This paper implements a multiplier approach to exhibit conservation laws in magneto-optic waveguides that maintain anti-cubic as well as generalized anti-cubic forms of the nonlinear refractive index. Three conservation laws are retrieved for each form of nonlinearity. They are power, linear momentum and Hamiltonian. The conserved quantities are computed from their respective densities.

We consider the 2-body problem in the sphere \(\mathbb{S}^{2}\). This problem is modeled by a Hamiltonian system with \(4\) degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of \(2\) degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series

We consider the planar three-body problem with generalized potentials. Some nonintegrability results for these systems have been obtained by analyzing the variational equations along homothetic solutions. But we cannot apply it to several exceptional cases. For example, in the case of inverse-square potentials, the variational equations along homothetic solutions are solvable. We obtain sufficient

In this paper, we consider regular topological flows on closed \(n\)-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles interacting via a cotangent potential, we show there are two families of relative equilibria, called

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e. g., a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities

The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression

A study is made of the stability of triangular libration points in the nearly-circular restricted three-body problem in the spatial case. The problem of stability for most (in the sense of Lebesgue measure) initial conditions in the planar case has been investigated earlier. In the spatial case, an identical resonance takes place: for all values of the parameters of the problem the period of Keplerian

Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are considered. Algorithms for constructing a Lax pair for equations of these hierarchies are presented. Lax pairs for ordinary differential equations of the fifth, seventh and eleventh orders corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given. The Lax pairs allow us to solve these equations

We study a linear cocycle over the irrational rotation \(\sigma_{\omega}(x)=x+\omega\) of the circle \(\mathbb{T}^{1}\). It is supposed that the cocycle is generated by a \(C^{2}\)-map \(A_{\varepsilon}:\mathbb{T}^{1}\to SL(2,\mathbb{R})\) which depends on a small parameter \(\varepsilon\ll 1\) and has the form of the Poincaré map corresponding to a singularly perturbed Hill equation with quasi-periodic

Let \(M\) be a closed manifold and \(L\) an exact magnetic Lagrangian. In this paper we prove that there exists a residual set \(\mathcal{G}\) of \(H^{1}\left(M;\mathbb{R}\right)\) such that the property $${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$ with \({\widetilde{\mathcal{M}}}\left(c\right)\)

In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the figure-eight choreography, and explore some symmetric periodic orbits of this system which turns out to be nonautonomous. We use reversing symmetries to study both theoretically and numerically a certain type of symmetric periodic orbits

We analyse the patterns of the current epidemic evolution in various countries with the help of a simple SIR model. We consider two main effects: climate induced seasonality and recruitment. The latter is introduced as a way to palliate for the absence of a spatial component in the SIR model. In our approach we mimic the spatial evolution of the epidemic through a gradual introduction of susceptible

For the curved \(n\)-body problem, we show that the set of ordinary central configurations is away from singular configurations in \(\mathbb{H}^{3}\) with positive momentum of inertia, and away from a subset of singular configurations in \(\mathbb{S}^{3}\). We also show that each of the \(n!/2\) geodesic ordinary central configurations for \(n\) masses has Morse index \(n-2\). Then we get a direct

A dynamical system with a cosymmetry is considered. V. I. Yudovich showed that a noncosymmetric equilibrium of such a system under the conditions of the general position is a member of a one-parameter family. In this paper, it is assumed that the equilibrium is cosymmetric, and the linearization matrix of the cosymmetry is nondegenerate. It is shown that, in the case of an odd-dimensional dynamical

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two-degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical

The presence of higher-index saddles on a multidimensional potential energy surface is usually assumed to be of little significance in chemical reaction dynamics. Such a viewpoint requires careful reconsideration, thanks to elegant experiments and novel theoretical approaches that have come about in recent years. In this work, we perform a detailed classical and quantum dynamical study of a model

A simple proof and a detailed analysis of the nonergodicity for multidimensional harmonic oscillator systems with the Nosé – Hoover-type thermostat are presented. The origin of the nonergodicity is the symmetries in the multidimensional target physical system, and it differs from that in the Nosé – Hoover thermostat with the one-dimensional harmonic oscillator. A new and simple deterministic method

