Computerized formal solutions of dynamical systems with two degrees of freedom and an application to the contopoulos potential—I : The exact resonance case

We develop the theory of the Lindstedt method for the solution of two-degree-of-freedom perturbed oscillators. We assume here a one-to-one resonance between the two basic frequencies of the oscillators. The Lindstedt solutions are developed in the neighborhood of the origin which is assumed to be a stable equilibrium point of the system.

We are able to develop the solutions to a fairly high order (eight) in the small parameter ε thanks to a systematic computerization of all the lengthy literal algebraic developments which are involved in such a project. The present article is the first of a series which aims at the application of the Lindstedt method in problems of non-linear oscillations with several degrees of freedom. We especially want to construct general algorithms and recurrence relations that can be implemented with a package of Poisson series manipulation programs.

The present formulation of the Lindstedt method also gives precise information about the existence and the exact number of solutions near an equilibrium point because certain algebraic conditions between the coefficients of the series must be satisfied. The detailed interpretation of these conditions allows an exact denumeration of all the distinct periodic modes of oscillation.

In Section 2 of the article, we first give the theory of the Lindstedt method, followed in Section 3 by a detailed application to a famous dynamical system: the Contopoulos system. We show, with the use of the second-order Lindstedt method, that the Contopoulos system has exactly six families of periodic solutions near the equilibrium at the origin. Four of these families are generally stable, while two of them are unstable. In Sections 4–7. these six families are discussed in detail, as well as their formal series expansions, the numerical verifications and the stability properties.     

Numerical experiments on the statistical behaviour of dynamical systems with a few degrees of freedom

Numerical experiments concerning the arising statistical laws in a system of nonlinear interacting waves are described. A chain of particles coupled by nonlinear springs has been used as a model. Various statistical properties of the chain have been investigated numerically. A comparison with the theory of the Korteweg-de Vries equation is given.     

Moduli and canonical forms for linear dynamical systems II: The topological case

In this paper we study real linear dynamical systems \(\dot x = Fx + Gu,y = Hx,x \in R^n \) = state space,u ∈ R ^m = input space,y ∈ R ^p = output space, under the equivalence relation induced by base change in state space; or in other words we study triples of matrices with real coefficients (F, G, H) of sizesn × n, n × m, p × n respectively, under the action(F, G, H.) →(TFT ^?1,TG, HT ^?1) ofGL _n (R), the group of invertible realn × n matrices. One of the central questions studied is: “do there exist continuous canonical forms for this equivalence relation?”. After various trivial obstructions to the existence of such forms have been removed the answer is very roughly: no ifm ≥ 2, p ≥ 2, yes ifm = 1, orp = 1. For a precise statement cf. theorem 1.7. Existence or nonexistence of continuous canonical forms is related to the existence of a universal family of real linear dynamical systems. More precisely continuous canonical forms exist if such a universal family exists and if the underlying vector bundle of this family is the trivial vector bundle. In the case studied we show that a universal family in the appropriate sense does exist. The methods used are purely (differential) topological and in particular do not involve any algebraic geometry. There is a corresponding algebraic theory over any fieldk instead ofR which is the subject of part III of this series of papers.     

Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections

In this paper, we define solutions for hybrid systems with prespecified hybrid inputs. Unlike previous work where solutions and inputs are assumed to be defined on the same domain a priori, we consider the case where intervals of flow and jump times of the input are not necessarily synchronized with those of the state trajectory. This happens in particular when the input is the output of another hybrid system, for instance, in the context of observer design or reference tracking. The proposed approach relies on reparametrizing the jumps of the input in order to write it on a common domain. The solutions then consist of a pair made of the state trajectory and the reparametrized input. Our definition generalizes the notions of solutions of continuous‐time and discrete‐time systems with inputs. We provide an algorithm that automatically performs the construction of solutions for a given hybrid input. In the context of hybrid interconnections, we show how the solutions of the individual systems can be linked to the solutions of a closed‐loop system. Example illustrate the notions and the proposed algorithm.     

