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

We consider a family of simple flows in tori that display chaotic behavior in a wide sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed point which is of parabolic type. However, the dynamics returns infinitely many times near the fixed point due to quasi-periodicity. A preliminary example is given for maps introduced in a paper containing many examples of

This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant

The general Jacobi symplectic variables generated by a combinatorial algorithm from the full binary tree T(N) are used to formulate some nonheliocentric gravitational N-body problems in perturbation form. The resulting uncoupled term HU for (N - 1) independent Keplerian motions and the perturbation term UP are both explicitly dependent on the partial ordering induced by the tree T(N). This leads to

This paper reports conservation laws for highly dispersive optical solitons in birefringent fibers. Three forms of nonlinearities are studied which are Kerr, polynomial and nonlocal laws. Power, linear momentum and Hamiltonian are conserved for these types of nonlinear refractive index.

The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found

A (multidimensional) spherical periscope is a system of two ideal mirrors that reflect every ray of light emanating from some point back to this point. A spherical periscope defines a local diffeomorphism of the space of rays through this point, and we describe such diffeomorphisms. We also solve a similar problem for (multidimensional) reversed periscopes, the systems of two mirrors that reverse the

The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational

This paper is concerned with the motion of an unbalanced dynamically symmetric sphere rolling without slipping on a plane in the presence of an external magnetic field. It is assumed that the sphere can consist completely or partially of dielectric, ferromagnetic, superconducting and crystalline materials. According to the existing phenomenological theory, the analysis of the sphere’s dynamics requires

This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincaré methods

We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with a strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all Casimirs linear in momenta on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation on the dual of the Lie algebra. On this basis we show that extremal controls are

We study here the asymptotic condition \(\dot E = - \mu {g_n}b_A^2 = 0\) for an eccentric rolling and sliding ellipsoid with axes of principal moments of inertia directed along geometric axes of the ellipsoid, a rigid body which we call here Jellett’s egg (JE). It is shown by using dynamic equations expressed in terms of Euler angles that the asymptotic condition is satisfied by stationary solutions

Slow-fast dynamics and resonant phenomena can be found in a wide range of physical systems, including problems of celestial mechanics, fluid mechanics, and charged particle dynamics. Important resonant effects that control transport in the phase space in such systems are resonant scatterings and trappings. For systems with weak diffusive scatterings the transport properties can be described with the

Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point.We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom

Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painlevé’s test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada-Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada-Kotera

We review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.

This paper considers the Radhakrishnan — Kundu — Laksmanan (RKL) equation to analyze dispersive nonlinear waves in polarization-preserving fibers. The Cauchy problem for this equation cannot be solved by the inverse scattering transform (IST) and we look for exact solutions of this equation using the traveling wave reduction. The Painlevé analysis for the traveling wave reduction of the RKL equation

We study asymptotic solution of the Cauchy problem for the linearized system of relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a space-like surface in the Minkowsky space. We prove that the solution can be decomposed into three modes, corresponding to different routsb of the equations of characteristics. One of these roots is twice degenerate and the there

We consider the roaming mechanism for chemical reactions under the nonholonomic constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian isokinetic thermostat applied to Chesnavich’s model for an ion-molecule reaction. Through an analysis of phase space structures we show that imposing the nonholonomic constraint does not prevent the system from exhibiting roaming

We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus

We investigate stability properties of a type of periodic solutions of the N-vortex problem on general domains Ω ⊂ ℝ2. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point a0 ∈ Ω of the Robin function associated to the Dirichlet Laplacian of Ω. Under a linear stability condition on the initial rotating configuration, which can

We consider the 2D Schrödinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system

We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj–Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent

We consider the possibility of using Dirac’s ideas of the deformation of Poisson brackets in nonholonomic mechanics. As an example, we analyze the composition of external forces that do no work and reaction forces of nonintegrable constraints in the model of a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is applied, the general mechanical energy, the invariant measure

We consider a Hamiltonian system depending on a parameter which slowly changes with rate ε ≪ 1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant

We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott–Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the

We consider the n-dimensional Chaplygin sphere under the assumption that the mass distribution of the sphere is axisymmetric. We prove that, for initial conditions whose angular momentum about the contact point is vertical, the dynamics is quasi-periodic. For n = 4 we perform the reduction by the associated SO(3) symmetry and show that the reduced system is integrable by the Euler-Jacobi theorem.

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials

The motion of a rigid body about a fixed point in a homogeneous gravitational field is investigated. The body is not dynamically symmetric and its center of gravity lies on the perpendicular, raised from the fixed point, to one of the circular sections of an ellipsoid of inertia. A body with such mass geometry may precess regularly about a nonvertical axis (Grioli’s precession). The problem of the

The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th birthday we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has a centre of mass offset

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries.

This paper addresses the problem of the rolling of a spherical shell with a frame rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire system is at the geometric center of the shell.For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists

The sum of elliptic integrals simultaneously determines orbits in the Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. The algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas the algebra

We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case when the sphere is 3-dimensional. As the 3-sphere is a group it acts on itself by left and right multiplication and these together generate the action of the SO(4) symmetry on the sphere. This gives rise to a notion of left and right momenta for the problem, and

We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t0 ∈ ℝ. Let X+, X− denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t >

This article is devoted to the results of phase topology research on a generalized mathematical model, which covers such two problems as the dynamics of two point vortices enclosed in a harmonic trap in a Bose – Einstein condensate and the dynamics of two point vortices bounded by a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-into-one and four-into-one tori bifurcations

This paper presents results of numerical statistical analysis of the effect of short-term localized noise of different intensity on the amplitude chimera lifetime in an ensemble of nonlocally coupled logistic maps in a chaotic regime. It is shown that a single and rather weak noise perturbation added only to the incoherence cluster of the amplitude chimera after its switching to the phase chimera mode

This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.We consider the two most widespread friction

We consider the motion of a 2π-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter e and other parameters and is autonomous at e = 0. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations

Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence

This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice

The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group G is Liouville integrable. We derive this property from the fact that the coadjoint orbits of G are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension.We also give normal forms for left-invariant