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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 pages

Applying Prediction-Based Control (PBC) $x_{n+1}=(1-\alpha_n)f(x_n)+\alpha_n x_{n}$ with stochastically perturbed control coefficient $\alpha_n=\alpha+\ell \xi_{n+1}$, $n\in \mathbb N$, where $\xi$ are bounded identically distributed independent random variables, we globally stabilize the unique equilibrium $K$ of the equation $ x_{n+1}=f(x_n) $ in a certain domain. In our results, the noisy control $\alpha+\ell \xi$ provides both local and global stability, while the mean value $\alpha$ of the control does not guarantee global stability, for example, the deterministic controlled system can have a stable two-cycle, and non-controlled map be chaotic. In the case of unimodal $f$ with a negative Schwarzian derivative, we get sharp stability results generalizing Singer's famous statement `local stability implies global' to the case of the stochastic control. New global stability results are also obtained in the deterministic settings for variable $\alpha_n$ and, generally, continuous but not differentiable at $K$ map $f$., Comment: 27 pages, 10 figures

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-\tau (t))=0,~~t\geq s\text{ }\ \end{equation*} where $\tau :\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}% \rightarrow \mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$, Comment: 23 pages

This paper studies the problem of event-triggered impulsive control for discrete-time systems. A novel periodic event-triggering scheme with two tunable parameters is presented to determine the moments of updating impulsive control signals which are called event times. Sufficient conditions are established to guarantee asymptotic stability of the resulting impulsive systems. It is worth mentioning that the event times are different from the impulse times, that is, the control signals are updated at each event time but the actuator performs the impulsive control tasks at a later time due to time delays. The effectiveness of our theoretical result with the proposed scheme is illustrated by three examples.

We consider a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is generally delay-dependent. Maximal and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations., Comment: 24 pages, 3 figures

This study focuses on event-triggered control of nonlinear discrete-time systems with time delays. Based on a Lyapunov-Krasovskii type input-to-state stability result, we propose a novel event-triggered control algorithm that works as follows. The control inputs are updated only when a certain measurement error surpasses a dynamical threshold depending on both the system states and the evolution time. Sufficient conditions are established to ensure that the closed-loop system maintains its asymptotic stability. It is shown that the time-dependent portion in the dynamical threshold is essential to derive the lower bound of the times between two consecutive control updates. As a special case of our results, we demonstrate the performance of the designed event-triggering algorithm for a class of linear control systems with time delays. Numerical simulations are provided to demonstrate the effectiveness of our algorithm and theoretical results.

We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the H\'{e}non and the Lozi maps., Comment: 25 pages, 10 figures

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))+\int_{g(t)}^t K(t,s) x(s)ds=0, $$ where $h_k(t)\leq t$, $g(t)\leq t$, $a_k(\cdot)$ and the kernel $K(\cdot,\cdot)$ are oscillatory and, generally, discontinuous functions. The proofs are based on establishing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for some bounded right-hand side. Explicit tests are applied to models of population dynamics, such as controlled Hutchinson and Mackey-Glass equations. The results are illustrated with numerical examples, and connection to known tests is discussed., Comment: 20 pages

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically., Comment: 28 pages, 9 figures

This note studies stability of event-triggered control systems with the event-triggered control algorithm proposed in [1]. We construct a novel Halanay-type inequality, which is used to show that sufficient conditions of the main results in [1] ensure stability of the event-triggered control systems that was missing in [1]. It is also shown that a positive parameter in the proposed event-triggering condition in [1] can be freely selected to exclude Zeno behavior from the event-triggered control system. An illustrative example is investigated to demonstrate the theoretical results of this study with numerical simulations. [1] K. Zhang, B. Gharesifard, and E. Braverman, Event-triggered control for nonlinear time-delay systems, IEEE Transactions on Automatic Control, vol. 67, no. 2, pp. 1031-1037, 2022.

