The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency regimes. A model is proposed in the form of a network of five globally coupled oscillators, characterized by varying degree of excitement of individual oscillators. Illustrations are given for the birth of the tori of increasing dimension by successive quasi-periodic Hopf bifurcation.

Nephrons (functional units of the kidney) may be described by means of the system of order differential equations. This provides an opportunity to describe dynamics of both the individual and coupled nephrons by using the theory of dynamical systems and the bifurcation theory. Considering a model of a pair of vascular coupled nephrons the present paper examines the effect that the non-identity of nephrons, i. e. non-identity of peak-to-peak variations in their arteriolar radii in autonomous state, has on the behavior of the coupled system. We investigate the appearance possibility of so-called broadband synchronization region, where the stronger nephron starts to suppress the autonomous oscillations of the weaker nephron. We investigate also the appearance possibility of the regime of total oscillator death, where oscillations of both nephrons are abolished.

Phenomenon of coherence resonance and external synchronization of noise-induced stochastic oscillations in hard excitation oscillator are studied by means of natural experiments. Regions of synchronization on parameter plane are constructed. Experiments on synchronization in hard excitation oscillator without noise are carried out.

Forced synchronization of a simple model of a three-mode self-oscillator which demonstrates appearance of self-modulation is considered. External driving at the fundamental frequency as well as at the self-modulation satellite frequency is considered. Structures of synchronization domains on the driving frequency — driving amplitude parameter plane and mechanisms of transition to the synchronous regime are investigated for the cases of single-mode and selfmodulated operation of the free-running oscillator.

A particle steady motions in vicinity of dynamically symmetric precessing rigid body are studied in assumption that the body gravitational field is modeled as gravitational field of two centers being on imaginary distance. Such particle motion equations are a variant of motion equations of the Generalized Restricted Circular Problem of Three Bodies (GRCP3B). The number of Coplanar Libration Points, i.e. the particle equilibria in the plane passing through the body axis of dynamical symmetry and through the axis of precession are established. (This number is odd and can be equal to 5, 7 or 9). CLPs evolution are studied at changing values of the considered system parameters. Moreover, two Triangular Libration Points, i. e. the particle equilibria in the axis crossing the body mass center orthogonally to axes of precession and dynamical symmetry are found.

A highly restricted problem of a relative equilibrium for two bodies is considered. They are put into a newtonian gravitational field and tied with inextensible massless fiber. One body is exposed to an air resistance. Stability of relative equilibria is under investigation.

A new integrable system describing the rolling of a rigid body with a spherical cavity over a spherical base is considered. Previously the authors found the separation of variables for this system at the zero level of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.