Numerical-analytic method for finding solutions of systems with distributed parameters and integral condition

Using a generalization of the numerical-analytic method, we establish sufficient conditions for the existence of solutions of systems of partial differential equations with integral condition. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 110–119, January–March, 2009.     

Investigation of even systems of nonlinear functional differential equations with restrictions

We establish consistency conditions for even systems of nonlinear functional differential equations with restrictions and substantiate the applicability of an iterative method to these problems. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 252–260, April–June, 2008.     

Non-smooth modelling of electrical systems using the flux approach

The non-smooth modelling of electrical systems, which allows for idealised switching components, is described using the flux approach. The formulations and assumptions used for non-smooth mechanical systems are adopted for electrical systems using the position–flux analogy. For the most important non-smooth electrical elements, like diodes and switches, set-valued branch relations are formulated and related to analogous mechanical elements. With the set-valued branch relations, the dynamics of electrical circuits are described as measure differential inclusions. For the numerical solution, the measure differential inclusions are formulated as a measure complementarity system and discretised with a difference scheme, known in mechanics as time-stepping. For every time-step a linear complementarity problem is obtained. Using the example of the DC–DC buck converter, the formulation of the measure differential inclusions, state reduction and their numerical solution using the time-stepping method is shown for the flux approach.     

A version of the projection-iterative method for systems of linear differential equations with delay of neutral type and restrictions

We establish consistency conditions for systems of linear differential equations with constant delay of neutral type and restrictions. The applicability of the projection-iterative method to these problems is justified. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 564–573, October–December, 2006.     

On invariant tori of weakly connected systems of differential equations

We consider a family of systems of differential equations depending on a sufficiently small parameter, whose zero value corresponds to a couple of independent systems. We use the method of Green-Samoilenko function for the construction of an invariant manifold of the perturbed system and present some examples of application. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 468–489, October–December, 2005.     

Averaging method in some problems of optimal control

We substantiate the application of the averaging method to the optimal-control problem for systems of differential equations in the standard Bogolyubov form. An ϵ-optimal control is constructed. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 512–519, October–December, 2008.     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

Critical case for singularly perturbed Noether boundary-value problems

We construct an asymptotic expansion of a solution for singularly perturbed linear systems of ordinary differential equations of the Noether type in the critical case. We successively determine all terms of the asymptotic expansion by the method of boundary functions and pseudoinverse matrices. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 45–54, January–March, 2007.     

Relationship between invariant sets of systems of differential equations and the corresponding difference equations

We study the relationship between invariant sets of systems of differential equations and the corresponding difference equations in terms of sign-constant Lyapunov functions. For systems of differential equations, we obtain a converse result concerning the existence of a positive-definite Lyapunov function whose zeros coincide with a given invariant manifold. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 280–285, April–June, 2006.     

Setting up Lyapunov functions for the class of systems with quasiperiodic coefficients

The paper proposes a method to set up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients. This function is used to establish asymptotic stability conditions for a class of linear systems Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 121–130, December 2008.     

On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators

We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators. Published in Neliniini Kolyvannya, Vol. 10, No. 4, pp. 560–573, October–December, 2007.     

New approach to the investigation of the existence of periodic solutions of systems of differential equations and their construction

Methods developed for the solution of general equations with restrictions are applied to the construction of periodic solutions of systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 55–70, January–March, 2007.     

On periodic solutions of nonlinear autonomous differential systems in the critical case

We propose a new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear autonomous systems of ordinary differential equations in the critical case. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 73–82, January–March, 2009.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

On approximation of systems of differential-difference equations of neutral type by systems of ordinary differential equations

We construct a scheme of approximation of a system of differential-difference equations of neutral type by systems of ordinary differential equations and investigate the convergence conditions of this scheme. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 328–335, July–September, 2007.     

Describing function based methods for predicting chaos in a class of fractional order differential equations

This paper deals with two different methods for predicting chaotic dynamics in fractional order differential equations. These methods, which have been previously proposed for detecting chaos in classical integer order systems, are based on using the describing function method. One of these methods is constructed based on Genesio–Tesi conjecture for existence of chaos, and another method is introduced based on Hirai conjecture about occurrence of chaos in a nonlinear system. These methods are restated to use in predicting chaos in a fractional order differential equation of the order between 2 and 3. Numerical simulation results are presented to show the ability of these methods to detect chaos in two fractional order differential equations with quadratic and cubic nonlinearities.     

Variational principles and conservation laws to the Burridge–Knopoff equation

We generate conservation laws for the Burridge–Knopoff equation which model nonlinear dynamics of earthquake faults by a new conservation theorem proposed recently by Ibragimov. One can employ this new general theorem for every differential equation (or systems) and derive new local and nonlocal conservation laws. Nonlocal conservation laws comprise nonlocal variables defined by the adjoint equations to the Burridge–Knopoff equation.