On the Recent Progress in Effective Dimension Estimates for Delay Equations

The main aim of this paper is to study the stability problem for neutral-type Hopfield neural networks possessing multiple time delays in the states of the neurons and multiple neutral delays in time derivative of states of the neurons. By using a suitable Lyapunov functional, a novel sufficient stability condition is obtained for global asymptotic stability of neutral-type neural networks with multiple delays. The derived stability criterion can be expressed in terms of the parameters of the neural network model which totally relies on some simple relationships established between the network parameters and it is completely independent of time delays and neutral delays. Hence, this new global asymptotic stability condition can be easiliy tested and verified by using some algebraic mathematical properties. We will also make a comparision between the result of this paper and previously published corresponding results. This comparision will indicate the advanteges of our proposed stability condition over the previously reported stability conditions. Since obtaining stability conditions for neutral type neural networks with multiple delays is a difficult task to achieve due to the insufficient mathematical methods and techniques, the result given in this paper can be considered an important and alternative result for this class of neutral type neural systems.

New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ, is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope (t0−t)−5/2, with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.

Today, the influence of dynamic changes in geopolitical and geo-economics’ conditions requires new requirements to ensure the national security of the state and its sustainable development. One of the central places in the structure of the national security of Ukraine is its economic security, which guarantees the independence of the country, the conditions of stability and effective functioning of society. In this regard, tracking of trends in the economic sphere, detection of the main risks and threats to economic security, identification of measures dealing with reduction of their negative impact are relevant both from a scientific and practical point of view. This article considers the key threats to the economic security of the state in the context of globalization, the long-term influence of which reduces the economic security of the state, makes it vulnerable to external challenges, and significantly weakens its position on the world stage.The importance of ensuring the state economic security and counteracting the negative impact of threats that may harm it has been proved. It has been noted that ensuring of economic security is a priority of the state. The essence of economic security of the state and its importance for national security in general has been described. The current legislative documents, which identify the main real and potential threats to the national security of Ukraine at the present stage, including in the economic sphere have been considered. The interference of threats to the state economic security, which are divided according to the scope of their occurrence into internal and external, has been assessed. It has been determined that external threats are associated with adverse processes in the world economy, and internal threats are dealing with the activities of public authorities, macroeconomic and microeconomic instability of the market economy, they are stipulated to those processes that occur during economic policy. It has been noted that the current conditions for the development of the state security, due to new challenges and threats, need to improve and implement new measures to combat the negative influence of major threats to the national economic security. As a result of the study, the main measures to counter the main threats to the state economic security were identified.

Radio frequency plasma enhanced chemical vapor deposition (rf-PECVD) is an efficient technique for preparing hydrogenated amorphous silicon (a-Si:H) layers used in thin film silicon solar cells. The most important parameters in a PECVD system are the chamber pressure, substrate temperature, partial gas flow, plasma power, electrode spacing, and deposition time, by which the physical and electrical parameters of the deposited layers, such as phase, quality, thickness, doping concentration, energy gap, defect density, and mobility can be controlled. Moreover, some of the film parameters are indirectly related to the others. As a result, it is always difficult to describe the relations between the deposition parameters and the specifications of the deposited layer in closed-form equations, therefore, prediction of the properties of the deposited films remains an issue. In addition, the helpful facilities for in-situ monitoring of the film under deposition are very limited. In this research, based on a broad experimental data reported in the literature, it has been found that in many cases the performance of a PECVD system can be described in equation or graph forms, and effective estimations may be established for prediction of its performance. A flowchart is also presented to explain the steps required for adjusting the input parameters of the system for the fabrication of a desired layer, which can reduce the complexity of interrelations among the variables, leading to a more accurate and flexible process design.

Multicriteria optimization of cold-formed thin-walled beams with generalized open shape under different loads

The essential control characteristics for a two and three-loop acceleration autopilot of tactical missile were presented from a frequency analysis perspective. The basic limitation of right-half-phase pole and zero from airframe aerodynamics was analyzed using an open-loop crossover frequency inequality, which showed a consistent conclusion with the final overall stability condition in nature. To reveal the feedback control capability, the rate feedback, rate plus attitude feedback, and linear acceleration feedback, were calculated and simplified based on engineering requirement, respectively. Then an equivalent actuator dynamics were introduced to compute the stability condition at high frequency. Such that the global stability condition was obtained in explicit formulation. The results showed that the autopilot control capability is dominated by actuator bandwidth, and a compromise should be determined between the flight performance and the actuator requirement for a statically instable missile.

