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We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do no...

This paper introduces the more general result on existence, uniqueness and boundedness for solutions of nonstandard Volterra type integral equation on an arbitrary time scales. We use Lipschitz type function and the Banach’s fixed point theorem at functional space endowed with a suitable Bielecki type norm. Furthermore, it allows to get new sufficient conditions for boundedness and continuous dependence of solutions.

We study Hyers-Ulam stability of a nonlinear Volterra integral equation on unbounded time scales. Sufficient conditions are obtained based on the Banach fixed point theorem and Bielecki type norm.

Using Green type map we can find sufficient conditions under which an impulsive quasilinear equation is dynamically equivalent to its corresponding linear equation. This result extends Grobman Hartman theorem for equations without ordinary dichotomy.

There will be presented asymptotic formulas for solutions of the equation y'' + (1 + φ (x))y = 0, 0 < x0 < x < ∞ , where function is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function φ . The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner von Neumann potential. A class of potentials generalizing that of Wigner von Neumann will be presented.

The gyrotron is a microwave source whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. Powerful gyrotrons can be used to heat nuclear fusion plasma. In addition, they have found a wide utility in plasma diagnostics, plasma chemistry, radars, extra-high-resolution spectroscopy, high-temperature processing of materials, medicine, etc. However, the main application of gyrotrons is in electron cyclotron resonance heating in tokamaks and stellarators. Equations describing gyrotron operation are ordinary differential equations and Schrödinger type partial differential equations. The present paper provides a survey of the analytical and numerical results concerning these equations obtained by our group in the last decade.

Using Green type map we can find sufficient conditions under which a quasilinear equation is dynamically equivalent to its corresponding linear equation. This result extends Grobman - Hartman theorem for equations without ordinary dichotomy.

In this paper we study numerical problems arising in solving the single mode gyrotron equation. Using the method of finite differences analytical and numerical solutions are obtained. Quasistationary solutions and corresponding eigenvalues and eigenfunctions of this problem are investigated. Girotrono lygties vienos modos skaitiniai sprendimai Santrauka Straipsnyje nagrinėjami skaitiniai sprendiniai gauti tiriant girotrono lygties vieną modą. Analitiniai ir skaitiniai sprendiniai gauti taikant baigtinių skirtumų metodą. Ištirti kvazistacionarieji sprendiniai ir atitinkamos tokio uždavinio tikrinės reikšmės ir tikrinės funkcijos. First Published Online: 14 Oct 2010

The onset of stochastic oscillations in gyrotrons is studied by means of the self-consistent theory describing nonstationary processes. Complicated alternating sequences of regions of stationary, automodulation, and chaotic oscillations are found in the plane of the generalized gyrotron variables: cyclotron resonance mismatch and dimensionless current. The results of the investigations are important in connection with attempts to increase the output power of gyrotrons by raising the current.

The Hamiltonian approach is used to analyze the equation describing the electron interaction in gyrotron resonators with realistic RF field profiles. A detailed numerical study of the behavior of electron trajectories for some specific values of parameters controlling the interaction is performed. It is found that in some cases chaos-like motions of electrons are possible.

A detailed mathematical analysis of the equation describing the electron interaction with the high-frequency field in a gyrotron resonator is presented. Electron trajectories in the phase space are classified. It is proven that in the cold-cavity approximation when the high-frequency field is represented by a Gaussian-type function, the solutions of the gyrotron equation are asymptotically equal to the solutions of the corresponding unforced equation. This means that chaos, which, in principle, can develop in a resonator for some values of control parameters, can be only transient, i.e., electrons again follow regular trajectories once they leave the interaction space. Understanding of the distribution of electron trajectories is important, both from the theoretical and the practical point of view. As an example, detailed numerical computations of electron trajectories are performed for those values of control parameters which correspond to the maximum efficiency.

Conjugacy technique is applied to analysis asymptotic equivalence of nonautonomous linear and semilinear difference equations in Banach space.

The conjugacy of discrete semidynamical system and its partially decoupled discrete semidynamical system in Banach space is proved in the neighbourhood of the trivial invariant manifold.

Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrödinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was numerically examined in our earlier work. In this paper we consider the stationary and nonstationary problems in large time interval with complex oscillating initial conditions. We use the implicit finite difference schemes and the method of lines realized with MATLAB. Two versions of gyrotron equation are investigated. We consider different methods for modelling new and old versions of the gyrotron equations. The main physical result is the possibility to determine the maximal value of the wave amplitude and the electron efficiency coefficient.

The main result of this paper is on decoupling by means of a topological transformation of a homeomorphism which admits invariant sets. A theorem of conjugacy of a given possible noninvertible mapping, which admits an invariant set, to a simpler mapping than the given one in terms of decoupling is also proved. The results are valid in an arbitrary complete metric space.

A stationary problem with the integral boundary condition arising in the mathematical modelling of a gyrotron is numerically investigated. The Chebyshev's polynomials of the second kind are used as the tool of calculations. The main result with physical meaning is the possibility to determine the maximal value of electrons efficiency. First published online:14 Oct 2010

Numerical aspects for solving of certain problem arising in gyrotron theory are discussed. Particularly, finite-difference schemes using quasistationarization and method of lines were applied and the relevant results analyzed.

Gyrotron is a special tube generating powerful radiowaves in the millimeter wave range. Gyrotrons are mainly used to heat nuclear fusion plasma, in order to induce controlled thermonuclear reactions on earth. In addition, they have found a wide utility in radars and the high‐temperature processing of materials. Differential equations describing gyrotron operation are analyzed from the mathematical point of view. Phase portraits of electron trajectories in realistic resonators are determined. First Published Online: 14 Oct 2010