Volume 23, Number 2
Volume 23, Number 2, 2018
Hillebrand M., PatersonJones G., Kalosakas G., Skokos C.
Abstract
In modeling DNA chains, the number of alternations between Adenine–Thymine (AT) and Guanine–Cytosine (GC) base pairs can be considered as a measure of the heterogeneity of the chain, which in turn could affect its dynamics. A probability distribution function of the number of these alternations is derived for circular or periodic DNA. Since there are several symmetries to account for in the periodic chain, necklace counting methods are used. In particular, Pólya’s Enumeration Theorem is extended for the case of a group action that preserves partitioned necklaces. This, along with the treatment of generating functions as formal power series, allows for the direct calculation of the number of possible necklaces with a given number of AT base pairs, GC base pairs and alternations. The theoretically obtained probability distribution functions of the number of alternations are accurately reproduced by Monte Carlo simulations and fitted by Gaussians. The effect of the number of base pairs on the characteristics of these distributions is also discussed, as well as the effect of the ratios of the numbers of AT and GC base pairs.

Kudryashov N. A.
Abstract
The standard FitzHugh–Nagumo model for description of impulse from one neuron to another is considered. The system of equations is transformed to a nonlinear secondorder ordinary differential equation. It is shown that the differential equation does not pass the Painlevé test in the general case and the general solution of this equation does not exist. The simplest solutions of the system of equations are found. The secondorder differential equation is transformed to another asymptotic equation with the general solution expressed via the Jacobi elliptic function. This transformation allows us to obtain the asymptotic solutions of the FitzHugh–Nagumo model. The perturbed FitzHugh–Nagumo model is studied as well. Taking into account the simplest equation method, the exact solutions of the perturbed system of equations are found. The asymptotic solutions of the perturbed model are presented too. The application of the exact solutions for construction of the neural networks is discussed.

Anastassiou S., Bountis A., Bäcker A.
Abstract
We investigate different aspects of chaotic dynamics in Hénon maps of dimension higher than 2. First, we review recent results on the existence of homoclinic points in 2d and 4d such maps, by demonstrating how they can be located with great accuracy using the parametrization method. Then we turn our attention to perturbations of Hénon maps by an angle variable that are defined on the solid torus, and prove the existence of uniformly hyperbolic solenoid attractors for an open set of parameters.We thus argue that higherdimensional Hénon maps exhibit a rich variety of chaotic behavior that deserves to be further studied in a systematic way.

Kuznetsov S. P.
Abstract
The main goal of the article is to suggest a twodimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energyconserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.

Hu S., Santoprete M.
Abstract
In this paper, we study a nonholonomic mechanical system, namely, the Suslov problem with the Clebsch–Tisserand potential. We analyze the topology of the level sets defined by the integrals in two ways: using an explicit construction and as a consequence of the Poincaré–Hopf theorem. We describe the flow on such manifolds.

Broer H. W., Hanßmann H., Wagener F. O.
Abstract
Nearresonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasiperiodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasiperiodic tori. The parameter values for which the Diophantine conditions are not fulfilled form socalled resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasiperiodicity, for larger parameter ranges. For a 1dimensional parameter space this allows to close almost all resonance gaps.
