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## Planar Penning trap with combined resonance and top dynamics on quadratic algebra

Russian Journal of Mathematical Physics. 2015. Vol. 22. No. 4. P. 463-468.

We study dynamics of a charge in the planar Penning trap in which the direction of magnetic ﬁeld does not coincide with the trap axis. Under some combined resonance condition on the deviation angle and magnitudes of magnetic and electric ﬁelds, the trajectories of a charge are near-periodic. We describe the reduction to a top-like system with one degree of freedom on the space with quadratic Poisson brackets and study the stability of the equilibrium points of this system.

Keywords: resonanceрезонансequilibrium pointsалгебра симметрийусреднениеnanotrapssymmetry algebraнаноловушкиaveragingточки покоя

Publication based on the results of:

M.V. Karasev, E.M. Novikova, Theoretical and Mathematical Physics 2014 Vol. 179 No. 3 P. 729-746

We discuss the choice of physical parameters of a quantum Penning nanotrap under the action of a perturbing inhomogeneous Ioffe magnetic field and also the role of frequency resonance modes. We present a general scheme for constructing the asymptotic behavior of the eigenstates by the generalized geometric quantization method and obtain the reproducing measure in ...

Added: October 16, 2015

М.В. Карасев, Е.М. Новикова, Теоретическая и математическая физика 2014 Т. 179 № 3 С. 406-425

The choice of physical parameters of the quantum Penning nanotrop with a perturbing in homogeneous magnetic Ioffe field is discussed, as well as the role of the resonance frequency modes. The general scheme for constructing the asymptotics of eigenstates by themethods of generalized geometric quantization is presented. The reproducing measure in theintegral representation of eigenfunctions ...

Added: March 8, 2014

М.В. Карасев, Е.М. Новикова, Наноструктуры. Математическая физика и моделирование 2015 Т. 13 № 2 С. 55-92

We study the planar Penning traps in a resonance mode. The axial symmetry of the system is violated by deviation of the magnetic field from the trap axis at a small angle (the small parameter in the given model). The geometry of planar electrodes and their electric potentials are made consistent to reach a combined ...

Added: February 25, 2016

M.V. Karasev, E.M. Novikova, Russian Journal of Mathematical Physics 2013 Vol. 20 No. 3 P. 283-294

The Penning trap Hamiltonian (hyperbolic oscillator in a homogeneous magnetic field) is considered in the basic three-frequency resonance regime. We describe its non-Lie algebra of symmetries. By perturbing the homogeneous magnetic field, we discover that, for special directions of the perturbation, a secondary hyperbolic resonance appears in the trap. For corresponding secondary resonance algebra, we ...

Added: November 17, 2013

Alexander M. Krasnosel’skii, Discrete and Continuous Dynamical Systems - Series S 2013 Vol. 6 No. 4 P. 999-1016

Autonomous higher order differential equations with scalar nonlinearities, periodic with respect to the main phase variable under appropriate generic conditions, have an infinite sequence of isolated cycles with amplitudes growing to infinity and periods converging to some specific value T. ...

Added: February 11, 2013

Krasnosel'skii Alexander M., Discrete and Continuous Dynamical Systems 2013 Vol. 33 No. 1 P. 239-254

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations L(p)x=f(x)+b(t), p=d/dt, with 2pi-periodic forcing b and periodic f we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of 2pi-periodic solutions. ...

Added: February 11, 2013

Samovol V. S., Математические заметки 2012 Т. 92 № 5 С. 731-746

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. The reducibility of such systems to pseudonormal form is studied. ...

Added: December 13, 2012

Protasov V., Systems and Control Letters 2016 Vol. 90 P. 54-60

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon ...

Added: February 22, 2016

Ivanov A. I., Орлов О., Демин А. В. et al., М. : Слово, 2012

В работе использован подход, заключающийся в применении результатов теории функций комплексного переменного для нахождения внутренних причин колебательных процессов тонуса вегетативной нервной системы (ВНС) человека. Установлены отдельные причины возникновения колебаний тонуса ВНС, численно характеризующегося индексов Кредо. Найдено объяснение возникновения отдельных сильно выделяющихся значений индекса Кредо как резонансных явлений. Пособие предназначено для научных работников, физиологов и врачей, ...

Added: February 8, 2013

Pereskokov A., Математические заметки 2012 Т. 92 № 4 С. 583-596

The eigenvalue problem for the perturbed resonant oscillator is considered. A method for constructing asymptotic solutions near the boundaries of spectral clusters using a new integral representation is proposed. The problem of calculating the averaged values of differential operators on solutions near the cluster boundaries is studied. ...