Exponential stability of the second order linear delay differential equation in $x$ and $u$-control $$ \ddot{x}(t)+a_1(t)\dot{x}(h_1(t))+a_2(t)x(h_2(t))+a_3(t)u(h_3(t))=0 $$ is studied, where indirect feedback control $\dot{u}(t)+b_1(t)u(g_1(t))+b_2(t)x(g_2(t))=0$ connects $u$ with the solution. Explicit sufficient conditions guarantee that both $x$ and $u$ decay exponentially., Comment: 7 pages

This paper studies impulsive stabilization of nonlinear systems. We propose two types of event-triggering algorithms to update the impulsive control signals with actuation delays. The first algorithm is based on continuous event detection, while the second type makes decision about updating the impulsive control inputs according to periodic event detection. Sufficient conditions are derived to ensure asymptotic stability of the impulsive control systems with the designed event-triggering algorithms. Lower bounds of the time period between two consecutive events are also obtained, so that the closed-loop impulsive systems are free of Zeno behavior. That is to say that the pulse phenomena are excluded from the event-triggered impulsive control systems, in the community of impulsive differential equations. An illustrative example demonstrates effectiveness of the proposed algorithms and our theoretical results.

This article studies the event-triggered control problem of general nonlinear systems with time delay. A novel event-triggering scheme is presented with two tunable design parameters, based on a Lyapunov functional result for the input-to-state stability of time-delay systems. The proposed event-triggered control algorithm guarantees the resulting closed-loop systems to be globally asymptotically stable, uniformly bounded, and/or globally attractive for different choices of these parameters. Sufficient conditions on the parameters are derived to exclude Zeno behavior. Two illustrative examples are studied to demonstrate our theoretical results.

The paper deals with the global asymptotic stability of general nonlinear time-delay systems with delay-dependent impulses through the Lyapunov-Krasovskii method. We derive a unified stability criterion which can be applied to a variety of impulsive systems. The cases when each of the continuous dynamics and the impulsive component is either stabilizing or destabilizing are investigated. Both theoretically and numerically, we demonstrate that the obtained result is more general than those existing in the literature.

Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equations $$ \dot{x_i}(t)=A_i(t)x_i(h_i(t)) +\sum_{j=1}^n \sum_{k=1}^{m_{ij}} B_{ij}^k(t)x_j(h_{ij}^k(t)) + \sum_{j=1}^n\int\limits_{g_{ij}(t)}^t K_{ij}(t,s)x_j(s)ds,~i=1,\dots,n. $$ Here $x_i$ are unknown vector functions, $A_i, B_{ij}^k, K_{ij}$ are matrix functions, $h_i,h_{ij}^k, g_{ij}$ are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations., Comment: 14 pages

Explicit exponential stability tests are obtained for the scalar neutral differential equation $$ \dot{x}(t)-a(t)\dot{x}(g(t))=-\sum_{k=1}^m b_k(t)x(h_k(t)), $$ together with exponential estimates for its solutions. Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations, exponential estimates are obtained here for the first time. Both the coefficients and the delays are assumed to be measurable, not necessarily continuous functions., Comment: 9 pages, 1 figure

Exponential stability and solution estimates are investigated for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$ are delayed arguments. Stability tests are applicable to a wide class of linear neutral systems with time-varying coefficients and delays. In addition, explicit exponential estimates for solutions of both homogeneous and non-homogeneous neutral systems are obtained for the first time. These inequalities are not just asymptotic estimates, they are valid on every finite segment and evaluate both short- and long-term behaviour of solutions., Comment: 16 pages, no figures. Accepted to Journal of Mathematical Analysis and Applications

We stabilize a prescribed cycle or an equilibrium of the difference equation using pulsed stochastic control. Our technique, inspired by the Kolmogorov's Law of Large Numbers, activates a stabilizing effect of stochastic perturbation and allows for stabilization using a much wider range for the control parameter than would be possible in the absence of noise. Our main general result applies to both Prediction-Based and Target-Oriented Controls. This analysis is the first to make use of the stabilizing effects of noise for Prediction-Based Control; the stochastic version has been examined in the literature, but only the destabilizing effect of noise was demonstrated. A stochastic variant of Target-Oriented Control has never been considered, to the best of our knowledge, and we propose a specific form that uses a point equilibrium or one point on a cycle as a target. We demonstrate our results numerically on the logistic, Ricker and Maynard Smith models from population biology., Comment: 14 pages, 7 figures