Towards an integrated formal method for verification of liveness properties in distributed systems: with application to population protocols

State-based formal methods [e.g. Event-B/RODIN (Abrial in Modeling in Event-B—system and software engineering. Cambridge University Press, Cambridge, 2010; Abrial et al. in Int J Softw Tools Technol Transf (STTT) 12(6):447–466, 2010)] for critical system development and verification are now well established, with track records including tool support and industrial applications. The focus of proof-based verification, in particular, is on safety properties. Liveness properties, which guarantee eventual, or converging computations of some requirements, are less well dealt with. Inductive reasoning about liveness is not explicitly supported. Liveness proofs are often complex and expensive, requiring high-skill levels on the part of the verification engineer. Fairness-based temporal logic approaches have been proposed to address this, e.g. TLA Lamport (ACM Trans Program Lang Syst 16(3):872–923, 1994) and that of Manna and Pnueli (Temporal verification of reactive systems—safety. Springer, New York, 1995). We contribute to this technology need by proposing a fairness-based method integrating temporal and first-order logic, proof and tools for modelling and verification of safety and liveness properties. The method is based on an integration of Event-B and TLA. Building on our previous work (Méry and Poppleton in Integrated formal methods, 10th international conference, IFM 2013, Turku, Finland, pp 208–222, 2013. doi: 10.1007/978-3-642-38613-8_15), we present the method via three example population protocols Angluin et al. (Distrib Comput 18(4):235–253, 2006). These were proposed as a theoretical framework for computability reasoning about Wireless Sensor Network and Mobile Ad-Hoc Network algorithms. Our examples present typical liveness and convergence requirements. We prove convergence results for the examples by integrated modelling and proof with Event-B/RODIN and TLA. We exploit existing proof rules, define and apply three new proof rules; soundness proofs are also provided. During the process we observe certain repeating patterns in the proofs. These are easily identified and reused because of the explicit nature of the reasoning.     

Infinite horizon optimal control for nonlinear interconnected large‐scale dynamical systems with an application to optimal attitude control

This paper presents a novel extended modal series method for solving the infinite horizon optimal control problem of nonlinear interconnected large‐scale dynamic systems. In this method, the infinite horizon nonlinear large‐scale two‐point boundary value problem (TPBVP), derived from Pontryagin's maximum principle, is transformed into a sequence of linear time‐invariant TPBVPs. Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of a uniformly convergent series. Moreover, in special cases, the proposed procedure facilitates the application of parallel processing, which improves its computational efficiency. In this study, an iterative algorithm is also presented, which has a low computational complexity and a fast convergence rate. Just a few iterations are required to obtain a suboptimal trajectory‐control pair. Finally, effectiveness of the proposed approach is verified by solving the optimal attitude control problem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Mathematical model of an industrial robot with articulated configuration and six degrees of freedom

This paper analyses the load influence on a d.c. drive motor servo system for all six degrees of freedom of industrial robots with an articulated configuration. Through the analyses, the analytical equations of total inertia and gravitation moments of an industrial robot servo system for external coordinate function were obtained. A mathematical model of an electro-mechanical industrial robot system was also developed.     

Adaptive sliding mode control of chaotic dynamical systems with application to synchronization

We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results.     

Adaptive sliding mode control of chaotic dynamical systems with application to synchronization

We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results.     

Tracking control of a closed-chain five-bar robot with two degrees of freedom by integration of an approximation-based approach and mechanical design

The trajectory tracking problem of a closed-chain five-bar robot is studied in this paper. Based on an error transformation function and the backstepping technique, an approximation-based tracking algorithm is proposed, which can guarantee the control performance of the robotic system in both the stable and transient phases. In particular, the overshoot, settling time, and final tracking error of the robotic system can be all adjusted by properly setting the parameters in the error transformation function. The radial basis function neural network (RBFNN) is used to compensate the complicated nonlinear terms in the closed-loop dynamics of the robotic system. The approximation error of the RBFNN is only required to be bounded, which simplifies the initial "trail-and-error" configuration of the neural network. Illustrative examples are given to verify the theoretical analysis and illustrate the effectiveness of the proposed algorithm. Finally, it is also shown that the proposed approximation-based controller can be simplified by a smart mechanical design of the closed-chain robot, which demonstrates the promise of the integrated design and control philosophy.     

