This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.

This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $\Omega\gg\omega$. Our analysis is limited to the following parametric choices:
(i) $\omega_0^2$, $\beta$, $m_0$, $\lambda>0$ (type-1 single-well),
(ii) $\omega_0^2>0$, $\beta <0$, $2< m_0 <3$, $1< \lambda <2$ (type-2 single-well),
(iii) $\omega_0^2>0$, $\beta <0$, $0< m_0 <2$, $0<\lambda<1$ (type-3 single-well).
The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $\lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.

Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.

Vyacheslav Zigmundovich Grines, Doctor of Physics and Mathematics, an outstanding mathematician, an ordinary professor at the National Research University “Higher School of Economics”, died on July 2, 2023, at the age of 76.

The far-reaching consequences of ecological interactions in the dynamics of biological communities remain an intriguing subject. For decades, competition has been a cornerstone in ecological processes, but mounting evidence shows that cooperation does also contribute to the structure of biological communities. Here, we propose a simple deterministic model for the study of the effects of facilitation and competition in the dynamics of such systems. The simultaneous inclusion of both effects produces rich dynamics and captures the context dependence observed in the formation of ecological communities. Our findings reproduce relevant aspects in plant succession and highlight the role of facilitation mechanisms in species coexistence and conservation efforts.

Motion of a particle modeling a spacecraft with a solar sail along a handrail joining two heliocentric space stations is considered under the assumption that the sail is a perfect reflecting plane that can be located at any angle with respect to the direction of solar rays, the particle does not leave the plane of the orbit of the stations, the handrail is a tether that realizes an ideal unilateral constraint whose boundary is some ellipse, and the particle motion is sufficiently fast with respect to the orbital motion of the stations to neglect noninertiality of the orbital frame of reference. The equations of particle motion are written in dimensionless form without parameters, and the existence of an energy integral for the case of the sail orientation depending only on the spacecraft location is established. This integral is used for complete integration of the equations of motion for the particle relocations along the constraint boundary. The optimal length of the tether for the fastest relocation of a particle between the most remote points of the constraint boundary is computed for the case of the sail being orthogonal to the solar rays throughout the motion. Such a relocation time is computed in dimensionless form and for some real and hypothetical situations. A set of pairs of points in the constraint boundary between which relocation along the constraint boundary with zero initial and final velocities and with the invariably oriented sail is possible is constructed depending on the eccentricity of the ellipse. The result is presented as several plots that illustrate the evolution of the pairs’ regions as the eccentricity of the ellipse changes.

The possibility of keeping a spacecraft with a solar sail near an unstable triangular libration point of a minor planet or a binary asteroid is studied under the assumption that only the gravitation and the solar radiation influence the spacecraft motion. The case where the solar sail orientation remains unchanged with respect to the frame of reference of the heliocentric orbit of the asteroid mass center is considered. This means that the angle between the solar sail normal and ecliptic, as well as the angle between this normal and the solar rays at the current point, does not change during the motion. The spacecraft equations of motion are deduced under assumptions of V.V. Beletsky’s generalized restricted circular problem of three bodies, but taking into account the Sun radiation. The existence of a manifold of initial conditions for which it is possible to choose the normal direction that guarantees the spacecraft bounded motion near the libration point is established. Moreover, the dimension of this manifold coincides with that of the phase space of the problem at which the libration point belongs to the manifold boundary. In addition, some proposals for stabilization of the spacecraft motions are formulated for trajectories beginning in the manifold.

In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping a 3-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a two-locus system. We find the set of all (a continuum set of) fixed points and show that each fixed point is nonhyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point.

The $\varphi^4$ theory is widely used in many areas of physics, from cosmology and elementary particle physics to biophysics and condensed matter theory. Topological defects, or kinks, in this theory describe stable, solitary wave excitations. In practice, these excitations, as they propagate, necessarily interact with impurities or imperfections in the on-site potential. In this work, we focus on the effect of the length and strength of a rectangular impurity on the kink dynamics. It is found that the interaction of a kink with an extended impurity is qualitatively similar to the interaction with a well-studied point impurity described by the delta function, but significant quantitative differences are observed. The interaction of kinks with an extended impurity described by a rectangular function is studied numerically. All possible scenarios of kink dynamics are determined and described, taking into account resonance effects. The inelastic interaction of the kink with the repulsive impurity arises only at high initial kink velocities. The dependencies of the critical and resonant velocities of the kink on the impurity parameters are found. It is shown that the critical velocity of the repulsive impurity passage is proportional to the square root of the barrier area, as in the case of the sine-Gordon equation with an impurity. It is shown that the resonant interaction in the $\varphi^4$ model with an attracting extended impurity, as well as for the case of a point impurity, in contrast to the case of the sine-Gordon equation, is due to the fact that the kink interacts not only with the impurity mode, but also with the kink’s internal mode. It is found that the dependence of the kink final velocity on the initial one has a large number of resonant windows.

This article is concerned with investigating the nonlinear dynamics of the cylindrical resonator of a wave solid-state gyroscope. The nonlinearity of oscillations caused by the nonlinear properties of electrostatic control sensors is considered. This nonlinearity is derived by taking into account the finite ratio of resonator flexure to the small gap of electrostatic control sensors. The equations of the electromechanical system that in interconnected form describe the nonlinear mechanical oscillations of the gyroscope resonator and electrical oscillations in the control circuit are derived. The resulting differential equations belong to the class of Tikhonov systems, since the equation of electrical processes in the control circuit is singularly perturbed. By taking into account the low electrical resistance of the oscillation control circuit, which determines a small parameter at the derivative in the singularly perturbed equation of electrical processes, the nonlinear oscillations of the wave solid-state gyroscope resonator are studied. The small parameter method is used to obtain a mathematical model of the resonator dynamics, which jointly takes into account the nonlinearity of the resonator oscillations and the electrical resistance of the oscillation control circuit. A special method is proposed to reduce the nonlinear equations of the resonator dynamics to the standard form of the system of differential equations for averaging and the equations of the dynamics of the wave solid-state gyroscope resonator are averaged. It is shown that, in the case of nonlinear oscillations, consideration of the electrical resistance of the oscillation control circuit does not affect the angular velocity of the gyroscope drift, but causes slight dissipation of the oscillations, which also leads to an insignificant correction of the resonant frequency.