Difference equations, such as a Ricker map, for an increased value of the parameter, experience instability of the positive equilibrium and transition to deterministic chaos. To achieve stabilization, various methods can be applied. Proportional Feedback control suggests a proportional reduction of the state variable at every $k$th step. First, if $k \neq 1$, a cycle is stabilized rather than an equilibrium. Second, the equation can incorporate an additive noise term, describing the variability of the environment, as well as multiplicative noise corresponding to possible deviations in the control intensity. The present paper deals with both issues, it justifies a possibility of getting a stable blurred $k$-cycle. Presented examples include the Ricker model, as well as equations with unbounded $f$, such as the bobwhite quail population models. Though the theoretical results justify stabilization for either multiplicative or additive noise only, numerical simulations illustrate that a blurred cycle can be stabilized when both multiplicative and additive noises are involved., Comment: 9 pages, 8 figures

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$ x^{\prime}(t)+c(t)x(\tau(t))=0,~~t\geq 0, $$ where $c$ is locally integrable of any sign, $\tau(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}\tau(t)=\infty$, we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the $3/2-$stability criterion to the case of measurable parameters, improving $1+1/e$ to the sharp $3/2$ constant., Comment: 22 pages, one figure, submitted to Journal of Differential Equations on June 20, 2019

Under natural assumptions, an unstable equilibrium of a difference equation can be stabilized by a bounded multiplicative noise, identically distributed at each step. This includes stabilization of an otherwise unstable positive equilibrium of Ricker, logistic, and Beverton-Holt maps. Introduction of a multiplicative noise also allows to destabilize a stable equilibrium in a sense that all solutions stay away from this point, almost surely. In our examples a noise has symmetric, discrete or continuous, distribution with support $[-1,1]$, including Bernoulli and uniform continuous distribution. We obtain conditions on the noise amplitudes in each case that allow to either stabilize or destabilize an equilibrium. Computer simulations illustrate our results., Comment: 28 pages, 12 figures. Accepted to Physica D

In this paper, we give explicit exponential estimates $\displaystyle |x(t)|\leq M e^{ -\gamma (t-t_0) }$, where $t\geq t_0$, $M>0$, for solutions of a linear scalar delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m b_k(t)x(h_k(t))=f(t),~~ t\geq t_0,~ x(t)=\phi(t),~t\leq t_0. $$ We consider two different cases: when $\gamma>0$ (corresponding to exponential stability) and the case of $\gamma <0$ when the solution is, generally, growing. In the first case, together with the exponential estimate, we also obtain an exponential stability test, in the second case we get estimation for solution growth. Here both the coefficients and the delays are measurable, not necessarily continuous., Comment: 15 pages, 3 figures. Accepted to "Applied Mathematics and Computation"

We consider a system of several nonlinear equations with a distributed delay and obtain absolute asymptotic stability conditions, independent of the delay. The ideas of the proofs are based on the notion of a strong attractor. The results are applied to Hopfield neural networks, Nicholson's blowflies type system, and compartment models of population dynamics., Comment: 21 pages; submitted

We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays $$ x(t+1)-x(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))=0, \quad a_k(t) \geq 0, ~~h_k(t) \leq t, \quad t \geq 0, \quad k=1, \dots, m. $$ We prove that for a fixed $h(t)\not\equiv t$, a positive solution may exist for $a_k$ exceeding any prescribed $M>0$, as well as for constant positive $a_k$ with $h_k(t) \leq t-n$, where $n \in {\mathbb N}$ is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with $\inf x(t)>0$ on any bounded interval should be considered rather than oscillation. Sufficient conditions when such solutions exist or do not exist are obtained. We also present an analogue of the Gr\"{o}nwall-Bellman inequality for equations with continuous time, and examine the question when the equation has no positive non-increasing solutions. Counterexamples illustrate the role of variable delays., Comment: 12 pages

