Discrete-continuous Dynamical Systems Research Articles

To the issue of decomposition of mathematical models of disturbed motion of complex discrete-continuous dynamic systems

Problem. Usually, complex dynamic systems are characterized by a high order of equations in their mathematical models and intricate interconnections among subsystems that constitute the complex dynamic system. Naturally, the analysis and synthesis of complex dynamic systems require the use of powerful computational systems with vast memory and high processing speed. Goal. The aim is quantitative assessment of the influence of the continuum part of a dynamic system on the behavior of its discrete part and the possibility to perform decomposition of the mathematical model of perturbed motion of a discrete-continuum system based on this assessment. Methodology. The measures are aimed at addressing issues that are based on the discrete-continuum capabilities and their decomposition. Results. For three types of complex dynamic systems described by infinite-dimensional systems of ordinary differential equations, quantitative assessments of the influence of the continuum part on the discrete part of the system have been proposed. These assessments determine the decomposition of the mathematical models of perturbed motion for such systems. Originality. Simplified mathematical models for three types of discrete-continuous dynamic systems were obtained. Practical value. The obtained results can be recommended for the study of the specialized course on the motion characteristics of cargo trucks during the transportation of liquids in tanks. Through optimization using the MATLAB software package, it is possible to simulate various parameters of both the cargo truck and the cargo being transported.

An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515)

<p style='text-indent:20px;'>This paper is an Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515).</p>

Lyapunov analysis of the spatially discrete-continuous system dynamics

The spatially discrete-continuous dynamical systems, that are composed of a spatially extended medium coupled with a set of lumped elements, are frequently met in different fields, ranging from electronics to multicellular structures in living systems. Due to the natural heterogeneity of such systems, the calculation of Lyapunov exponents for them appears to be a challenging task, since the conventional techniques in this case often become unreliable and inaccurate. The paper suggests an effective approach to calculate Lyapunov exponents for discrete-continuous dynamical systems, which we test in stability analysis of two representative models from different fields. Namely, we consider a mathematical model of a 1D transferred electron device coupled with a lumped resonant circuit, and a phenomenological neuronal model of spreading depolarization, which involves 2D diffusive medium. We demonstrate that the method proposed is able reliably recognize regular, chaotic and hyperchaotic dynamics in the systems under study.

Sufficient optimality conditions in hierarchical models of nonuniform systems

We give a brief survey of the studies developed on the basis of general Krotov's sufficient optimality conditions in an important research direction related to nonuniform systems. We construct a two-layered model of the network structure. The upper level of this model is an abstract network of operators; the lower level contains continuous dynamical models. For such a network, we pose an optimization problem and find general sufficient optimality conditions as generalizations of sufficient conditions for discrete-continuous dynamical systems. We survey possible applications of this direction and consider an example of optimizing nature preservation activities in a river basin.

Cosine effect on shallow water equations and mathematical properties

This paper presents a viscous Shallow Water type model with new Coriolis terms, and some limits according to the values of the Rossby and Froude numbers. We prove that the extension to the bidimensional case of the unidimensional results given by [J.–F. Gerbeau, B. Perthame. Discrete Continuous Dynamical Systems, (2001)] including the Coriolis force has to add new terms, omitted up to now, depending on the latitude cosine, when the viscosity is assumed to be of the order of the aspect ratio. We show that the expressions for the waves are modified, particularly at the equator, as well as the Quasi-Geostrophic and the Lake equations. To conclude, we also study the mathematical properties of these equations.

Linear coupled component automata for MILP modeling of hybrid systems

We first introduce a novel modeling framework, called linear coupled component automata (LCCA), to facilitate the modeling of discrete-continuous dynamical systems with piecewise constant derivatives. Second, we provide a procedure for transforming models in this framework to mixed-integer linear programming (MILP) constraints. Traditionally, such systems have been modeled directly with MILP constraints. We show with an example that our framework significantly simplifies model formulation and allows the complex MILP constraints to be produced systematically.

Nonmonotone systems decomposable into monotone systems with negative feedback

Motivated by the work of Angeli and Sontag [Monotone control systems, IEEE Trans. Automat. Control 48 (2003) 1684–1698] and Enciso and Sontag [On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems, Discrete Continuous Dynamical Systems, to appear] in control theory, we show that certain finite and infinite dimensional semi-dynamical systems with “negative feedback” can be decomposed into a monotone “open-loop” system with “inputs” and a decreasing “output” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé [A criterion of global convergence to equilibrium for differential systems with an application to Lotka–Volterra systems, Rapport de Recherche 894, INRIA] and Cosner [Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka–Volterra models, Dynam. Cont., Discrete Impulsive Systems 3 (1997) 283–303] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence.

Existence and nonexistence of time-global solutions to damped wave equation on half-line

On the half-line R + = ( 0 , ∞ ) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent ρ c ( N , k ) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space D N , k = R + k × R N - k is given by ρ c ( N , k ) = 1 + 2 / ( N + k ) (J. Appl. Math. Phys. 39 (1988) 135–149; Arch. Rational Mech. Anal. 109 (1990) 63–71). Since the damped wave equation is expected to be close to the heat equation (J. Differential Equations 191 (2003) 445–469; Math. Z. 244 (2003) 631–649), the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation. However, there is no blow-up result on the halved space for the damped wave equation. In this paper, the exponent ρ c ( 1 , 1 ) = 2 is shown to be critical for the existence and nonexistence of time-global solution to both the semilinear heat equation and damped wave equation on the half-line R + , together with the derivation of the blow-up time. For the proof the explicit formulas of solutions are used in a similar fashion to those in Li and Zhou (Discrete Continuous Dynamic Systems 1 (1995) 503–520).

Strong stability of elastic control systems with dissipative saturating feedback

We will consider, with a focus on saturating feedback control laws, two problems associated with damping in a bounded acoustic cavity Ω⊂ R 3 . Our objective is to verify (compare (Discrete Continuous Dynamical Systems 7 (2001) 319, Math. Control Signals Systems 2 (1989) 265) that these are strongly stable: for every finite-energy solution, the acoustic energy goes to zero as t→∞. We will, in each case, formulate the problem in terms of a contraction semigroup of nonlinear operators on an appropriate Hilbert space and compare this with the corresponding semigroups without saturation—following Avalos and Lasiecka (Semigroup Forum 57 (1998) 278) in using the spectral methods of Arendt and Batty (Trans. Amer. Math. Soc. 8 (1988) 837) to show strong stabilization for those linear semigroups.