Differential Equations Of Neutral Type Research Articles

Dynamics of a Coupled Chua’s Circuit with Lossless Transmission Line

This paper proposes a coupled-circuit system composed of two Chua’s circuits with lossless transmission lines. By applying the Kirchhoff’s voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point.

Neutral fuzzy fractional functional differential equations

In this paper, a class of fuzzy fractional functional differential equations of neutral type is investigated. The existence, uniqueness and iterative formula in the case of linear equations are obtained. Meanwhile, the continuous dependence of solutions on initial data, parameters and functions involved in right hand side of equation is also discussed. Several examples are given to illustrate the applications of our results.

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

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Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

On the Theory of Positional Differential Games for Neutral-Type Systems

For a dynamical system whose motion is described by neutral-type differential equations in Hale’s form, we consider a minimax–maximin differential game with a quality index evaluating the motion history realized up to the terminal time. The control actions of the players are subject to geometric constraints. The game is formalized in classes of pure positional strategies with a memory of the motion history. It is proved that the game has a value and a saddle point. The proof is based on the choice of an appropriate Lyapunov–Krasovskii functional for the construction of control strategies by the method of an extremal shift to accompanying points.

Dichotomy of Linear Partial Differential Equations of Neutral Type in Banach Spaces

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback theory, we introduce a new transformation of neutral-type equations which plays a key role in our investigation.

Oscillation criteria for higher-order neutral type difference equations

In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form ∆ rn ∆k−1 yn + pnyτn + qnf yσn = 0, n ≥ n0, 0.1 where rn, pn, qn ∈ [n0, ∞ , rn > 0, qn > 0; 0 ≤ pn ≤ p0 0; τσ = στ ; f u u ≥ m > 0 for u = 0. Moreover, we provide some examples to illustrate our main results.

NEW RESULTS ON WATER HAMMER STABILITY

The present paper starts from a model with distributed parameters (i.e. described by hyperbolic partial differential equations with nonstandard (derivative) boundary conditions) of a hydroelectric power plant with tunnel, surge tank and penstock. The association of a system of functional differential equations of neutral type and the oneto-one correspondence between the solutions of the two mathematical objects is given. Further, it is given the deduction - via singular perturbations - of the nonlinear ordinary differential equations for modeling the surge tank in order to discuss its stability under constant power delivery of the hydraulic turbine. Some other unsolved problems are pointed out.

Dynamics of a Hybrid Circuit System With Lossless Transmission Line

In this paper, a hybrid circuit system composed of a Chua's circuit with lossless transmission line and a RC oscillating circuit is considered. First, the equations describing the hybrid circuit system are reduced to a set of hybrid equations made up of a delayed neutral-type differential equation and a differential equation. By constructing the appropriate Lyapunov function, a novel condition of global stability is derived, which depends on the lower bound of fi(x)/x. Using the length of transmission line as the bifurcation parameter, the conditions of local stability and Hopf bifurcation are obtained. Finally, the realization of our proposed circuit is given and two examples are given to verify the obtained theoretical analysis by using Matlab and Multisim14.0.

On oscillatory second-order nonlinear delay differential equations of neutral type

On oscillatory second-order nonlinear delay differential equations of neutral type

Estimates for Solutions to Neutral Differential Equations with Periodic Coefficients of Linear Terms

Under study are the systems of nonlinear delay differential equations of neutral type with periodic coefficients of linear terms. We establish sufficient conditions of exponential stability of the zero solution, point out the attraction domains of the zero solution, and provide estimates for solutions characterizing the stabilization rate at infinity.

On the stability of systems of linear differential equations of neutral type with distributed delay

On the stability of systems of linear differential equations of neutral type with distributed delay

On the Stability of Systems of Linear Differential Equations of Neutral Type with Distributed Delay

We consider one class of systems of nonautonomous linear differential equations of neutral type with distributed delay. We obtain sufficient conditions for the exponential stability of the zero solution and conditions on perturbations of the coefficients under which the exponential stability of the zero solution is preserved. Using a Lyapunov—Kjasovskiĭ functional of a special kind, we prove some estimates that characterize the exponential decay of solutions at infinity.

Stability of coupled functional differential-difference equations

General coupled functional differential-difference equations are considered. A new approach to exponential stability of such equations is presented. Consequently, some novel criteria for exponential stability of coupled functional differential-difference equations are given. As applications, the obtained results are used to derive explicit criteria for exponential stability of nonlinear functional differential equations of neutral type and those of singular delay differential equations. Some illustrative examples are provided.

ON STABILITY WITH DERIVATIVE OF A CLASS OF DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

We consider an important class of differential-difference equations of neutral type, for which we study the asymptotic properties of a solution. We present necessary and sufficient conditions of exponential stability. The conditions have a geometric form of a region in the parameter space. The behavior of a solution at the boundaries of the region is analyzed separately. The loss of stability on the boundary can occur in various ways, namely through the appearance of stationary solutions, and through the appearance of periodic modes. Along with the asymptotic properties of a solution, we study analogous properties of its derivative.

Existence of strictly decreasing positive solutions of linear differential equations of neutral type

The paper is concerned with a linear neutral differential equation \begin{document}$ \dot y(t) = -c(t)y(t-\tau(t))+d(t)\dot y(t-\delta(t)) $\end{document} where \begin{document}$ c\colon [t_0,\infty)\to (0,\infty) $\end{document} , \begin{document}$ d\colon [t_0,\infty)\to [0,\infty) $\end{document} , \begin{document}$ t_0\in {\Bbb{R}} $\end{document} and \begin{document}$ \tau, \delta \colon [t_0,\infty)\to (0,r] $\end{document} , \begin{document}$ r\in{\mathbb{R}} $\end{document} , \begin{document}$ r>0 $\end{document} are continuous functions. A new criterion is given for the existence of positive strictly decreasing solutions. The proof is based on the Rybakowski variant of a topological Wazewski principle suitable for differential equations of the delayed type. Unlike in the previous investigations known, this time the progress is achieved by using a special system of initial functions satisfying a so-called sewing condition. The result obtained is extended to more general equations. Comparisons with known results are given as well.

Control of a Time Delay System Arising From Linearized Conservation Laws

A structure of energy cogeneration with distributed parameters is considered, modeled by an initial boundary value problem for hyperbolic conservation laws with a nonlinear and nonstandard boundary condition. Under a standard simplifying assumption leading to linear partial differential equations, the model is studied by associating a system of functional differential and difference equations of neutral type: considered are basic theory (existence, uniqueness, and continuous data dependence), invariant sets (positiveness of some state variables), equilibria, and their inherent stability (without control). These properties are illustrated by simulation results. Furthermore, a control Lyapunov functional is constructed, and feedback stabilizing structure is designed. This paper ends with a conclusion section, where open problems, such as stability by the first approximation/robustness and stability preservation under singular perturbations, are pointed out.

Existence of nonoscillatory solutions for second-order nonlinear differential equations of neutral type

In this work, we consider the existence of nonoscillatory solutions of second order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations. We use the Lebesgue’s dominated convergence theorem and Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

Stable Functionals of Neutral-Type Dynamical Systems

We consider a controlled dynamical system under conditions of disturbances. Its motion is described by functional differential equations of neutral type in the form of J. Hale. A functional of the motion history is said to be stable with respect to this system if there exists a control strategy that guarantees the monotonicity of this functional for any disturbances. We study various nonlocal and infinitesimal conditions for the stability of functionals.