Dynamical Systems in Theoretical Perspective

Ordinary differential equations arise in a variety of applications, including climate modeling, electronics, predator-prey modeling, etc., and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system’s solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. We present significant improvements of an algorithm recently suggested by the authors to compute complete Lyapunov functions. Previously this methodology was incapable to fully detect chain-recurrent sets in dynamical systems with high differences in speed. In the new approach we replace the system under consideration with another one having the same solution trajectories but such that they are traversed at a more uniform speed. The qualitative properties of the new system such as attractors and repellers are the same as for the original one. This approach gives a better approximation to the chain-recurrent set of the system under study.

Based on the nonloocal elasticity theory, this paper deals with the dynamic instability analysis of cantilevered single-walled carbon nanotube with concentrated mass, located at a generic position, and subject to a follower force at the free end. Accounting for the small scale effect, the governing equations of motion are derived using an alternative Hamilton’s variational principle and the governing equations are solved numerically employing the Cell-Discretization Method (CDM) in which the nanotube is reduced to a set of rigid bars linked together by means of elastic constraints. The resulting discrete system takes into account nonlocal effects, added mass, and position of added mass, and follower force direction. A comparative analysis is performed in order to verify the accuracy and validity of the proposed numerical method. The effects of the nonlocal parameter and dimensionless mass on the dynamic instability of single-walled carbon nanotube are shown and discussed in details. The effect of a sub-tangential follower force on the stability of cantilever single-walled carbon nanotube is studied. Finally, the validity of the proposed analysis is confirmed by comparing the present results with those obtained from the litertaure and listed in bibliography.

This work presents a study of the mechanism of physical crosslinking of hyaluronic acid in the presence of common phospholipids in synovial joint organ systems. Molecular dynamic simulations have been executed to understand the formation of hyaluronan networks at various phospholipid concentrations. The results of the simulations suggest that the mechanisms exhibit subdiffusion characteristics. Transportation quantities derive as a function of time during numerical calculations of mean square displacement, and observations of sublinear growth were noted. Coarse-grained models are deployed to obtain a mathematical description where a random walker and several subdiffusion schemes of its motion describe the models. The findings of this study may establish mechanisms of biopolymer network formations in normal and pathologic synovial fluid and help elucidate the mechanism of facilitated AC biolubrication.

A mathematical solenoid is a geometric object which can be presented either in an abstract way as an inverse limit or in a geometric way as nested intersections of solid tori. In dynamical systems solenoids were introduced by Smale as hyperbolic attractors of a diffeomorphism of a three-dimensional manifold. The topological complexity of a solenoid can be expressed by a topological entropy which is equal to an upper capacity of some Carathéodory structure, in the sense of Pesin. We consider topological and measure-theoretical approach to dynamical properties of a solenoid and discuss homogeneous measures. In general case there is no invariant measure for a solenoid, therefore one can not say neither about measure-theoretical entropy nor about a measure of maximal entropy of a solenoid. We define $$\delta -$$ δ - measure-theoretical entropy of a solenoid, in sense of Katok, which is related to the topological entropy.

We present a generalization of results obtained by X. Mao in his book “Stochastic Differential Equations and Applications” (2008). When studying what Mao calls “almost sure exponential stability”, essentially a negative upper bound on the almost sure Lyapunov exponents, he works with Lyapunov functions that are twice continuously differentiable in the spatial variable and continuously differentiable in time. Mao gives sufficient conditions in terms of such a Lyapunov function for a solution of a stochastic differential equation to be almost surely exponentially stable. Further, he gives sufficient conditions of a similar kind for the solution to be almost surely exponentially unstable. Unfortunately, this class of Lyapunov functions is too restrictive. Indeed, R. Khasminskii showed in his book “Stochastic Stability of Differential Equations” (1979/2012) that even for an autonomous stochastic differential equation with constant coefficients, of which the solution is stochastically stable and such that the deterministic part has an unstable equilibrium, there cannot exists a Lyapunov function that is differentiable at the origin. These restrictions are inherited by Mao’s Lyapunov functions. We therefore consider Lyapunov functions that are not necessarily differentiable at the origin and we show that the sufficiency conditions Mao proves can be generalized to Lyapunov functions of this form.

