Behavior of Trajectories of Time-Invariant Systems

Finitely many embedded localizing sets are constructed for invariant compact sets of a time-invariant differential system. These localizing sets are used to divide the state space into three subsets, the least localizing set and two sets called sets of the first kind and the second kind. We prove that the trajectory passing through a point of the set of the first kind remains in this set and tends to infinity. For a trajectory passing through a point of the set of the second kind, there are three possible types of behavior: it either goes to infinity or, at some finite time, enters the least localizing set, or has a nonempty ω-limit set contained in the intersection of the boundary of one of the constructed localizing sets with the universal section of the corresponding localizing function.     

Nonlinear dynamic analysis of a hybrid squeeze-film damper-mounted rigid rotor lubricated with couple stress fluid and active control

The hybrid squeeze-film damper bearing with active control is proposed in this paper and the lubricating with couple stress fluid is also taken into consideration. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincaré maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the non-dimensional speed ratio s = 3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action. Besides, the rotor dynamic responses of the system will be more stable by using couple stress fluid.     

Analysis of a second-order hybrid system with linear structure

The paper is devoted to the investigation of the behavior of trajectories of a hybrid system on a plane consisting of a set of two-dimensional systems of linear differential equations and a discrete component—a rule for jumping from one system to another one when the trajectory of the system reaches the boundary of the domain of its consideration. The notion of quasiperiodic and asymptotically quasiperiodic trajectory is introduced; the structure of the set of initial positions that quasiperiodic or asymptotically quasiperiodic trajectories can be issued from is investigated. The Zeno behavior (an infinite number of jumps between systems of differential equations in a finite amount of time) is studied; sufficient conditions for a trajectory not to exhibit this type of behavior are formulated.     

Simulation and feed-forward control of a flexible parallel manipulator

The importance of lightweight constructions are steady increasing since they promise a low energy consumption together with higher movement speeds. However these demand modern, model-based feed-forward control designs. Especially the undesired vibrations due to the reduced overall stiffness of such manipulators have to be taken into account. A convenient way to model the dynamical behavior of systems that perform large, nonlinear motions superposed with small, elastic deformations is the floating frame of reference approach in a flexible multibody system. The application of the Newton-Euler-Formalism together with D'Alembert's principle to parallel manipulators results in a set of differential-algebraic equations. Therefore, the consideration of the trajectory tracking problem with so-called servo constraints appears to be promising. In case of a non-flat system, the arising set of differential-algebraic equations, which consists of the system dynamics, the holonomic loop closing constraint equations and the servo constraints embodies nontrivial dynamics. With an oblique projection, the embedded set of ordinary differential equations describing the internal dynamics can be obtained. The stability properties of these dynamics determines the complexity of the feed-forward control design, as two-point boundary value problems might have to be solved. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Feedback Integrators

A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves the first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge–Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and Störmer–Verlet schemes.     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

Linearly Constrained Global Optimization and Stochastic Differential Equations

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajectory is achieved by projecting its dynamics onto the set defined by the linear equality constraints. A barrier term is used for the purpose of forcing the trajectory to stay within the box constraints. Using Laplace’s method we give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the projected diffusion process and show that its weak limit is given by Π. Numerical experiments using standard test problems from the literature are reported. Our results suggest that the method is robust and applicable to large-scale problems.     

SIR DYNAMICS WITH ECONOMICALLY DRIVEN CONTACT RATES

The susceptible‐infected‐recovered (SIR) model has greatly evidenced epidemiology despite its apparent simplicity. Most applications of the SIR framework use a form of nonlinear incidence to describe the number of new cases per instant. We adapt theorems to analyze the stability of SIR models with a generalized nonlinear incidence structure. These theorems are then applied to the case of standard incidence and incidence resulting from adaptive behavioral response based on epidemiological‐economic theory. When adaptive behavior is included in the SIR model multiple equilibria and oscillatory epidemiological dynamics can occur over a greater parameter space. Our analysis, based on the epidemiological‐economic incidence, provides new insights into epidemics as complex adaptive systems, highlights important nonlinearities that lead to complex behavior, and provides mechanistic motivation for a shift away from standard incidence, and outlines important areas of research related to the complex‐adaptive dynamics of epidemics.     

混沌记忆系统的自适应反馈控制和轨迹跟踪控制

研究了混沌记忆系统的自适应反馈控制和基于反馈线性化的轨迹跟踪控制问题.首先,通过绘制系统的时域波形图和混沌吸引子图验证系统的复杂的动力学行为;然后,分别应用自适应反馈控制方法和基于反馈线性化的轨迹跟踪控制方法设计控制器,对系统施加控制;最后,通过数值仿真验证控制器的有效性.     

On the solution of the trajectory survival problem for a nonlinear dynamical system

Nonlinear control systems possessing the flatness property are encountered in many applied mathematical models. In this paper, a trajectory survival problem is considered for a specific nonlinear system that possesses the above property. A method based on the properties of the system is proposed for constructing a control that solves the trajectory survival problem when the controlled object moves to the goal set within a bounding set containing an obstacle. Results of numerical calculations of the control and the trajectory of a system with a given initial position are presented.     

THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS

Abstract

In this paper we study the welldefinedness of the central path associated to a nonlinear convex semidefinite programming problem with smooth objective and constraint functions. Under standard assumptions, we prove that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point. The monotonic behavior of the primal and dual logarithmic barriers and of the primal and dual objective functions along the trajectory is also discussed. The existence and optimality of cluster points is established and finally, under the additional assumption of analyticity of the data functions, the convergence of the primal-dual trajectory is proved.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Tracking within a time interval on the basis of data supplied by finite observers

We solve the tracking control problem, in which one should bring a trajectory of a system of linear ordinary differential equations into a neighborhood of a trajectory of another system within a given time interval. After getting into this neighborhood, one should keep the trajectory of the first subsystem in it for a time interval of given duration. For the control synthesis, we use incomplete and imprecise information on the online deviation of one trajectory from the other, which is obtained in real time from linear equations of observation. We consider distinct structures of observers, which substantially affect the solution of control problems for such systems. The equations of dynamics and admissible measurements contain uncertainty for which one knows only some hard pointwise constraints. To solve the main problem, we use an approach that can be reduced to the construction of auxiliary information sets and weakly invariant sets with a subsequent “aiming” of one set at a tube. We suggest an efficient method for an approximate solution on the basis of ellipsoidal calculus techniques. The results of the algorithm operation are illustrated by an example of the solution of a tracking control problem for two fourth-order subsystems.     

Scaling dynamics for solvable coagulation equations with dust and gel

We study limiting behavior of rescaled size distributions that evolve by Smoluchowski's rate equations for coagulation, with rate kernel K=2, x+y or xy. We find that the dynamics naturally extend to probability distributions on the half-line with zero and infinity appended, representing populations of clusters of zero and infinite size. The “scaling attractor” (set of subsequential limits) is compact and has a Levy-Khintchine-type representation that linearizes the dynamics and allows one to establish several signatures of chaos. In particular, for any given solution trajectory, there is a dense family of initial distributions (with the same initial tail) that yield scaling trajectories that shadow the given one for all time. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy guaranteed cost controller for trajectory tracking in nonlinear systems

In this paper, we propose a fuzzy logic based guaranteed cost controller for trajectory tracking in nonlinear systems. Takagi–Sugeno (T–S) fuzzy model is used to represent the dynamics of a nonlinear system and the controller design is carried out using this fuzzy model. State feedback law is used for building the fuzzy controller whose performance is evaluated using a quadratic cost function. For designing the fuzzy logic based controller which satisfies guaranteed performance, linear matrix inequality (LMI) approach is used. Sufficient conditions are derived in terms of matrix inequalities for minimizing the performance function of the controller. The performance function minimization problem with polynomial matrix inequalities is then transformed into a problem of minimizing a convex performance function involving standard LMIs. This minimization problem can be solved easily and efficiently using the LMI optimization techniques. Our controller design method also ensures that the closed-loop system is asymptotically stable. Simulation study is carried out on a two-link robotic manipulator tracking a reference trajectory. From the results of the simulation study, it is observed that our proposed controller tracks the reference trajectory closely while maintaining a guaranteed minimum cost.     

Deriving Behavior of Boolean Bioregulatory Networks from Subnetwork Dynamics

In the well-known discrete modeling framework developed by R. Thomas, the structure of a biological regulatory network is captured in an interaction graph, which, together with a set of Boolean parameters, gives rise to a state transition graph describing all possible dynamical behaviors. For complex networks the analysis of the dynamics becomes more and more difficult, and efficient methods to carry out the analysis are needed. In this paper, we focus on identifying subnetworks of the system that govern the behavior of the system as a whole. We present methods to derive trajectories and attractors of the network from the dynamics suitable subnetworks display in isolation. In addition, we use these ideas to link the existence of certain structural motifs, namely circuits, in the interaction graph to the character and number of attractors in the state transition graph, generalizing and refining results presented in [10]. Lastly, we show for a specific class of networks that all possible asymptotic behaviors of networks in that class can be derived from the dynamics of easily identifiable subnetworks.      

Trajectory stabilization of a model car via fuzzy control

This paper deals with trajectory stabilization of a computer simulated model car via fuzzy control. Stability conditions of fuzzy systems are given in accordance with the definition of stability in the sense of Lyapunov. First, we approximate a computer simulated model car, whose dynamics is nonlinear, by T-S (Takagi and Sugeno) fuzzy model. Fuzzy control rules, which guarantee stability of the control system under a condition, are derived from the approximated fuzzy model. The simulation results show that the fuzzy control rules effectively realize trajectory stabilization of the model car along a given reference trajectory from all initial positions under a condition and the dynamics of the approximated fuzzy model agrees well with that of the model car.     

Emulating complex business cycles by using an electronic analogue

Long run growth of the US national economic system, for example, reveals a strong oscillatory behavior due to complex interactions of aggregates. However, modelizations of such dynamics often assume that instability is the outcome of linear and additive cycles determined by exogenous shocks. In this work, a modelization of endogenous nonlinear and inseparable cycles is retained to explain the highly complex business cycle phenomenon. Bouali’s system is built to this scope. Its numerical simulations exhibit a rich repertoire of nonlinear dynamical phenomena, but this paper introduces its electronic implementation. The robust plug and play chaotic circuit is designed to be easily realized using standard components in a rigorous, fast and inexpensive way. We find that experimental results display periodicity, bifurcations and chaos that match with high accuracy the corresponding theoretical values.     

Barbashin and krasovskii’s asymptotic stability theorem in application to control systems on smooth manifolds

We introduce the notion of so-called standard control system, whose phase space is a finite-dimensional smooth manifold satisfying a number of conditions; in particular, it is supposed to be connected, orientable, and having a countable atlas. For a given standard control system, we consider a set of time translations and construct the closure of this set in the topology of uniform convergence on compact sets. In these terms, we study the conditions of uniform local reachability of a given trajectory. The main result is formulated in terms of a modified Lyapunov function. A simple example is considered.