The role of second-order saddle in the isomerization dynamics was investigated by considering the \(E-Z\) isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding

Previous work has shown that by using the path integral representation of quantum mechanics and by mapping discrete electronic states to continuous Cartesian variables, it is possible to derive an exact quantum “mapping variable” ring-polymer (MV-RP) Hamiltonian. The classical molecular dynamics generated by this MV-RP Hamiltonian can be used to calculate equilibrium properties of multi-level quantum

Real-life epidemic situations are modeled using systems of differential equations (DEs) by considering deterministic parameters. However, in reality, the transmission parameters involved in such models experience a lot of variations and it is not possible to compute them exactly. In this paper, we apply B-spline wavelet-based generalized polynomial chaos (gPC) to analyze possible stochastic epidemic

We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge \(q\) . The pendulum oscillates above a fixed point with a charge \(Q.\) The dynamics is studied as a system in the small parameter \(\epsilon\) given by the amplitude of the suspension point. The system depends on two other parameters, \(\alpha\) and \(\beta,\) the first related to the frequency

This paper continues the discussion started in [ 10 ] concerning Arnold’s legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit “global” Arnold’s KAM theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure

We present an algorithm for constructing analytically approximate integrals of motion in simple time-periodic Hamiltonians of the form \(H=H_{0}+\varepsilon H_{i}\), where \(\varepsilon\) is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip

In the present paper, high-order nonlinear Schrödinger equations in non-Kerr law media with different laws of nonlinearities are studied. In this respect, after considering a complex envelope and distinguishing the real and imaginary portions of the models, describing the propagation of solitons through nonlinear optical fibers, their soliton solutions are obtained using the well-organized new Kudryashov

General constructions of differential equations with autotransforms are presented. Differential operators connecting a linear case of the Grad – Shafranov equation and the axisymmetric Helmholtz equation are found. Infinite families of exact solutions to both equations are derived.

The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanical-mathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of

We propose a simple deterministic, differential equation-based, SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 pandemic. The main hypothesis is the existence of two populations of susceptible persons, one which obeys confinement and for which the infection rate does not exceed 1, and a population

This paper studies the conditions under which the tippe top inverts in the presence of vibrations of the base along the vertical. A vibrational potential is constructed by averaging and it is shown that, when this potential is added to the system, the Jellett integral is preserved. This makes it possible to apply the modified Routh method and to find the effective potential to whose critical points

Asymptotically stable attractors are only a particular case of a large family of invariant compacta whose global topological structure is regular. We devote this paper to investigating the shape properties of this class of compacta, the nonsaddle sets. Stable attractors and unstable attractors having only internal explosions are examples of nonsaddle sets. The main aim of this paper is to generalize

This paper deals with the analysis of Hamiltonian Hopf bifurcations in three-degree-of-freedom systems, for which the frequencies of the linearization of the corresponding Hamiltonians are in \(\omega:3:6\) resonance (\(\omega=1\) or \(2\)). We obtain the truncated second-order normal form that is not integrable and expressed in terms of the invariants of the reduced phase space. The truncated first-order

In this paper, we study gradient-like flows without heteroclinic intersections on an \(n\)-sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies

One of the earliest formulations of dynamics of nonholonomic systems traces back to 1895 and is due to Chaplygin, who developed his analysis under the assumption that a certain number of the generalized coordinates do not occur either in the kinematic constraints or in the Lagrange function. A few years later Voronec derived equations of motion for nonholonomic systems removing the restrictions demanded

The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh – Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by

A nonlinear fourth-order differential equation with arbitrary refractive index for description of the pulse propagation in an optical fiber is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for solutions of the equation using the traveling wave reduction. We present a novel method for finding soliton solutions of nonlinear evolution

A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the in-circle or inscribed circle. In this paper we classify all planar four-body central configurations, where the four bodies are at the vertices of a tangential trapezoid.

