with a circle of sphere latitude (assuming that points where the symmetry axis passes through the sphere are the poles). On the limit cycle the governing equations become linear in two oscillating magnetization vector components orthogonal to the axis, while the third one along the axis remains constant. In this paper we analyze how this effective linearity manifests itself when two such oscillators are mutually coupled via their magnetic fields. Using the phase approximation approach, we reveal that the system can exhibit bistability between synchronized and nonsynchronized oscillations. For the synchronized one the Adler equation is derived, and the estimates for the boundaries of the bistability area are obtained. The twodimensional slices of the basins of attraction of the two coexisting solutions are considered. They are found to be embedded in each other, forming a series of parallel stripes. Charts of regimes and charts of Lyapunov exponents are computed numerically. Due to the effective linearity the overall structure of the charts is very simple; no higher-order synchronization tongues except the main one are observed.

Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also, we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well-known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite-time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite-time Lyapunov exponents, we demonstrate that average expanding and contracting properties specific to pseudohyperbolicity are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.