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In this paper, ongoing studies to investigate nonlinear ordinary differential equations (ODE) are extended by presenting a new concept in non-dimensionalization process. This concept is illustrated with a practical example of nonlinear ODEs, which cannot reliably solve using the numerical methods. In this paper, two perturbation techniques are used to solve the problem. Effect of varying the dimensionless initial displacement on the accuracy of solution is investigated. It is shown that if the process of non-dimensionalizing is done appropriately, the calculated results will be extremely accurate. Moreover, a new concept called "behavior of results" is proposed to find accurate results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of an exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations is investigated. Based on the Lyapunov method, a new delay-dependent criterion for exponential stability is established in terms of LMI (linear matrix inequalities). Numerical examples are carried out to support the effectiveness of our results. [ABSTRACT FROM AUTHOR]

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation $\backslash epsilon^2\; y\text{'}\text{'}=(1+\backslash epsilon^\; 2\; \backslash psi(x,\; \backslash epsilon))y$ with a small parameter $\backslash epsilon$, where $\backslash psi$ is analytic and even with respect to $\backslash epsilon$. It is well known that it has two formal solutions of the form $y^\backslash pm(x,\backslash epsilon)=e^\{\backslash pm\; x/\backslash epsilon\}h^\backslash pm(x,\backslash epsilon)$, where $h^\; \backslash pm(x,\backslash epsilon)$ is a formal series in powers of $\backslash epsilon$ whose coefficients are functions of $x$. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if $\backslash psi$ satisfies certain conditions, in particular concerning its $x$-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations $\backslash epsilon^r\; y\text{'}\; =\; A(x,\backslash epsilon)\; y$ in the neighborhood of a nonturning point and for second-order (analytic) equations $\backslash epsilon\; y\text{'}\text{'}\; -\; 2xy\text{'}-g(x,\backslash epsilon)\; y=0$ exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution $\backslash hat\; y(x,\backslash epsilon)=\backslash sum\; y\_n(x)\backslash epsilon^n$ such that the coefficients have no poles at $x=0$. [ABSTRACT FROM AUTHOR]

In the present paper, we prove a theorem on the continuous dependence of the solution on perturbations of the initial data (the initial time, the initial value of the trajectory, the initial function) and the right-hand side; the perturbation of the right-hand side of the differential equation is assumed to be small in the integral sense. Such theorems play an important role in the investigation of optimal control problems [1-3]. The theorem proved here is a straightforward generalization of theorems in [1-4]. The well-posedness of the Cauchy problem for various classes of differential equations for the case in which the perturbation of the right-hand side is assumed to be small in the integral sense was considered in [5-9]. [ABSTRACT FROM AUTHOR]

Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley's results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even further. We also hope that our results provide a better bridge between the different fields. Furthermore we show how to interpret Conley's conditions in terms of averaging. The result are illustrated by the two-dimensional van der Pol equation and by a three-dimensional Morris-Lecar model. [ABSTRACT FROM AUTHOR]

The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider a nonoscillatory half-linear second order differential equation and suppose that we know its solution h. Using this solution we construct a function d such that the equation is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459–472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278–2295]. Finally, we give two examples to illustrate the obtained results. [Copyright &y& Elsevier]

Time scales have been introduced in order to unify the theories of differential and difference equations and in order to extend these cases to many other so-called dynamic equations. In this paper we consider a linear dynamic equation on a time scale together with a perturbed equation. We show that, if certain exponential dichotomy conditions are satisfied, then for any solution of the perturbed equation there exists a solution of the unperturbed equation that asymptotically differs from the solution of the perturbed equation no more than the order of the perturbation term. In order to show this perturbation theorem, we use many properties of the exponential function on time scales and derive several bounds for certain monomials that appear in the dynamic version of Taylor's formula. [ABSTRACT FROM AUTHOR]

The article considers delay systems with permanent perturbations. A new sight of the notion of smallness of perturbations was suggested. The trivial solution is said to be stable under permanent perturbations small on the set M. The criteria of stability under permanent perturbations small on a set was discussed.