The aim of the study is to analyze an isotropic plate in terms of its dynamic stability (or its instability), by applying tools that are mainly used in the vibrations theory of dynamical systems e.g. in the theory of bifurcation and chaos. The results achieved by using tools such as phase portraits, Poincaré maps, FFT analysis, the largest Lyapunov exponents were compared with the results obtained by using the Volmir criterion.

The paper studies dynamical behaviour of Jeffcott rotor supported by a hydrodynamic bearings. It uses different analytical formulations for hydrodynamic bearing forces acting on Jeffcott rotor. The model is nonlinear due to the presence of hydrodynamic bearings and can show different subharmonic behaviour like oil whip and oil whirl. Such a system is subjected to dynamical analysis using numerical continuation aimed at detection of nonlinear phenomena like bifurcations and unstable behaviours with respect to basic system parameters.

We introduce a fractional-order model for the coinfection of the immunodeficiency virus and tuberculosis, in the presence of drug resistant tuberculosis strains and treatment for both diseases. We compute the reproduction number of the model. Numerical simulations show the different dynamics of the model for variation of relevant parameters. Moreover, the order of the fractional derivative plays an important role in the severity of the epidemics.

As a result of classification of second order ordinary differential equations without movable branch points, a number of the so-called Painlevé equations was obtained. Among them, six irreducible equations are the best known. They led to the recognition of new functions, called the Painlevé transcendents. The Painlevé equations have numerous applications in modern mathematics and mathematical physics. The solutions of these equations, as they are meromorphic in the complex plane can be studied from the perspective of value distribution and growth theory, with such values as defect, deviation or multiplicity index estimated.

Some nonlinear and discrete optimal control problems with phase constraints on a fixed but sufficiently large time interval are considered as singularly perturbed problems. In continuous-time case the state equations are reduced to singularly perturbed equations on a finite time interval and, in discrete-time case, the state equations have the form of systems with a small step. Using the technique for singularly perturbed systems, the formal asymptotic expansions by the corresponding small parameter are constructed which contain the structural information about the solution. That is usually sufficient for most applications to obtain an initial approximation to control in the global optimum neighborhood. The obtained algorithms can be applied to mathematical economics and technical objects control problems with phase and control constraints, and with turnpike effects in the trajectories, where the turnpike trajectories can be discontinuous. The use of traditional algorithms for these problems is inefficient due to the large increase of computational difficulty.

Results of the analysis of dynamic impact effect for vehicle light alloy wheels of various types, which may occur in various road situations (head-on crash, drift, collision with another car) are given. This study applied to simulate the impact behavior caused by a dynamic loading of vehicle wheels by impact testing according to the scheme of certification tests with static and dynamic strain measurement for definition of deformation fields and impact stresses. New approach to creation of FEM model of virtual impact tests of wheels with use of program complex of nonlinear dynamics Ls-Dyna is developed and validation of models by comparison with results of dynamic strain-gaging is carried out.

This work presents the modeling and simulation of a manipulator robot with three degrees of freedom and considering its structures with rigid behavior. The concepts of kinematics for the mathematical deduction and the Lagrangian mechanics were used to obtain the dynamic models of the manipulator and the DC actuators with permanent magnet. Due to nonlinearity and dynamics characteristics, both the states observer and the control used were based on State Dependet Ricatti Equation (SDRE). The simulations made for constant performance parameters demonstrated the effectiveness of the optimal control applied to the manipulator and to the chosen DC actuator models. The applications of trajectories to the manipulator enrich the applicability of the project and the results obtained with the techniques chosen show his efficiency.

Two distinctive features of challenging control engineering problems are commonly taken into consideration in design of dynamical, mechatronics systems, namely operation ranges, of such systems with nonlinear effects, which are not always near to equilibrium states, as well as a fairly high level of uncertainties of their physical description with which controllers have to cope despite a lack of knowledge on the all system parameters although physical modeling allows to identify their particular nonlinear effects. It should be noted that usage of nonlinear physical modeling in real-time control systems can be computationally very demanding. Hence, it seems to be suitable to use robust control methods based on linearized models with adaptive updating algorithms. However, usually strong nonlinearities can reduce the effectiveness of control methods, and thus of adaptive control algorithms. The controller gains can be often updated by using the estimated parameters. In this contribution the adaptive control systems for automotive applications, which are based on indirect (or self-tuning) controller strategies are discussed. The modeling issue of indirect optimal controller strategies is illustrated by the application example.

