arXiv subjects

Elena Braverman

Publications and source records attributed to Elena Braverman.

At least 55 records · Page 3Linked to original sources

arXiv2019

Stabilization of structured populations via vector target oriented control

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.

math.DS↗

arXiv2019

On different types of stability for linear delay dynamic equations

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.

math.DS↗

arXiv2017

A cyclic system with delay and its characteristic equation

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.

math.CA↗

arXiv2017

Iterative oscillation tests for difference equations with several non-monotone arguments

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.

math.DS↗

arXiv2016

Stability tests for second order linear and nonlinear delayed models

For the nonlinear second order Lienard-type equations with time-varying delays $$ \ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^l s_k(t,x(h_k(t)))=0, $$ global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor-Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications.

math.DS↗

arXiv2016

Boundedness and persistence of delay differential equations with mixed nonlinearity

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.

math.DS↗

arXiv2016

Stability conditions for scalar delay differential equations with a nondelay term

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.

math.DS↗

arXiv2016

Stabilization with target oriented control for higher order difference equations

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.

math.DS↗

arXiv2016

Stabilization of second order nonlinear equations with variable delay

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.

math.DS↗

arXiv2016

On stability of cooperative and hereditary systems with a distributed delay

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].

math.DS↗

arXiv2016

Stochastic difference equations with the Allee effect

For a truncated stochastically perturbed equation $x_{n+1}=\max\{ f(x_n)+lχ_{n+1}, 0 \}$ with $f(x) f(H)>m$. As the amplitude grows, an interval $(m-\varepsilon, m+δ)$ of initial values arises and expands, such that with a certain probability, $x_n$ sustains in $[m, H]$, and possibly eventually gets into the interval $(0,m-\varepsilon)$, with a positive probability. Lower estimates for these probabilities are presented. If $H$ is large enough, as the amplitude of perturbations grows, the Allee effect disappears: a solution persists for any positive initial value.

math.DS↗

arXiv2016

Stabilization of difference equations with noisy proportional feedback control

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 $x_{n+1}= f\left( (ν+ \ellχ_{n+1})x_n \right)$ or an additive noise $x_{n+1}=f(λx_n) +\ellχ_{n+1}$. 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 $(x^{\ast}-\varepsilon,x^{\ast}+\varepsilon)$.

math.DS↗

arXiv2016

Stabilisation of difference equations with noisy prediction-based control

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 $$x_{n+1}=f(x_n)-\left( α+ lξ_{n+1} \right) (f(x_n)-x_n), \quad n=0, 1, \dots$$ if they arise from stochastic variation of the control parameter, or additive $$x_{n+1}=f(x_n)-α(f(x_n)-x_n) +lξ_{n+1}, \quad n=0, 1, \dots $$ 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.

math.DS↗

arXiv2016

On convergence of solutions to difference equations with additive perturbations

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.

math.DS↗

arXiv2014

Image Inpainting Using Directional Tensor Product Complex Tight Framelets

In this paper we are particularly interested in the image inpainting problem using directional complex tight wavelet frames. Under the assumption that frame coefficients of images are sparse, several iterative thresholding algorithms for the image inpainting problem have been proposed in the literature. The outputs of such iterative algorithms are closely linked to solutions of several convex minimization models using the balanced approach which simultaneously combines the $l_1$-regularization for sparsity of frame coefficients and the $l_2$-regularization for smoothness of the solution. Due to the redundancy of a tight frame, elements of a tight frame could be highly correlated and therefore, their corresponding frame coefficients of an image are expected to close to each other. This is called the grouping effect in statistics. In this paper, we establish the grouping effect property for frame-based convex minimization models using the balanced approach. This result on grouping effect partially explains the effectiveness of models using the balanced approach for several image restoration problems. Inspired by recent development on directional tensor product complex tight framelets (TP-CTFs) and their impressive performance for the image denoising problem, in this paper we propose an iterative thresholding algorithm using a single tight frame derived from TP-CTFs for the image inpainting problem. Experimental results show that our proposed algorithm can handle well both cartoons and textures simultaneously and performs comparably and often better than several well-known frame-based iterative thresholding algorithms for the image inpainting problem without noise. For the image inpainting problem with additive zero-mean i.i.d. Gaussian noise, our proposed algorithm using TP-CTFs performs superior than other known state-of-the-art frame-based image inpainting algorithms.

cs.IT↗

arXiv2013

New Global Exponential Stability Criteria for Nonlinear Delay Differential Systems with Applications to BAM Neural Networks

We consider a nonlinear non-autonomous system with time-varying delays $$ \dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which has a large number of applications in the theory of artificial neural networks. Via the M-matrix method, easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained. Application of the main theorem requires just to check whether a matrix, which is explicitly constructed by the system's parameters, is an $M$-matrix. Comparison with the tests obtained by K. Gopalsamy (2007) and B. Liu (2013) for BAM neural networks illustrates novelty of the stability theorems. Some open problems conclude the paper.

math.DS↗

arXiv2012

Structured stability radii and exponential stability tests for Volterra difference systems

Uniform exponential (UE) stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space. The state space $\X$ is supposed to be an abstract Banach space. We work both with non-fading phase spaces $c_0 (\ZZ^-, \X)$ and $\ll^\infty (\ZZ^-, \X)$ and with exponentially fading phase spaces of the $\ll^p$ and $c_0$ types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform $\wh K (ζ)$ of the convolution kernel $K (\cdot)$, in terms of the input-state operator and of the resolvent (fundamental) matrix. These criteria do not impose additional positivity or compactness assumptions on coefficients $K(j)$. Time-varying (non-convolution) difference equations are studied via structured UE stability radii $\r_\t$ of convolution equations. These radii correspond to a feedback scheme with delayed output and time-varying disturbances. We also consider stability radii $\r_\c$ associated with a time-invariant disturbance operator, unstructured stability radii, and stability radii corresponding to delayed feedback. For all these types of stability radii two-sided estimates are obtained. The estimates from above are given in terms of the Z-transform $\wh K (ζ)$, the estimate from below via the norm of the input-output operator. These estimates turn into explicit formulae if the state space $\X$ is Hilbert or if disturbances are time-invariant. The results on stability radii are applied to obtain various exponential stability tests for non-convolution equations. Several examples are provided.

math.DS↗

arXiv2010

Bohl-Perron type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) $ł^p$-input $ł^q$-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to $ł^q$ when non-homogeneous terms are in $ł^p$. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted $ł^r$-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and $ł^p$-input $ł^q$-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. $ł^1$-input $ł^\infty$-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.

math.DS↗