We consider the dynamics of a ring of nonlocally coupled logistic maps when varying the coupling coefficient. We introduce the coupling function, which characterizes the impact of nonlocal neighbors and study its dynamics together with the dynamics of the whole ensemble. Conditions for the transition from complete chaotic synchronization to partial one are analyzed and the corresponding theoretical estimation of the bifurcation parameter $\sigma$ is given. Conditions for the appearance of phase and amplitude chimera states are also studied.

We present results of analytical and numerical investigation of the nonstationary planar dynamics of a string with uniformly distributed discrete masses without preliminary tension and taking into account the bending stiffness. Each mass is coupled to the ground by lateral springs without tension which have (effectively) a characteristic that is nonlinearizable in the case of planar motion. The most important limiting case corresponding to low-energy transversal motions is considered taking into account geometrical nonlinearity. Since such excitations are described by approximate equations where cubic elastic forces contribute the most, oscillations take place under conditions close to the acoustic vacuum. We obtain an adequate analytical description of resonant nonstationary processes in the system under consideration, which correspond to an intensive energy exchange between its parts (clusters) in the domain of low frequencies. Conditions of energy localization are given. The analytical results obtained are supported by computer numerical simulations. The system considered may be used as an energy sink of enhanced effectiveness.

We consider the Morris–Lecar neuron model with a parameter set corresponding to class 1 excitability. We study the effect of random disturbances on the model in the parametric zone where the only attractor of the deterministic system is a stable equilibrium. We show that under noise the stochastic generation of large amplitude oscillations occurs in the system. This phenomenon is confirmed by changes in distributions of random trajectories and interspike intervals. This effect is analyzed using the stochastic sensitivity function technique and the method of confidence domains. We suggest a criterion for the estimation of threshold values of noise intensity leading to the stochastic generation of oscillations.

The dynamics of a dense ensemble of interacting active Brownian particles is studied. Nonlinear negative friction is described in the sense of Rayleigh; particles are interconnected via Morse potential forces. Such a chain can be considered as an ensemble of interconnected Rayleigh oscillators.
The stationary modes (attractors) of chains with periodic boundary conditions looks like cnoidal waves. They are characterized by a uniform distribution of the density maxima of particles in the chain. However, when the chain starts with random initial conditions, a state of nonuniformly distribution of density maxima arises first. This state is metastable and the transition to a stable mode corresponds to a long transition process.
Characteristics of metastable states, regularities and probability of their occurrence and their lifetimes are studied by methods of computer simulation.

We discuss an algorithmic construction of the auto Bäcklund transformations of Hamilton–Jacobi equations and possible applications of this algorithm to finding new integrable systems with integrals of motion of higher order in momenta. We explicitly present Bäcklund transformations for two Hamiltonian systems on the plane separable in parabolic and elliptic coordinates.

The motion of a heavy point on the surface of a rotating sphere is considered. It is assumed that the rotation axis does not coincide with the vertical diameter of the sphere and the angular velocity of the sphere is constant. The Lagrange equations for this system are derived. Sets of relative equilibria are found and their dependence on the parameters of the system is studied in extreme cases when the magnitude of the angular velocity or the distance between the rotation axis and the center of the sphere is large. The results are represented in graphic form. The same graphic series are also numerically plotted in the general case.

In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.

In the present work, we prove the existence of fractional monodromy in a large class of compact Lagrangian fibrations of four-dimensional symplectic manifolds. These fibrations are considered in the neighbourhood of the singular fibre $\Lambda^0$, that has a single singular point corresponding to a nonlinear oscillator with frequencies in $1 : (−2)$ resonance. We compute the matrices of monodromy defined by going around the fibre $\Lambda^0$. For all fibrations in the class and for an appropriate choice of the basis in the one-dimensional homology group of the torus, these matrices are the same. The elements of the monodromy matrix are rational and there is a non-integer element among them. This work is a continuation of the analysis in [20, 21, 39] where the matrix of fractional monodromy was computed for most simple particular fibrations of the class.