Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

We introduce a new non-abelian quantum synchronisation model over the unitary group, represented as a gradient flow, where state matrices asymptotically converge to a common one up to phase translation. We provide a sufficient framework leading to quantum synchronisation based on Riccati-type differential inequalities. In addition, uniform time-delayed interaction is considered for modelling realistic communication, and we demonstrate that quantum synchronisation is persistent when a small time delay is allowed. Finally, numerical simulation is performed to visualise qualitative behaviours and support theoretical results.

Networked dynamical systems, i.e., systems of dynamical units coupled via nontrivial interaction topologies, constitute models of broad classes of complex systems, ranging from gene regulatory and metabolic circuits in our cells to pandemics spreading across continents. Most of such systems are driven by irregular and distributed fluctuating input signals from the environment. Yet how networked dynamical systems collectively respond to such fluctuations depends on the location and type of driving signal, the interaction topology and several other factors and remains largely unknown to date. As a key example, modern electric power grids are undergoing a rapid and systematic transformation towards more sustainable systems, signified by high penetrations of renewable energy sources. These in turn introduce significant fluctuations in power input and thereby pose immediate challenges to the stable operation of power grid systems. How power grid systems dynamically respond to fluctuating power feed-in as well as other temporal changes is critical for ensuring a reliable operation of power grids yet not well understood. In this work, we systematically introduce a linear response theory (LRT) for fluctuation-driven networked dynamical systems. The derivations presented not only provide approximate analytical descriptions of the dynamical responses of networks, but more importantly, also allow to extract key qualitative features about spatio-temporally distributed response patterns. Specifically, we provide a general formulation of a LRT for perturbed networked dynamical systems, explicate how dynamic network response patterns arise from the solution of the linearised response dynamics, and emphasise the role of LRT in predicting and comprehending power grid responses on different temporal and spatial scales and to various types of disturbances. Understanding such patterns from a general, mathematical perspective enables to estimate network responses quickly and intuitively, and to develop guiding principles for, e.g., power grid operation, control and design.

We establish new oscillation criteria for nonlinear differential equations of second order. The results here make some improvements of oscillation criteria of Butler, Erbe, and Mingarelli [2], Wong [8, 9], and Philos and Purnaras [6].

Synchronisation is one of the most interesting collective motions observed in large-scale complex networks of interacting dynamical systems. We consider global synchronisation for networks of nonlinearly coupled identical cells with time delays, using an approach where the synchronisation problem is converted to solving an homogeneous linear system. This approach is extended to fit networks under more general coupling topologies, and we derive four delay-dependent and delay-independent criteria that ensure the coupled dynamical network is globally synchronised. Some examples show that the four criteria are not mutually inclusive, and numerical simulations also demonstrate our theoretical results.

The exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.

We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C. G. Philos.

In this work, oscillatory and asymptotic behaviours of all solutions of higher-order neutral differential equations are compared with first-order delay differential equations, depending on two different ranges of the coefficient associated with the neutral part. Some simple examples are given to compare our results with the existing results in the literature and to illustrate the significance and applicability of our new results. Our results generalise, improve and correct some of the existing results in the literature.

The existence of $2\pi$-periodic solutions of the second-order differential equation \[ x''+f(x)x'+ax^+-bx^-+g(x)=p(t), \qquad n\in \mathbb{N},\] where $a, b$ satisfy $1/\sqrt{a}+1/\sqrt{b}=2/n$ and $p(t)=p(t+2\pi)$, $t\in \mathbb{R}$, is examined. Assume that limits $\lim_{x\to\pm\infty}F(x)=F(\pm\infty)$ ($F(x)=\int_0^xf(u) du$) and $\lim_{x\to\pm\infty}g(x)=g(\pm\infty)$ exist and are finite. It is proved that the equation has at least one $2\pi$-periodic solution provided that the zeros of the function $\Sigma_1$ are simple and the zeros of the functions $\Sigma_1, \Sigma_2$ are different and the signs of $\Sigma_2$ at the zeros of $\Sigma_1$ in $[0,2\pi/n)$ do not change or change more than two times, where $\Sigma_1$ and $\Sigma_2$ are defined as follows: \[\Sigma_1(\theta)=\frac{n}{\pi}\left[\frac{g(+\infty)}{a}-\frac{g(-\infty)}{b} \right]-\frac {1}{2\pi}\int_0^{2\pi}p(t)\varphi(t+\theta)\,dt,\qquad \theta\in [0, 2\pi/n],\]\[\Sigma_2(\theta)=\frac{n}{\pi}[F(+\infty)-F(-\infty)]-\frac{1}{2\pi} \int_0^{2\pi}p(t)\varphi'(t+\theta)\,dt, \qquad\theta\in [0, 2\pi/n].\] Moreover, it is also proved that the given equation has at least one $2\pi$-periodic solution provided that the following conditions hold: \begin{eqnarray*} -\infty&<&\liminf_{x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}\le \limsup_ {x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}<+\infty,\\ 0&<&\liminf_{x\to\pm\infty}\frac{g(x)}{|x|^{q-2}x}\leq \limsup_{x\to\pm\infty} \frac{g(x)}{|x|^{q-2}x}<+\infty, \end{eqnarray*} with $1\leq p<q<2$.

Some sufficient conditions are given for all solutions of the nonlinear differential equation y″(x) +p(x)f(y) = 0 to be nonoscillatory, where p is positive and

for a quotient γ of odd positive integers, γ > 1.

Consider the second order nonlinear differential equation

y" + a(t)f(y) = 0

where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

The authors consider the nonlinear neutral delay differential equation

and obtain results on the asymptotic behavior of solutions. Some of the results require that P(t) has arbitrarily large zeros or that P(t) oscillates about — 1

A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t)+ a(t)f[x(t)]=0, where a ∈ C([t0∞,)), f∈C(R) with yf(y)>0 for y≠0 and and f is continously differentiable on R-{0} with f'(y)≥0 for all y≠O. In the special case of the differential equation (γ > 1) this criterion leads to an oscillation result due to Wong [9].