Proposed model consisting of two coupled van der Pol equations is considered as a description of the heart’s action potential. System of ordinary differential equations with time delay is used to recreate pathological behaviour in the heart’s conducting system such as slow/fast and slow/slow type of atrioventricular nodal reentrant tachycardia (AVNRT). In our study, introducing the feedback loops and couplings entails the creation of waves which can correspond to the re-entry waves occurring in the AVNRT. Our main aim is to study solutions of the given equations and take into consideration the influence of feedback and delays which occur in these pathological modes. Analytical results are illustrated by some numerical examples of the model dynamics.

Motion of a small-scale Darrieus counter-rotating vertical axis wind turbine (VAWT) in a steady wind flow is studied. The system consists of two turbines that rotate in opposite directions. The shaft of the first turbine is rigidly joined to the rotor of a generator and the shaft of the second turbine is rigidly joined to the stator. A closed few-parametric mathematical model that takes into account the changeable electrical load in the local circuit of the generator is constructed. The corresponding dynamical system is a two-frequency system. In order to describe operating modes of the model, the system is averaged over two angles under the assumption that both frequencies are bounded away from zero. It is shown that passage through resonances has no crucial effect on the system behavior in the considered range of the parameters of the model.

The paper is focused on the problem of lack of the uniform method of CPS systems description. In this paper a new, task-oriented, method of designing the CPS systems is proposed. The method is based on the process approach and continuous improvement of the design. This new method should be considered as extremely versatile and useful in the design and construction of the CPSs. The proposed method of the description of CPSs is based on the assumption that the task to be realized by the system is nothing more than the goal of one or more processes that should be implemented in the system. Processes, in turn, are collections of activities carried out by resource groups (components) and the tasks performed by one resource (a single element of the CPS). Activities and tasks can be carried out both in series and in parallel. Division for operations and tasks allows you to use the description of the process at different levels of detail in accordance with the requirements of the design phase of the CPS.

The paper presents a dynamic model of hypothetical missile-artillery system mounted on a moving object (e.g. mobile platform or warship). Model inputs are driving torques for the azimuth and elevation angle, and the angular and linear displacements of the set base relative to the given stationary coordinate system. The output of the model is the resulting position of the line of sight relative to the mentioned stationary coordinate system. The effect of disturbances (motion) from the moving object was studied and the reverse dynamic analysis of the presented system under the influence of the disturbances was performed. This has been investigated to check if the set’s drive systems will be able to work out the required torque in time to maintain the desired line of sight. Simulations were made using the SciLab environment. Some results of numerical simulation tests was presented in graphical form.

Usually, as the input data of the parametric identification methods in the frequency domain, the corresponding pairs of the “unit harmonic force excitation”—“steady state harmonic response” are considered. This paper deals with approximate identification of linear dynamical systems by time response on unknown initial displacement (or velocity) with the help of the Fourier transform. In this paper, basic analytical relationships and identification alternatives are analyzed. Formulae are completed and presented with special consideration given to the simplest one mass dynamical system.

This paper is devoted to developing the control algorithms of semi-active systems of vehicle suspensions. In order to accomplish the goal, a vehicle was equipped with controlled magneto-rheological (MR) dampers. The model of a vehicle with the controlled suspension was developed. It was assumed that controlling the force in the suspension will be based on two criteria: ride comfort and safety. On the basis of the adopted evaluation criteria, the algorithm for controlling the damping force of vehicle vibrations. The mathematical vehicle model and the control algorithm helped develop a simulation programme. The coefficients adopted in the model were determined empirically. The numerical study of the vehicle model with the controlled suspension was conducted in the Matlab/Simulink programme. As a result of the performed work, the control algorithm was developed, taking into account two conflicting criteria of the drive comfort and safety. A vehicle with the controlled suspension can be driven in the conditions of maximal comfort and safety. A compromise solution was suggested, where a weight factor of the influence of individual control criteria is introduced. In this paper, some sample results of the numerical tests of a vehicle with the MR dampers are presented.

In this review paper we describe some consequences of the shadowing property for global and local aspects of dynamics. We will put additional emphasis on approximation of invariant measures by ergodic measures with additional properties of their supports (minimality, positive entropy, mixing).