This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint

The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed. We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the

It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations, meaning that the perturbed system still has an invariant torus with prescribed frequencies. However, the degenerate torus is sensitive to perturbations. In this paper, we prove the persistence of two classes of hyperbolic-type

Energy band rearrangement along a control parameter in isolated molecules is studied through axially symmetric Hamiltonians describing the coupling of two angular momenta \(\mathbf{S}\) and \(\mathbf{L}\) of fixed amplitude. We focus our attention on the case \(S=1\) which, albeit nongeneric, describes the global rearrangement of a system of energy bands between two well-defined limits corresponding

In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics. We formulate depth and flatness in the context of one- and two-degree-of-freedom (DOF) Hamiltonian normal form for the saddle-node bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, García-Garrido et al. [2]

This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically

In chemical reactions, trajectories typically turn from reactants to products when crossing a dividing surface close to the normally hyperbolic invariant manifold (NHIM) given by the intersection of the stable and unstable manifolds of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never leave this manifold when propagated forward or backward in time. This still holds for driven

Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate “fractal” (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors “forward in time” and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics

The problem of rolling a nonholonomic bundle of two bodies is considered: a spherical shell with a rigid body rotating along the axis of symmetry, on which rotors spinning relative to this body are fastened. This problem can be regarded as a distant generalization of the Chaplygin ball problem. The reduced system is studied by analyzing Poincaré maps constructed in Andoyer – Deprit variables. A classification

The stability of linear mechanical systems with finite numbers of degrees of freedom subjected to potential and non-conservative positional forces is considered. The positive semi-definiteness of the potential energy is assumed. Three new stability criteria which are in a simple way related to the properties of the system matrices are derived. These criteria improve previously obtained results of the

A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i. e., the pendulum never falls

This paper numerically addresses optical dromions and domain walls that are monitored by Kundu – Mukherjee – Naskar equation. The Kundu – Mukherjee – Naskar equation is considered because this model describes the propagation of soliton dynamics in optical fiber communication system. The scheme employed in this work is Laplace – Adomian decomposition type. The accuracy of the scheme is \(O(10^{-8})\)

The behavior of specific dispersive waves in a new \((3+1)\)-dimensional Hirota bilinear (3D-HB) equation is studied. A Bäcklund transformation and a Hirota bilinear form of the model are first extracted from the truncated Painlevé expansion. Through a series of mathematical analyses, it is then revealed that the new 3D-HB equation possesses a series of rational-type solutions. The interaction of lump-type

We prove that hyperbolic billiards constructed by Bussolari and Lenci are Bernoulli systems. These billiards cannot be studied by existing approaches to analysis of billiards that have some focusing boundary components, which require the diameter of the billiard table to be of the same order as the largest curvature radius along the focusing component. Our proof employs a local ergodic theorem which

We study the dynamics of a simple pendulum attached to the center of mass of a satellite in an elliptic orbit. We consider the case where the pendulum lies in the orbital plane of the satellite. We find two linearly stable equilibrium positions for the Hamiltonian system describing the problem and study their parametric stability by constructing the boundary curves of the stability/instability regions

We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painlevé property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems

We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to

Nonlinear differential equations associated with the second Painlevé equation are considered. Transformations for solutions of the singular manifold equation are presented. It is shown that rational solutions of the singular manifold equation are determined by means of the Yablonskii-Vorob’ev polynomials. It is demonstrated that rational solutions for some differential equations are also expressed

A generalization of the Suslov problem with changing parameters is considered. The physical interpretation is a Chaplygin sleigh moving on a sphere. The problem is reduced to the study of a two-dimensional system describing the evolution of the angular velocity of a body. The system without viscous friction and the system with viscous friction are considered. Poincaré maps are constructed, attractors

For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n

Intermittency is a route to chaos when transitions between laminar and chaotic dynamics occur. The main attribute of intermittency is the reinjection mechanism, described by the reinjection probability density (RPD), which maps trajectories of the system from the chaotic region into the laminar one. The main results on chaotic intermittency strongly depend on the RPD. Recently a generalized power law

We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised