Binary N-Step Markov Chain as an Exactly Solvable Model of Long-Range Correlated Systems

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In our model, the conditional probability that the i-th symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The model allows exact analytical treatment. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity transformation of the chain parameters is presented. The diffusion equation governing the distribution function of the L-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.Comment: LaTeX2e, 16 pages, 9 figure

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

Decomposition of conditional probability for high-order symbolic Markov chains

The main goal of the paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear memory function monomials of different orders (from zero up to the chain order). This allows us to construct artificial sequences by method of successive iterations taking into account at each step of iterations increasingly more high correlations among random elements. At weak correlations, the memory functions are uniquely expressed in terms of the high-order symbolic correlation functions. The proposed method fills up the gap between two approaches: the likelihood estimation and the additive Markov chains. The obtained results might be used for sequential approximation of artificial neural networks training.Comment: 9 pages, 3 figure

Entropy of finite random binary sequences with weak long-range correlations

We study the N-step binary stationary ergodic Markov chain and analyze its differential entropy. Supposing that the correlations are weak we express the conditional probability function of the chain through the pair correlation function and represent the entropy as a functional of the pair correlator. Since the model uses the two-point correlators instead of the block probability, it makes it possible to calculate the entropy of strings at much longer distances than using standard methods. A fluctuation contribution to the entropy due to finiteness of random chains is examined. This contribution can be of the same order as its regular part even at the relatively short lengths of subsequences. A self-similar structure of entropy with respect to the decimation transformations is revealed for some specific forms of the pair correlation function. Application of the theory to the DNA sequence of the R3 chromosome of Drosophila melanogaster is presented.Comment: 9 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1411.2761, 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

Rank distributions of words in additive many-step Markov chains and the Zipf law

The binary many-step Markov chain with the step-like memory function is considered as a model for the analysis of rank distributions of words in stochastic symbolic dynamical systems. We prove that the envelope curve for this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations. The Zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the Markov sequence. A self-similarity in the rank distribution with respect to the decimation procedure is observed.Comment: 4pages, 3 figure

Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems

We give an example of a simple mechanical system described by the generalized harmonic oscillator equation, which is a basic model in discussion of the adiabatic dynamics and geometric phase. This system is a linearized plane pendulum with the slowly varying mass and length of string and the suspension point moving at a slowly varying speed, the simplest system with broken $T$-invariance. The paradoxical character of the presented results is that the same Hamiltonian system, the generalized harmonic oscillator in our case, is canonically equivalent to two different systems: the usual plane mathematical pendulum and the damped harmonic oscillator. This once again supports the important mathematical conclusion, not widely accepted in physical community, of no difference between the dissipative and Hamiltonian 1D systems, which stems from the Sonin theorem that any Newtonian second order differential equation with a friction of general nature may be presented in the form of the Lagrange equation.Comment: 12 pages, 1 figu

Continuous stochastic processes with non-local memory

We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into expression for the higher-order transition probability function and stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with prescribed pair correlation function. As illustration, some examples of numerical simulation of the processes with non-local memory are presented.Comment: 7 pages, 2 figure

Memory Functions of the Additive Markov chains: Applications to Complex Dynamic Systems

A new approach to describing correlation properties of complex dynamic systems with long-range memory based on a concept of additive Markov chains (Phys. Rev. E 68, 061107 (2003)) is developed. An equation connecting a memory function of the chain and its correlation function is presented. This equation allows reconstructing the memory function using the correlation function of the system. Thus, we have elaborated a novel method to generate a sequence with prescribed correlation function. Effectiveness and robustness of the proposed method is demonstrated by simple model examples. Memory functions of concrete coarse-grained literary texts are found and their universal power-law behavior at long distances is revealed.Comment: 5 pages, 5 figures, changes of minor nature, 1 figure adde

Equivalence of Markov's Symbolic Sequences to Two-Sided Chains

A new object of the probability theory, two-sided chain of events (symbols), is introduced. A theory of multi-steps Markov chains with long-range memory, proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to establish the correspondence between these chains and two-sided ones. The Markov chain is proved to be statistically equivalent to the definite two-sided one and vice versa. The results obtained for the binary chains are generalized to the chains taking on the arbitrary number of states.Comment: 5 page