Delayed loss of stability and excitation of oscillations in nonautonomous differential equations with retarded argument

Assume that zero is a stable equilibrium of an ODE ẋ = ƒ(푥, λ) for parameter values λ λ0. If we suppose that λ(t) varies slowly with t, then, under some conditions, the trajectories of the nonautonomous ODE ẋ = ƒ(푥, λ (t)) stay close to zero even long after λ(t) has crossed the value λ0. This phenomenon is called νdelayed loss of stabilityν and is well-known for ODEs. In this paper, we describe an analogous phenomenon for delay equations of the form ẋ(t) = ƒ(t, 푥 (t-1)). Further, we point out a difference between delay equations and ODEs: The inhomogeneity 풽 in the linear equation ẋ (t) = c푥 (t-1) + 풽(t) inevitably leads to an excitation of the most unstable modes of oscillation of the homogeneous equation, even if all segments 풽t are contained in a space of more rapidly decaying solutions for the homogeneous equation

Global exponential stability of nonautonomous neural network models with continuous distributed delays

For a family of non-autonomous differential equations with distributed delays, we give sufficient conditions for the global exponential stability of an equilibrium point. This family includes most of the delayed models of neural networks of Hopfield type, with time-varying coefficients and distributed delays. For these models, we establish sufficient conditions for their global exponential stability. The existence and global exponential stability of a periodic solution is also addressed. A comparison of results shows that these results are general, news, and add something new to some earlier publications.Fundação para a Ciência e a Tecnologia (FCT

Exponential dichotomy and stability of neutral type equations

AbstractA linear functional differential equation of neutral type with unbounded delay Lx(dds)Dx+Bx=0, where D and B are linear bounded retarded operators with exponentially fading memory, is considered. It is shown that if operator L is interpreted as operator from the space C into the special space C−1 of distributions, then its invertibility is equivalent to the presence of exponential dichotomy of the solutions of this equation. As applications, we prove the theorems on stability and instability in the first approximation for neutral functional differential equations of a general form

On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in 'Nonlinear Dynamics'. The final authenticated version is available online at https://doi.org/10.1007/s11071-020-05620-8

Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach

In this paper, a stability test procedure is proposed for linear nonhomogeneous fractional order systems with a pure time delay. Some basic results from the area of finite time and practical stability are extended to linear, continuous, fractional order time-delay systems given in state-space form. Sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems. A numerical example is given to illustrate the validity of the proposed procedure

Towards a resolution of the Buchanan-Lillo conjecture

Buchanan and Lillo both conjectured that oscillatory solutions of the first-order delay differential equation with positive feedback $x^{\prime }(t)=p(t)x(\tau (t))$, $t\geq 0$, where $0\leq p(t)\leq 1$, $0\leq t-\tau (t)\leq 2.75+\ln2,t\in \mathbb{R},$ are asymptotic to a shifted multiple of a unique periodic solution. This special solution was known to be uniform for all nonautonomous equations, and intriguingly, can also be described from the more general perspective of the mixed feedback case (sign-changing $p$). The analog of this conjecture for negative feedback, $p(t)\leq0$, was resolved by Lillo, and the mixed feedback analog was recently set as an open problem. In this paper, we investigate the convergence properties of the special periodic solutions in the mixed feedback case, characterizing the threshold between bounded and unbounded oscillatory solutions, with standing assumptions that $p$ and $\tau$ are measurable, $\tau (t)\leq t$ and $\lim_{t\rightarrow \infty }\tau (t)=\infty$. We prove that nontrivial oscillatory solutions on this threshold are asymptotic (differing by $o(1)$) to the special periodic solutions for mixed feedback, which include the periodic solution of the positive feedback case. The conclusions drawn from these results elucidate and refine the conjecture of Buchanan and Lillo that oscillatory solutions in the positive feedback case $p(t)\geq0$, would differ from a multiple, translation, of the special periodic solution, by $o(1)$.Comment: 27 page

Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made