Preface

The quest to ensure perfect dynamical properties and the control of different systems used in both human everyday life and engineering applications is currently the goal of numerous research conducted by different scientific centers and laboratories all over the world. To help support this effort, various scientific agencies and foundations have established funding for investigations regarding modeling, numerical simulation, and experimental verifications of proposed mathematical models of these systems. As a result of extensive research, numerous interesting solutions allowing for the simulation and analysis of real dynamical systems have been obtained. Different theorems and algorithms have been developed to analyze systems that are usually governed by ordinary or partial differential equations. Their dynamics and stability have been investigated by bifurcation diagrams, Lyapunov exponents, frequency spectra, Fourier transforms, wavelets, asymptotic methods, multiple scale methods, finite element methods, and many other mathematical approaches.

The aim of this book is to provide the reader with a selection of modern methods in the field of mathematical modeling, simulation, dynamics and control of lumped mass mechanical systems, structural members, continuous/discontinuous systems, and parametric physical systems. The book presents selected approaches suitable for the analysis of different kinds of dynamical systems, and is sectioned into nine chapters introducing the reader to different issues of dynamical systems theory. A brief description of each chapter follows.

In Chapter 1, Barrios et al. focus on studies related to the stability transition curves of coupled Mathieu equations utilizing numerical and theoretical methods. They extend previously developed theory and mathematical formalism used for Hamiltonian systems to carry out the investigation of two coupled and damped Mathieu equations used as an example, highlighting advantages of the presented formalism. Stability of the considered system is analyzed by using the appropriate stable/ unstable regions where parametric resonance phenomena are highlighted. Due to the symmetry of the used symplectic matrices, the parametric resonance zones are characterized yielding faster and more robust computations, and achieving a higher accuracy while estimating the stability transition curves. The proposed procedure can be extended to a higher number of coupled Mathieu equations.

In Chapter 2, Yan and Zeng consider the problem of the entropy of topological dynamical systems, which is an important notion for understanding the complexity of such systems. The authors study first the relationship among topological entropy, pseudo-orbit, and preimage entropies for topological dynamical systems regarding localization phenomena. Second, they introduce and investigate two entropy-like invariants for non-autonomous discrete dynamical systems, i.e. partial entropy and bundle-like entropy. As a result, the relation between the topological entropy for open covers and several preimage entropy invariants is studied, which is viewed as the local version of the Hurley inequality. Eventually, the authors show that topological entropy for open covers can be computed by measuring the exponential growth rate of the number of pseudo-orbits that end at a particular point.

In Chapter 3, Maksymovych and Podhorecki consider the problems of determination of stresses at cracks in bounded plates with holes of different shapes under the action of concentrated forces or distributed forces at their boundary. The presented approach is developed by using the singular integral equations determined based on the established interdependences between the Lekhnitskii potentials and stress/ strain. The numerical method for solving the obtained integral equations is developed based on the quadrature method for the systems exhibiting holes and cracks. The developed numerical algorithm is applied for calculating stresses at cracks in bounded plates with holes of various shapes due to concentrated forces or distributed forces. Eventually, the analysis of stresses at cracks in samples, which are used in experimental studies of crack fracture resistance, is performed.

Fahmy proposes a theory devoted to nonlinear generalized thermoelastic stresses in anisotropic circular cylindrical plate structures in Chapter 4. Due to strong nonlinearity of the considered problem and because of difficulties with finding its analytical solution, a novel computerized boundary element model is developed. To describe the thermal stresses in such structures, two-dimensional temperature nonlinear radiative heat conduction equations coupled with electron, ion, and phonon temperatures are used. To verify the proposed model, the threetemperature radiative heat conduction results are replaced with one-temperature heat conduction results. The presented data obtained by using the developed computerized boundary element model and a finite element model analyzed in COMSOL Multiphysics commercial software confirms the validity and accuracy of the proposed theory.

In Chapter 5, Steinberg and Kvasov present theoretical analysis of the distinctive characteristics of Cosserat plate-free vibrations by using the proposed dynamic model developed as an extension of the Reissner plate theory. During numerical validation of the model the authors detect and classify different characteristics of the plate vibrations, including additional high-resonance frequencies of a plate depending on the shape and orientation of the microelements incorporated into the Cosserat plates. They show that microfrequencies associated with the microrotatory inertia and its transverse variation of the ellipsoid elements have higher microfrequencies than the ball-shaped elements. In addition, the dependence of eigenfrequencies on the angles of rotation of the horizontal ellipsoid microelements is detected.

Chapter 6 looks at Grzelczyk and Awrejcewicz develop and investigate numerically a general kinematic model of a multilegged hybrid robot with crab-like and/or mammal-like legs. The locomotion process and stabilization of the position and orientation of the robot on planar, unstable, and vibrating ground are visualized in Mathematica. The authors focus especially on precise control of the position of the robot during walking in different directions and precisely control all six spatial degrees of freedom of the robot's body, as well as all the robot's legs. The obtained results show the possibility for full control of the robot position and orientation in space, as well as during the locomotion process. The control algorithm can also be successfully employed on unstable or vibrating ground to stabilize the spatial position of the robot.

In Chapter 7, Vidiella et al. analyze the numerical dynamics of a predator-prey biological ecosystem existing in nature and governed by nonlinear difference equations. Despite the relatively simple model of the investigated dynamical system, it is characterized by an enormous richness of dynamics, including extinctions,

coextinctions, ordered and chaotic coexistence, as well as hyperchaotic behavior detected by two positive Lyapunov exponents for some parameters' regions. The authors provide conditions for the global stability of the fixed points corresponding to the coextinctions of the predator-prey as well as for the extinction of predators and survival of prey. By iterating the analyzed dynamical system, a very complicated shape of the escaping regions, presumably with a highly entangled fractal topology, is detected and characterized.

Chapter 8 sees Wahrhaftig perform a dynamic analysis of vibration of a real slender reinforced concrete pole with variable geometry, with a concentrated force present at the free end of the structural element. During simulations they consider the selfweight of the structure and nonlinear properties of the material. Using two simulation models, the author demonstrates that the used rheological model characterized by three parameters is suitable for mathematical implementation, in comparison with the standard criteria. It stands for the possibility of adjustment of a simple model to the standard one and shows practical applications to calculate the first natural frequency as the critical buckling load. Eventually, it is noted that in further investigations, a programing routine for obtaining a finer adjustment of the curve between the viscoelastic rheological model and that of the model for creep as predicted by Eurocode criteria should be developed.

Finally, in Chapter 9, Oueslati et al. present a modeling study of heat transfer during the drying of a moist agricultural product placed in a hot air flow in a tunnel dryer with partial solar heating. To do this, they use a bond graph approach, which is an object-oriented graphical approach based on an energetic description between subsystems. Experimental drying tests are carried out to validate the developed model, taking into account the geometry of the dryer, the physical properties of building materials, agricultural product, and air properties. Especially, the influence of temperature and velocity of hot air are studied to evaluate the performance of the dryer. As a result of the optimization technique, the developed model can be adapted to different agricultural products providing their high quality, or to other drying processes.

We hope that the readers of this book will be attracted by the topics covered in the content, which are directed primarily towards research aimed at increasing their academic knowledge with competences related to the selected new mathematical theoretical approaches and original numerical tools related to a few problems of dynamical systems theory.

> Jan Awrejcewicz and Dariusz Grzelczyk Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Lodz, Poland