In this paper the integration of the differential equations of motion of a non-conservative dynamical system is replaced by an equivalent problem of finding a complete integral a quasi-linear partial differential equation of the first order. In second part, these complete integrals are combined with the two time scales perturbation method in the study of non-linear oscillatory motions. [Copyright &y& Elsevier]

In this paper a numerical method is suggested to solve a class of boundary value problems for a weakly coupled system of singularly perturbed second order ordinary differential equations of reaction-diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of the well known perturbation method. Then initial value problems and terminal value problems (TVPs) are formulated such that their solutions are the terms of this asymptotic expansion. These problems are happened to be singularly perturbed problems and therefore exponentially fitted finite difference schemes are used to solve these problems. Since the boundary value problem is converted into a set of initial and TVPs and an asymptotic expansion approximation is used, the present method is termed as an asymptotic initial value method. Necessary error estimates are derived and examples provided to illustrate the method. The present method is easy to implement and well suited for parallel computing. [Copyright &y& Elsevier]

This paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations. The results of D.D. Bainov and P.S. Simeonov (1993) are extended and dynamics different from the corresponding continuous system are found. It is shown that the system may have a unique positive periodic solution which is globally asymptotically stable, or go extinct when the two periods are rational dependent. When they are rational independent, the system has no periodic solutions, however, still has a global attractor or go extinct under some conditions. [Copyright &y& Elsevier]

Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. [ABSTRACT FROM AUTHOR]

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing. [ABSTRACT FROM AUTHOR]

The existence of mild solutions for second-order impulsive semilinear neutral functional differential equations with nonlocal conditions in Banach spaces is investigated. The results are obtained by using fractional power of operators and Sadovskii's fixed point theorem. [ABSTRACT FROM AUTHOR]

We establish that the reducibility exponent ( Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system , with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1 A+ Q ) with an arbitrary piecewise continuous perturbation Q satisfying the condition $$ \overline {\lim } _{t \to + \infty } t^{ - 1} \ln \left\| {Q(t)} \right\| \leqslant - \sigma $$ is reducible to the original system (1 A ) by some Lyapunov transformation. [ABSTRACT FROM AUTHOR]

We survey some results dealing with well-posedness of scalar optimization problems, with applications to the optimal control of ordinary differential equations under plant perturbations. Then we deal with the definition of a condition number for optimization problems. Corresponding distance theorems are discussed, aiming to generalize the Eckart-Young theorem to the infinite-dimensional setting. [ABSTRACT FROM AUTHOR]

Normwise backward errors and residual bounds for an approximate Hermitian positive semidefinite solution set to the periodic discrete-time algebraic Riccati equation are obtained. The results are illustrated by using simple numerical examples. [ABSTRACT FROM AUTHOR]

Considers generalized impulsive conditions for singularly perturbed impulse systems. Definition of the unknown constant vectors with the use of initial and impulsive conditions; Representation of impulsive conditions.

For many technical systems, the use of a refined network modeling approach leads to hyperbolic-type initial-boundary value problems of partial differential-algebraic equations (PDAEs). The boundary conditions of these systems are governed by time-dependent differential-algebraic equations (DAEs) that couple the PDAE system with the network elements that are modeled by DAEs in time only. In order to classify these systems, we extend some index notions that have already been introduced to treat parabolic-type problems. A perturbation index is considered that reflects the sensitivity of the mixed system to slight perturbations in the right-hand side of the PDAEs, as well as in the input signals of the DAE systems and initial values. In order to make an a posteriori analysis of semidiscretization in space and time, we introduce additionally a space and a method of lines (MOL) index. Here one is especially interested in whether the semidiscretized systems properly reflect the properties of the underlying systems. We will show that these indices may detect an artificial sensitivity with respect to perturbations, e.g., if the semidiscretization does not consider the information transport along the characteristics. [ABSTRACT FROM AUTHOR]

The computational singular perturbation (CSP) method is employed for the solution of stiff PDEs and for the acquisition of the most important physical understanding. The usefulness of the method is demonstrated by analyzing a transient reaction-diffusion problem. It is shown that in the regions where the solution exhibits smooth spatial slopes, a simple nonstiff system of equations can be used instead of the full governing equations. From the simplified system, which is numerically provided by CSP and whose structure varies with space and time, important physical information comes to light. The relation of this method to the class of asymptotic expansion methods is explored. It is shown that the CSP results are identical to the ones obtained by the asymptotic methods. The identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by CSP. Preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy. [ABSTRACT FROM AUTHOR]

Using the theory of n-widths, the approximability of solutions of singularly perturbed reaction-diffusion and convection-diffusion problems in one dimension is quantified. [ABSTRACT FROM AUTHOR]