Solving of a problem for systems subjected to random series of impulses is aimed at determining an approximate distribution of the strength of stochastic impulses forcing vibrations of an oscillator with damping. The difficulties that arose in connection with interpretation of experimental data forced us to search for a mathematical model, where algorithms were applied based on precise solutions. Under appropriate assumptions regarding random variables: the time of action of impulse and their strength, the deviation of the oscillator from its balanced position is a process which, in the limit as time tends to infinity, is stationary and ergodic. At the first stage of the simulation study discussed in this paper, classification of the elements of the structure of statistical series is necessary. The work was inspired by attempts at constructing a measuring device that would control granularity of the medium in a dust pipeline. The device had to signal appearance of big or small particles in excessive quantity in the transported dust.

Although the notion of an entropy has a global character, in many cases the value of an entropy depends on the behaviour of a function near some point. For that reason, in many papers various versions of a notion of “entropy point” are considered. We will examine properties and relations between full entropy points and focal entropy points. Moreover, we will introduce the notion of a $${\mathrm{full}}^*$$ full ∗ entropy point and unbalanced point and examine the possibility of graph approximation of some kind of functions by functions having either $${\mathrm{full}}^*$$ full ∗ entropy point or unbalanced point.

This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguish it from the other procedures for the problem. First of all a sliding mode is coped with differential–algebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. The second important feature is the calculation of cost and constraints functions gradients with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion. The third feature is the integration of the presented procedure with the Interactive Dynamic Optimization Server (IDOS) which is a computing environment dedicated to optimal control problems. IDOS user interface relies on Dynamic Optimization Modeling Language (DOML) which is an extension of Modelica language. In the paper we discuss the elements of DOML which help defining hybrid optimal control problems. The paper presents the application of the proposed procedure to an optimal control problem related to a mechanical system with dry friction.

Magnetic interactions are strongly non-linear, especially for small distances between magnets. Their implementation to the oscillator gives it the ability to display complex non-linear and chaotic behaviours. These phenomena under certain conditions can lead to widening of the vibration bandwidth of the system which in the case of energy harvesting systems increases their efficiency, especially under varying excitation conditions. In this paper we compare the numerical and experimental study of different systems of longitudinal magnetic oscillators with one and many degrees of freedom. Tested oscillator configurations differ in magnet parameters and system rigidity. In systems modelling we look for conditions in which the high-amplitude solutions occur over a wide frequency range. Predictions of models are next verified in experimental investigations.

Animal vibrissae are used as natural inspiration for artificial tactile sensors, e.g., the mystacial vibrissae enable rodents to perform several tasks in using these tactile hairs: object shape determination and surface texture discrimination. Referring to the literature, the Kinetic Signature Hypothesis states that the surface texture detection is a highly dynamic process. It is assumed that the animals gather information about the surface texture out of a spatial, temporal pattern of kinetic events. This process has to be analyzed in detail to develop an artificial tactile sensor with similar functionalities. Hence, we set up a mechanical model for theoretical investigations of the process. This model is analyzed in two different directions using numerical simulations: at first a quasi-static and then a fully dynamic description.

The use of compliant tensegrity structures in robotic applications offers several advantageous properties. In this work the dynamic behaviour of a planar tensegrity structure with multiple static equilibrium configurations is analysed, with respect to its further use in a two-finger-gripper application. In this application, two equilibrium configurations of the structure correspond to the opened and closed states of the gripper. The transition between these equilibrium configurations, caused by a proper selected actuation method, is essentially dependent on the actuation parameters and on the system parameters. To study the behaviour of the dynamic system and possible actuation methods, the nonlinear equations of motion are derived and transient dynamic analyses are performed. The movement behaviour is analysed in relation to the prestress of the structure and actuation parameters.

The aim of the paper is to study a synchronisation phenomenon as observed in a rotating structure consisting of three composite beams and a hub. The beams are made of eighteen carbon-epoxy prepreg material layers stacked in a specific sequence. In the performed analysis it is assumed one of the beams is de-tuned due to small misalignment of its reinforcing fibers orientation with regard to the two remaining nominal design blades. The non-classical effects like transverse shear, material anisotropy, non-uniform torsion and cross-section warping are taken into account in the mathematical model of the blades. The partial differential equations of motion of the structure are derived by the Hamilton principle; next the reduction to the ordinary differential ones is done by the Galerkin method. Finally, the equations are solved numerically and the resonance curves for the hub and the individual beams are plotted. In the performed studies two possible variants of the rotor excitation are considered: (a) driving torque expressed by a harmonic function or (b) torque given by a chaotic oscillator formula. The analysis of the synchronisation phenomenon of the hub and the blades motion is based on the study of the resonance curves and time histories in the prepared graphs. The analysis of the structure driven by chaotic oscillator revealed the existence of the strange chaotic attractor for every beam of the rotor; in the particular, nominal beams are fully synchronised, but the de-tuned one is synchronised with a small difference in amplitude.

