Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions

We investigate a nonlinear autonomous parabolic partial differential equation in one space variable subject to Neumann boundary conditions on a compact interval. The object of our study is to determine the asymptotic behavior of solutions. Our methods are borrowed from the Liapunov theory of stability for dynamical systems. We give conditions under which a solution has a nonempty ω-limit set. We show that any such ω-limit set consists solely of equilibrium solutions. We render criteria for asymptotic stability and for instability of an equilibrium solution. We examine the possibility of escape behavior.     

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.     

Identification of Linear Error-Models with Projected Dynamical Systems

Linear error models are an integral part of several parameter identification methods for feedforward and feedback control systems and lead in connection with the L ₂-norm to a convex distance measure which has to be minimised for identification purposes. The parameters are hereby often subject to specific restrictions whose intersections span a convex solution set with non-differentiability points on its boundary. For solving these well conditioned problems on-line the paper formulates the solution of the bounded convex minimisation problem as a stable equilibrium set of a proper system of differential equations. The vector field of the corresponding system of differential equations is based on a projection of the negative gradient of the distance measure. A general drawback of this approach is the discontinuous right-hand side of the differential equation caused by the projection transformation. The consequence are difficulties for the verification of the existence, uniqueness and stability of a solution trajectory. Therefore the first subject of this paper is the derivation of an alternative formulation of the projected dynamical system, which exhibits, in contrast to the original formulation, a continuous right-hand side and is thus accessible to conventional analysis methods. For this purpose the multi-dimensional stop operator is used and the existence, uniqueness and stability properties of the solution trajectories are established. The second part of this paper deals with the numerical integration of the projected dynamical system which is used for an implementation of the identification method on a digital signal processor for example. To demonstrate the performance the application of this on-line identification method to the hysteretic filter synthesis with the modified Prandtl-Ishlinskii approach is presented in the last part of this paper.     

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

Exponential and global stability of nonlinear dynamical systems relative to initial time difference

The exponential and global stability of nonlinear differential dynamical systems with different initial times are investigated. Several criteria for the stability of nonlinear dynamical systems relative to initial time difference are obtained by means of vector Lyapunov functions. The obtained criteria have been applied to a proposed differential dynamic system. The numerical simulation validates our conclusions.     

Grazing bifurcation and chaotic oscillations of vibro-impact systems with one degree of freedom

The bifurcations of dynamical systems, described by a second-order differential equation with periodic coefficients and an impact condition, are investigated. It is shown that a continuous change in the coefficients of the system, during which the number of impacts of the periodic solution increases, leads to the occurrence of a chaotic invariant set.     

Differential equations and solution of linear systems

Many iterative processes can be interpreted as discrete dynamical systems and, in certain cases, they correspond to a time discretization of differential systems. In this paper, we propose to derive iterative schemes for solving linear systems of equations by modeling the problem to solve as a stable state of a proper differential system; the solution of the original linear problem is then computed numerically by applying a time marching scheme. We discuss some aspects of this approach, which allows to recover some known methods but also to introduce new ones. We give convergence results and numerical illustrations. AMS subject classification 65F10, 65F35, 65L05, 65L12, 65L20, 65N06     

Stability of interconnected stochastic delay systems

A decomposition of a large-scale stochastic functional differential system into a family of isolated subsystems is made. A set of criteria is obtained under which the uniform asymptotic stability with probability one of the trivial solution process of the large-scale system is preserved. These criteria are expressed in terms of the stability of isolated subsystems and bounds on interconnection components. As a tool of our analysis, Razumikhin-type stability conditions for stochastic functional differential systems are also obtained. Several examples are given to show the applicability of our results.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator–prey model with mutual interference and Beddington–DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set‐valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Computed torque control of an under-actuated service robot platform modeled by natural coordinates

The paper investigates the motion planning of a suspended service robot platform equipped with ducted fan actuators. The platform consists of an RRT robot and a cable suspended swinging actuator that form a subsequent parallel kinematic chain and it is equipped with ducted fan actuators. In spite of the complementary ducted fan actuators, the system is under-actuated. The method of computed torques is applied to control the motion of the robot.The under-actuated systems have less control inputs than degrees of freedom. We assume that the investigated under-actuated system has desired outputs of the same number as inputs. In spite of the fact that the inverse dynamical calculation leads to the solution of a system of differential–algebraic equations (DAE), the desired control inputs can be determined uniquely by the method of computed torques.We use natural (Cartesian) coordinates to describe the configuration of the robot, while a set of algebraic equations represents the geometric constraints. In this modeling approach the mathematical model of the dynamical system itself is also a DAE.The paper discusses the inverse dynamics problem of the complex hybrid robotic system. The results include the desired actuator forces as well as the nominal coordinates corresponding to the desired motion of the carried payload. The method of computed torque control with a PD controller is applied to under-actuated systems described by natural coordinates, while the inverse dynamics is solved via the backward Euler discretization of the DAE system for which a general formalism is proposed. The results are compared with the closed form results obtained by simplified models of the system. Numerical simulation and experiments demonstrate the applicability of the presented concepts.     

Stability and boundedness criteria for nonlinear differential systems relative to initial time difference and applications

This paper establishes several criteria on stability and boundedness for nonlinear differential dynamical systems relative to initial time difference by employing a new comparison principle. The reported novel results complement the existing results. As an application, these results are applied to a nonlinear dynamical system to obtain the stability properties.     

ON A MULTI-DELAY LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH FEEDBACK CONTROLS AND PREY DIFFUSION

This article is focusing on a class of multi-delay predator-prey model with feedback controls and prey diffusion. By developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the globally attractivity of positive solution for the predator-prey system.Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Approach to the stabilization of unstable periodic solutions of autonomous systems of partial differential equations

We suggest an approach to the stabilization of unstable periodic solutions of autonomous systems of partial differential equations based on the introduction of a derivative system in which each periodic solution of the original system is stationary. By using the introduction of an additional space into the derivative system, we suggest to stabilize its stationary solution corresponding to a periodic solution of the original system. This approach permits effectively obtaining a complete ordered set of functions corresponding to an unstable cycle of the original system. We consider an example of stabilization of an unstable cycle in the Kuramoto-Tsuzuki system.     

小参数时变非线性系统的技术稳定性

分析一类含小参数的时变非线性系统关于给定状态约束集合的技术稳定性.根据向量微分比较原理和基本的单调性准则,利用向量V函数方法给出由系统系数表达的技术稳定性判据.并讨论了基于派生系统和线性化方法研究非线性系统技术稳定性的条件.另外,对于派生时变线性系统的指数稳定性给出了简单的代数判据.最后给出示例说明文中方法.     

Global exponential stability of almost periodic solution for a large class of delayed dynamical systems

Research on delayed neural networks with variable self-inhibitions, interconnection weights, and inputs is an important issue. In this paper, we discuss a large class of delayed dynamical systems with almost periodic self-inhibitions, inter-connection weights, and inputs. This model is universal and includes delayed systems with time-varying delays, distributed delays as well as combination of both. We prove that under some mild conditions, the system has a unique almost periodic solution, which is globally exponentially stable. We propose a new approach, which is independent of existing theory concerning with existence of almost periodic solution for dynamical systems.