The paper explores the influence of harvesting (or culling) on the outcome of the competition of two species in a spatially heterogeneous environment. The harvesting effort is assumed to be proportional to the space-dependent intrinsic growth rate. The differences between the two populations are the diffusion strategy and the harvesting intensity. In the absence of harvesting, competing populations may either coexist, or one of them may bring the other to extinction. If the latter is the case, introduction of any level of harvesting to the successful species guarantees survival to its non-harvested competitor. In the former case, there is a strip of "close enough" to each other harvesting rates leading to preservation of the original coexistence. Some estimates are obtained for the relation of the harvesting levels providing either coexistence or competitive exclusion., Comment: 25 pages, 11 figures

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $$ where $$ |a(t)|<1,~ b(t)\geq 0, ~h(t)\leq t, ~g(t)\leq t, $$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay., Comment: 28 pages. to appear in Czechoslovak Mathematical Journal, published online

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ (x(t)-a(t)x(g(t)))'+b(t)x(h(t))=0, $ where $|a(t)| \leq A_0 < 1$, $0

We obtain new explicit exponential stability conditions for linear scalar equations with positive and negative delayed terms $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))- \sum_{k=1}^l b_k(t)x(g_k(t))=0 $$ and its modifications, and apply them to investigate local stability of Mackey--Glass type models $$\dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(g(t))}-\gamma x(h(t))\right]$$ and $$\dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(h(t))}-\gamma x(t)\right].$$, Comment: 33 pages, 3 figures

The paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value., Comment: 28 pages, 7 figures

We present new explicit exponential stability conditions for the linear scalar neutral equation with two variable coefficients and delays $$ (x(t)-a(t)x(g(t)))'=-b(t)x(h(t)), $$ where $|a(t)|<1$, $b(t)\geq 0$, $h(t)\leq t$, $g(t)\leq t$, in the case when the delays $t-h(t)$, $t-g(t)$ are bounded, as well as an asymptotic stability condition, if the delays can be unbounded., Comment: 14 pages

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $ where $ 0\leq a(t)\leq A_0<1$, $0

Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given., Comment: The paper was originally open access. The purpose of this publication is correct the original mistake that occurred in Theorems 6 and 10 (in particular, the variables under the integral in (2.11) and (2.27)) in the published paper

We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also given to show applicability and sharpness of the new results., Comment: 33 pages

In contrast with unstructured models, structured discrete population models have been able to fit and predict chaotic experimental data. However, most of the chaos control techniques in the literature have been designed and analyzed in a one-dimensional setting. Here, by introducing target oriented control for discrete dynamical systems, we prove the possibility to stabilize a chosen state for a wide range of structured population models. The results are illustrated with introducing a control in the celebrated LPA model describing a flour beetle dynamics. Moreover, we show that the new control allows to stabilize periodic solutions for higher order difference equations, such as the delayed Ricker model, for which previous target oriented methods were not designed., Comment: 16 pages, 5 figures

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation., Comment: 22 pages, 1 figure, accepted for publication in a special issue of DCDS-S

For a wide class of second order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal, allows to stabilize unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilization by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points., Comment: 10 pages, 4 figures, published in 2015 in International Journal of Control, non-autonomous second order delay differential equations, stabilization, proportional feedback, a controller with damping, derivative control, sunflower equation

For a physical or biological model whose dynamics is described by a higher order difference equation $u_{n+1}=f(u_n,u_{n-1}, \dots, u_{n-k+1})$, we propose a version of a target oriented control $u_{n+1}=cT+(1-c)f(u_n,u_{n-1}, \dots, u_{n-k+1})$, with $T\ge 0$, $c\in [0,1)$. In ecological systems, the method incorporates harvesting and recruitment and for a wide class of $f$, allows to stabilize (locally or globally) a fixed point of $f$. If a point which is not a fixed point of $f$ has to be stabilized, the target oriented control is an appropriate method for achieving this goal. As a particular case, we consider pest control applied to pest populations with delayed density-dependence. This corresponds to a proportional feedback method, which includes harvesting only, for higher order equations., Comment: 18 pages, 3 figures, published in 2015 in Physics Letters A

The problem considered in the paper is exponential stability of linear equations and global attractivity of nonlinear non-autonomous equations which include a non-delay term and one or more delayed terms. First, we demonstrate that introducing a non-delay term with a non-negative coefficient can destroy stability of the delay equation. Next, sufficient exponential stability conditions for linear equations with concentrated or distributed delays and global attractivity conditions for nonlinear equations are obtained. The nonlinear results are applied tothe Mackey-Glass model of respiratory dynamics., Comment: 12 pages, 2 figures, published in 2015 in Applied Mathematics and Computation