Added: November 26, 2012

Karasev M., Novikova E., Vybornyi E., Mathematical notes 2016 Vol. 100 No. 6 P. 807-819

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under breaking its axial symmetry (inclination of the magnetic field) and turning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation ...

Added: October 22, 2016

Novikova E., Mathematical notes 2019 Vol. 106 No. 6 P. 940-956

The algebra of symmetries of the quantum three-frequency hyperbolic resonance oscillator is studied. It is shown that this algebra is determined by a nite set of generators with polynomial commutation relations. The irreducible representations of this algebra and the corresponding coherent states are constructed. ...

Added: October 28, 2019

Belenky A., Procedia Computer Science 2014 Vol. 31 P. 1150-1159

A part of a country’s electrical grid in which an electricity generator (which may consist of several base load power plants
and several peaking power plants) supplies electricity to a set of large customers of the grid, whereas the customers can a)
receive electricity from renewable sources of energy, b) store electricity in certain volumes, and c) ...

Added: September 29, 2014

Safonova D., Demina M.V., Kudryashov N., Regular and Chaotic Dynamics 2018 Vol. 23 No. 5 P. 569-579

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point ...

Added: October 26, 2018

Колпаков И. Ю., Вестник Пермского национального исследовательского политехнического университета. Прикладная математика и механика 2013 № 11 С. 32-47

In work conditions of solvability of a boundary value problem for equation with a perturbed argument are received. For receiving conditions of solvability the boundary value problem write down on the corresponding space in the form of one quasilinear operator equation to which the theorem of existence of the solution with a condition on area ...

Added: April 7, 2015

Samovol V. S., Доклады Академии наук 2012 Vol. 85 No. 1 P. 122-124

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth ...

Added: November 27, 2012

Семенякин Н. С., Modern Physics Letters A 2014 Vol. 29 No. 35

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev{V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all ...

Added: April 4, 2018

Novikova E., Наноструктуры. Математическая физика и моделирование 2017 Т. 16 № 2 С. 69-88

The scale of physical parameters determining the Penning planar trap with rectangular annular electrode is analyzed and a unifi ed relation between these parameters is obtained. This relation leads to the resonance oscillator in the leading part of the Hamiltonian for the electron in the trap. In the regime of basic hyperbolic resonance, an explicit ...

Added: January 30, 2018

Novikova E., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 3 P. 406-410

The quantum averaging method is applied to the Hamiltonian of the multifrequency resonance harmonic oscillator
perturbed by a differential operator with polynomial coefficients. The twisted product is used to transfer the averaging procedure to the space of graduated algebra of symbols. As a result, the averaged Hamiltonian is expressed in terms of generators of the quantum algebra of symmetries of the ...

Added: December 14, 2020

Gaydukov R., Сибирский журнал вычислительной математики 2022 Т. 15 № 2 С. 97-109

A viscous liquid flow along a semi-infinite plate with small periodic irregularities on the surface was
considered for large Reynolds numbers. The flow near the plate is described by Prandtl equations with
induced pressure which are non-classical PDE, because they contain a limiting term. The main goal is to
construct a numerical algorithm for solving these equations with ...

Added: June 10, 2020

Bogachev V., 2020 Vol. 75 No. 3 P. 393-425

Generalizations and refnements are given for results of Kozlov and Treschev on non-uniform averagings in the ergodic theorem in the case of operator semigroups on spaces of integrable functions and semigroups of
measure-preserving transformations. Conditions on the averaging measures are studied under which the averages converge for broad classes of integrable functions. ...

Added: October 23, 2020

Danilov V., European Journal of Mechanics - B/Fluids 2019 Vol. 74 P. 152-158

A stratified liquid with two layers separated by a fast oscillating interface in the case of Raleigh--Taylor instability
is considered. The averaged equations are derived, and it is shown that a mushy region of a certain density appears after averaging. The similarity between this fact and the case of unstable jump decay is discussed. ...

Added: October 21, 2018

Anikin A. Y., Brüning J., Dobrokhotov S. et al., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 265-276

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can ...

Added: September 18, 2019

Novikova E., Математические заметки 2021 Т. 109 № 5 С. 747-767

For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach for calculating the coefficients in the quantum averaging procedure is proposed. The twisted product introduced in the paper is used to transfer the averaging procedure to the space of the graduated algebra of symbols. As a result, the averaged Hamiltonian is expressed ...

Added: January 7, 2021