Entropy of random symbolic high-order bilinear Markov chains

The main goal of this paper is to develop an estimate for the entropy of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We present here the detailed analytical study of the entropy for the high-order Markov chain in the bilinear approximation. The appendix contains a short comprehensive introduction into the subject of study.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1412.369

Integrable order parameter dynamics of globally coupled oscillators

We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of the corresponding nonlinear system on an attracting manifold. We present a complete classification of phase portraits and bifurcations, obtain explicit expressions for invariant manifolds (a limit cycle among them) and derive analytical solutions for arbitrary initial data and different regimes