For a nonlinear equation with several variable delays $$ \dot{x}(t)=\sum_{k=1}^m f_k(t, x(h_1(t)),\dots,x(h_l(t)))-g(t,x(t)), $$ where the functions $f_k$ increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on $[0, \infty)$, as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide., Comment: 24 pages, published in Applied Mathematics and Computation, 2016

We consider a system $\displaystyle \frac{dx}{dt}=r_1(t) G_1(x) \left[ \int_{h_1(t)}^t f_1(y(s))~d_s R_1 (t,s) - x(t) \right], \frac{dy}{dt}=r_2(t) G_2(y) \left[ \int_{h_2(t)}^t f_2(x(s))~d_s R_2 (t,s) - y(t)\right]$ with increasing functions $f_1$ and $f_2$, which has at most one positive equilibrium. Here the values of the functions $r_i,G_i,f_i$ are positive for positive arguments, the delays in the cooperative term can be distributed and unbounded, both systems with concentrated delays and integro-differential systems are a particular case of the considered system. Analyzing the relation of the functions $f_1$ and $f_2$, we obtain several possible scenarios of the global behaviour. They include the cases when all nontrivial positive solutions tend to the same attractor which can be the positive equilibrium, the origin or infinity. Another possibility is the dependency of asymptotics on the initial conditions: either solutions with large enough initial values tend to the equilibrium, while others tend to zero, or solutions with small enough initial values tend to the equilibrium, while others infinitely grow. In some sense solutions of the equation are intrinsically non-oscillatory: if both initial functions are less/greater than the equilibrium value, so is the solution for any positive time value. The paper continues the study of equations with monotone production functions initiated in [Nonlinearity, 2013, 2833-2849]., Comment: 18 pages, 3 figures, published in 2015 in Nonlinearity

We consider difference equations with several non-monotone deviating arguments and nonnegative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficient oscillation conditions are obtained in an explicit iterative form. Additional results in terms of $\liminf $ are obtained for bounded deviations. Examples illustrating the oscillation tests are presented., Comment: 18 pages, 3 figures, combines published in 2015 in Journal of Difference Equations and Applications article and Erratum published in 2017

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behavior of the map function $f$ at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map $f$ at zero exceeds one or, independently of the behavior of $f$ at zero, when the amplitudes are not square summable., Comment: 22 pages, 4 figures, to appear in Journal of Difference Equations and Applications

Given a deterministic difference equation $x_{n+1}= f(x_n)$, we would like to stabilize any point $x^{\ast}\in (0, f(b))$, where $b$ is a unique maximum point of $f$, by introducing proportional feedback (PF) control. We assume that PF control contains either a multiplicative $x_{n+1}= f\left( (\nu + \ell\chi_{n+1})x_n \right)$ or an additive noise $x_{n+1}=f(\lambda x_n) +\ell\chi_{n+1}$. We study conditions under which the solution eventually enters some interval, treated as a stochastic (blurred) equilibrium. In addition, we prove that, for each $\varepsilon>0$, when the noise level $\ell$ is sufficiently small, all solutions eventually belong to the interval $(x^{\ast}-\varepsilon,x^{\ast}+\varepsilon)$., Comment: 20 pages, 19 figures

We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction-based control. These perturbations may be multiplicative $$x_{n+1}=f(x_n)-\left( \alpha + l\xi_{n+1} \right) (f(x_n)-x_n), \quad n=0, 1, \dots$$ if they arise from stochastic variation of the control parameter, or additive $$x_{n+1}=f(x_n)-\alpha(f(x_n)-x_n) +l\xi_{n+1}, \quad n=0, 1, \dots $$ if they reflect the presence of systemic noise. We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable:i.e. the presence of noise improves the known effectiveness of prediction-based control. Finally, we show that systemic noise has a "blurring" effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results., Comment: 21 pages, 11 figures, to appear in Physica D