Oscillations in multi-stable monotone systems with slowly varying feedback

The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.     

Oscillations in first-order systems with lag

With the help of the asymptotic method, oscillations of various types are investigated in detail in a first-order system with large lag.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1193–1201, September, 1991.     

On some resonance cases in quasilinear differential systems with slowly varying parameters

For a quasilinear differential system of the second order whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we find features of existence of a partial solution of analogous structure with slowly varying coefficients and frequency in the case of purely imaginary eigenvalues of the matrix of the linear part that satisfy a certain resonance relation.     

Asymptotic methods in the theory of optimum correction for systems with slowly varying parameters

This paper deals with the problem of optimum control for the case where the constraints are linear differential equations and the functional is of quadratic type. It is assumed that the matrix of these simultaneous equations dependsslowly on the time. The paper describes a method which permits to construct effectively asymptotic solutions of such a problem when the matrix of the characteristic equation which corresponds to Pontryagin's set of equations does not have any simple elementary divisors.Portions of this research were presented at the Colloquium on Advanced Problems and Methods for Space Flight Optimization, Liège, Belgium, 1967.     

Fast-slow dynamics in Logistic models with slowly varying parameters

Fast-slow behaviors in the Logistic models with slowly varying parameters are revealed by using singular perturbation method. We first rewrite the Logistic models with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then we apply matching to obtain the approximate solutions, which are explicit and analytical and compare very well with the numerically integrated ones. More importantly, we prove the uniform validity of the approximate solutions rigorously and give the error estimate between the approximate solutions and the exact solutions via the way of upper and lower solutions.     

On integral manifolds of oscillating systems with slowly varying frequencies

By using averages of functions, we construct the integral manifold of an oscillating system that passes through resonances in the course of its evolution. We investigate the smoothness of the integral manifold and obtain estimates for its partial derivatives. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 87–93, January, 1998.     

Stability analysis of nonlinear dynamic systems with slowly and periodically varying delay

On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results.     

Peak congestion in multi-server service systems with slowly varying arrival rates

In this paper we consider the M _t queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M _t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak. This revised version was published online in June 2006 with corrections to the Cover Date.     

A unified approach for impulsive lag synchronization of chaotic systems with time delay

In this paper, we propose a unified approach for impulsive lag-synchronization of a class of chaotic systems with time delay by employing the stability theory of impulsive delayed differential equations. Three well-known delayed chaotic systems are presented to illustrate our results. Also, the estimates of the stable regions for these systems are given, respectively.     

On the stability of oscillations representable by Fourier series with slowly varying parameters

For a quasilinear almost triangular differential system whose coefficients are representable by trigonometric series with slowly varying coefficients and frequency, we obtain conditions for the existence of a particular solution of similar structure and study the stability of this solution.     

Extending satisficing control strategy to slowly varying nonlinear systems

Based on the satisficing control strategy, a novel approach to design a stabilizing control law for nonlinear time varying systems with slowly varying parameters (slowly varying systems) is presented. The satisficing control strategy has been originally introduced for time-invariant systems; however, this technique does not have any stability proof for time varying systems. In this paper, first, a parametric version of the satisficing control strategy is developed. Then, by considering the time as a frozen parameter, the parametric satisficing control strategy is utilized. Finally, a theorem is presented which suggested a stabilizing satisficing control law for the slowly varying control systems. Moreover, in this theorem, the maximum admissible rate of change of the system dynamics is evaluated. The efficiency of the proposed approach is demonstrated by a computer simulation.