A simplified MMC model for the control of an arm with redundant degrees of freedom

The paper proposes a model for the control of a multisegmented arm with redundant degrees of freedom. On the basis of an earlier model, the so-called MMC net, a simplified version is proposed here which has several advantages. First, it can easily be scaled up for the 3D case. Second, for the linear version a complete convergence proof is possible. Third, the properties of the earlier model are unchanged, namely versatile control of the redundant system, immediate change from direct kinematics to inverse kinematics or any mixed control task, as well as robustness against singularities.     

Topology of the moduli space for reachable linear dynamical systems: The complex case

This paper deals with the algebraic topology of the space _n,m of complex reachable linear dynamical systems. Topological invariants such as the singular homology groups of _n,m are explicitly computed and they are shown to coincide with those of a certain Grassmann manifold. From this some new results on the topology of rational transfer matrices with fixed McMillan degree are obtained.     

Data dependent systems technique: a methodology with an application to human systems

The objective of this paper is to introduce the application of Data Dependent Systems (DDS) methodology to the field of ergonomics. Many current techniques in ergonomics utilize static models, which can have significant limitations. DDS is a stochastic modelling and analysis technique that can be used to capture the dynamics of a system through quantitative analysis of the available data. DDS has been successfully applied to the analysis of manufacturing processes and the surfaces generated by those processes. In this research, DDS was used to analyse time-based hand-skin temperature data for the evaluation of two types of glove liners to be used underneath latex gloves. DDS was able to capture the differences between the two glove liners and the two subjects. The implications of the results and the potential of the DDS methodology are discussed.     

On the phase-space dynamics of systems of spiking neurons. II: formal analysis

We begin with a brief review of the abstract dynamical system that models systems of biological neurons, introduced in the original article. We then analyze the dynamics of the system. Formal analysis of local properties of flows reveals contraction, expansion, and folding in different sections of the phase-space. The criterion for the system, set up to model a typical neocortical column, to be sensitive to initial conditions is identified. Based on physiological parameters, we then deduce that periodic orbits in the region of the phase-space corresponding to normal operational conditions in the neocortex are almost surely (with probability 1) unstable, those in the region corresponding to seizure-like conditions are almost surely stable, and trajectories in the region corresponding to normal operational conditions are almost surely sensitive to initial conditions. Next, we present a procedure that isolates all basic sets, complex sets, and attractors incrementally. Based on the two sets of results, we conclude that chaotic attractors that are potentially anisotropic play a central role in the dynamics of such systems. Finally, we examine the impact of this result on the computational nature of neocortical neuronal systems.     

From a B formal specification to an executable code: application to the relational database domain

This paper presents a formal approach for the development of trustworthy database applications. This approach consists of three complementary steps. Designers start by modeling applications using UML diagrams dedicated to database applications domain. These diagrams are then automatically translated into B specifications suitable not only for reasoning about data integrity checking but also for the derivation of trustworthy implementations. In this paper, we present a process based on the B refinement technique for the derivation of a SQL relational implementation, embedded in the JAVA language (JAVA/SQL), from a B specification obtained by the first translation phase.     

The merging of two gaseous bubbles with an application to underwater explosions

A numerical study of two gaseous bubbles merging into a single coalesced bubble as in underwater explosions is investigated in this paper. This explosive phenomenon is modeled using a boundary integral method. Two configurations, which are in-phase and out-of-phase explosions, are simulated and compared with available experimental results. The thickness of the liquid film between the two bubbles determines our coalescence criterion. Bubble shapes and periods of oscillation are predicted well, compared to those of the experiments.