On the application of the homotopy analysis method to limit cycles’ approximation in planar self-excited systems

In this paper, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. After changing general planar self-excited systems to the canonical forms by several linear transformations, the auxiliary linear operators and the initial guess of solutions are introduced. Hence, the homotopy analysis solving is set up. Importantly, in solving the higher-order deformation equations, the idea of a perturbation procedure of limit cycles’ approximation proposed in the setting of second-order self-excited equations is embedded. As an application, a Rosenzweig–MacArthur predator–prey model is studied in detail. By choosing the suitable convergence-control parameters, the accurately analytical approximations of the large amplitude limit cycles and their frequency of the model are obtained. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations.     

On Markov?CKrein characterization of the mean waiting time in M/G/K and other queueing systems

We propose a new research direction to reinvigorate research into better understanding of the M/G/K and other queueing systems??via obtaining tight bounds on the mean waiting time as functions of the moments of the service distribution. Analogous to the classical Markov?CKrein theorem, we conjecture that the bounds on the mean waiting time are achieved by service distributions corresponding to the upper/lower principal representations of the moment sequence. We present analytical, numerical, and simulation evidence in support of our conjectures.     

The Perron–Frobenius theorem – a proof with the use of Markov chains

The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.     

Methods for the investigation of dynamical systems with impulse action and “mortal” dynamical systems

The notions of dynamical systems with impulse action and mortal dynamical systems are introduced. Their connection with the idealizations of ordinary dynamical systems is considered. General methods for the investigation of these systems are worked out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1605–1613, November, 1991.     

On the use of the Borel–Cantelli lemma in Markov chains

In the present paper, we propose technical generalizations of the Borel–Cantelli lemma. These generalizations can be further used to derive strong limit results for Markov chains. In our work, we obtain some strong limit results.     

Some properties of biorthogonal polynomials and their application to Padé approximations

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N?1/N],N ε ?, for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ wheref(z) is a function with known Padé approximants of the indicated orders,T _j [f;z] are Taylor polynomials of degreej for the functionf(z), and α _{m, M} = $\overline {1,M} $ are constants.     

Laplace transforms of the Hulthén Green’s function and their application to potential scattering

We derive closed-form representations for the single and double Laplace transforms of the Hulthén Green’s function of the outgoing wave multiplied by the Yamaguchi potential and write them in the maximally reduced form. We use the expression for the double transform to compute the low-energy phase shifts for the elastic scattering in the systems α–nucleon, α–He³, and α–H³. The calculation results agree well with the experimental data.     

General approach to filtering with fractional brownian noises — application to linear systems

At first a general approach is proposed to filtering in systems where the observation noise is a fractional Brownian motion. It is shown that the problem can be handled in terms of some appropriate semimartingale and analogs of the classical innovation process and fundamental filtering theorem are obtained. Then the problem of optimal filtering is completely solved for Gaussian linear systems with fractional Brownian noises. Closed form simple equations are derived both for the mean of the optimal filter and the variance of the filtering error. Finally the results are explicited in various specific cases     

Uniform Hölder continuity of approximate solutions to parabolic systems and its application

By means of a minimizing scheme, we constructively define approximate solutions to parabolic systems. On the basis of Campanato theory, these approximate solutions are shown to be locally Hölder equi-continuous with respect to an approximation parameter. As an application, we obtain a weak solution to an initial-boundary value problem with the same Hölder continuity.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

Existence of periodic solutions and their asymptotic stability to the Navier–Stokes equations with the Coriolis force

We consider the time-periodic problem for the Navier–Stokes equations in the rotational framework. We prove the unique existence of time-periodic solutions for the prescribed external force. Furthermore, we also show the asymptotic stability of small time-periodic solutions provided the initial disturbance is sufficiently small.     

ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions

Let us consider m(KS) below for all N?2 and general exponents m and q. In particular, the 2-D semi-linear case such as N=2, m=1 and q=2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in which coincides with a scaling invariant class of weak solutions associated with m(KS). In addition, the Hausdorff dimension of its singular set is zero if and .     

Factorized time-dependent distributions for certain multiclass queueing networks and an application to?enzymatic processing networks

Motivated by applications in biological systems, we show for certain multiclass queueing networks that time-dependent distributions for the multiclass queue-lengths can have a factorized form which reduces the problem of computing such distributions to a similar problem for related single-class queueing networks. We give an example of the application of this result to an enzymatic processing network.