The article discusses the most important stages of modelling the processes taking place in the BLDC motor working in the faulty state. The purpose of the analyses was to determine the relations between magnetically induced vibrations and mechanical damages, which can occur on a real object. Several chapters of this work focus on the successive stages of modelling. Vibrations created as a result of the fluctuation of the rotor magnetic field were analysed, as of the source of the field of the greatest magnitude, dominating inside the motor and playing the key role, from the point of view of the generated vibrations. The obtained results have been discussed and compared with the results obtained from the real object. Usefulness of the conducted analyses in support of diagnostics of this type of motors has been indicated.

Thin linearly elastic Kirchhoff-Love-type open circular cylindrical shells having a functionally (transversally) graded macrostructure and a tolerance-periodic microstructure in circumferential direction are objects of consideration. At the same time, the shells have constant structure in axial direction. On the microscopic level, the geometrical, elastic and inertial properties of these shells are determined by highly oscillating, non-continuous and tolerance-periodic functions in circumferential direction. On the other hand, on the macroscopic level, the averaged (effective) properties of the shells are described by functions being smooth and slowly varying along circumferential direction. The aim of this note is to study some problems of micro-dynamics of these shells, e.g. micro-vibrations depending on a cell size, The micro-dynamic problems will be analysed in the framework of the averaged asymptotic-tolerance model. Contrary to the exact shell equations with highly oscillating, non-continuous and tolerance-periodic coefficients, governing equations of the averaged model mentioned above have continuous and slowly varying coefficients depending also on a cell size. An important advantage of this model is that it makes it possible to investigate micro-dynamics of the tolerance-periodic shells independently of their macro-dynamics.

We propose a study of the flow in a tube with wavy wall adopting Malevich - Mityushev - Adler’s method, and find a correction to Hagen-Poiseuille’s solution. The problem is to be solved by expanding the velocity and pressure fields in Fourier series involving an infinite set of unknown coefficients. The boundary surface is expanded in Taylor’s series. A perturbation expansion in terms of the powers of the small parameter $$\varepsilon $$ ε of the full set of Stokes’ equations yields a cascade of boundary value problems which are solved at each step in closed form. Even in the first order approximation $$O(\varepsilon )$$ O ( ε ) , new results are obtained.

Model-based control methods are very attractive in the field of robotics as their tracking performance can exceed the classical controllers (such as the independent joint PID controllers). Using the dynamic model of the manipulator, however, requires detailed knowledge about the manipulator’s dynamic parameters such as link masses and inertias or joint friction properties. These parameters are not always easily identifiable and, to some degree, might vary between robots of one kind (e.g. slight differences in masses/inertias) or during the robot operation (e.g. friction changes related to the temperature). Thus, the identified model might not always be suitable for the desired control tasks. A possible method to overcome the aforementioned problems is to use the adaptive control scheme. In that approach, the parameters of the model are constantly updating their values in real-time to assure good tracking performance. This paper deals with the implementation of such an adaptive controller for the KUKA LWR 4+ robot. Using the KUKA’s communication protocols, a C++ implementation of the outer-loop adaptive controller (which feeds the KUKA controller with the desired joint torques) was created and its quality evaluated.

The dynamic behavior of multi-span uniform continuous beam excited by moving stochastic load is studied. In this paper we consider two models of moving load, namely: load described by space-time stochastic process and random train of concentrated forces moving with constant velocity. It is assumed that forces have random amplitudes and their appearance on the beam is described by point stochastic process (Poisson process). Solution of the problem in terms of expected values, variances and cumulants of the higher order (for the second case of load) was obtained by introducing dynamic influence function. In determination of the dynamic influence function Volterra integral equations was applied. Solution is illustrated with two numerical examples of 2- and 3-span beam.