The dynamics of food consumption by higher trophic level is complicated as its availability, choice, and consequently their development. The top predator interference has a strong influence on the dynamic structure of the tritrophic food chain system. The proposed article deals with the characteristic features of a three‐tire modified Leslie–Gower model with a one‐prey two‐predator system with the assumption of multiple gestation delays. The intermediate predator of the food chain is taken as a specialist type, whereas the growth of top predator is assumed by sexual reproduction. Two different response functions named Monod–Haldane (M‐H) and Beddington–DeAngelis (B‐D) are considered for the modeling of system dynamics. The population densities of intermediate and top predators are assumed to be affected by two different gestation delays. The boundedness and positive equilibria and their stability conditions are examined theoretically for the delay‐free system. The local and global stability conditions of the delayed system are also determined. With the help of normal form theory and central manifold arguments, the properties of bifurcating periodic solutions are carried out. The numerical computation is performed to validate our analytical findings that show the different dynamical outcomes such as periodic and chaotic dynamics. The Hopf‐bifurcation scenario for different parameters of the model system is well studied. Further, the stability behavior of multiple delays for the different cases is investigated. The system shows chaotic behavior when the gestation delay of intermediate predators is large enough. It is also observed that the top predator density is highly fluctuating through the creation of chaos by the inclusion of time delay. Finally, the time delay destabilizing effect is noticed for three‐way interactions among prey, intermediate predator, and top predator, highlighting its significance in chaotic dynamics.

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

In this paper, we study the global dynamic behavior of a non-autonomous delayed Nicholson's blowflies model with a linear harvesting term. Under proper con- ditions, we employ a novel argument to establish a criterion on the global exponential stability of positive almost periodic solutions for the model. Moreover, we also provide a numerical example to support the theoretical results.

Generalized Aizerman–Myshkis problem for abstract differential-delay equations

New LMI-based conditions for global exponential stability to a class of Cohen–Grossberg BAM networks with delays

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane. The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species. The first Manuscript was on the study of the following equation: [Mathematical equations can not be displayed here, refer to PDF] Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation: [Mathematical equations can not be displayed here, refer to PDF] The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the di erence equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium. In my second Manuscript, I considered the difference equation: [Mathematical equations can not be displayed here, refer to PDF] where C;D and F are positive numbers and the initial conditions x-1 and x0 are non-negative numbers. Equation (2) which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium. The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts: Theorem

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all work conserving policies.However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all work conserving policies, then any corresponding queueing network is not rate stable under the class of all work conserving policies. We establish the result by building a particular work conserving scheduling policy which makes any corresponding stochastic process transient. An important corollary of our result is that the condition of the form ρ* < 1, which was proven in [7) to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for any corresponding queueing network. Here ρ* is a certain computable parameter of the network involving virtual station and push start conditions.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies.\n¶\nHowever, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ<sup>*</sup>≤1, which was proven in [Oper. Res. 48 (2000) 721–744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ<sup>*</sup> is a certain computable parameter of the network involving virtual station and push start conditions.

[1] We study the nature and stability of low-frequency waves in collisionless, gyrotropic (P⊥ ≠ P∥), multi-component plasmas, characterized by arbitrary distribution functions. The ambient medium is assumed to be stratified perpendicular to the magnetic field, with a scale height much larger than the wavelength of our waves, and the whole system is allowed to rotate. We show that the waves are generally non-adiabatic, and we identify the four physical causes of their non-adiabatic behavior. We also derive a global necessary condition for stability, which, quite remarkably, consists of the simple superposition of a criterion against instabilities triggered by thermal pressure anisotropies and a criterion against instabilities triggered by stratification. In the case of a nonrotating bi-Maxwellian plasma, our stability condition turns out to be not only necessary, but also sufficient.