Discontinuity and Complexity in Nonlinear Physical Systems

The new concepts of self-adjoint equations formulated in Gandarias (J Phys A: Math Theor 44:262001, 2011) and Ibragimov (J Phys A: Math Theor 44:432002, 2011) are applied to some classes of third order equations. Then, from Ibragimov’s theorem on conservation laws, conservation laws for two generalized equations of KdV type and a potential Burgers equation are established.

Ibragimov introduced the concepts of self-adjoint and quasi-self-adjoint equations. Gandarias generalized these concepts and defined the concept of weak self-adjoint equations. In this paper we consider a family of Benjamin-Bona-Mahony-Burgers equations and we determine the subclass of equations which are self-adjoint, quasi-self-adjoint and weak self-adjoint. By using a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for these equations.

In the last decades, much effort has been dedicated to analytical aspects of the fractional differential equations. The Adomian decomposition method and the variational iteration method have been developed from ordinary calculus and become two frequently used analytical methods. In this article, the recent developments of the methods in the fractional calculus are reviewed. The realities and challenges are comprehensively encompassed.

Local fractional Fourier series is a generalized Fourier series in fractal space. The local fractional calculus is one of useful tools to process the local fractional continuously non-differentiable functions (fractal functions). Based on the local fractional derivative and integration, the present chapter is devoted to the theory and applications of local fractional Fourier analysis in generalized Hilbert space. We recall the local fractional Fourier series, the Fourier transform, the generalized Fourier transform, the discrete Fourier transform and fast Fourier transform in fractal space.

This chapter presents optimization of fractional order PI ^λ D ^μ control parameters by using response surface methodology. The optimization process is observed on a fractional order diffusion system subject to input hysteresis which is defined with Riemann–Liouville fractional derivative. The system is transferred to a fractional order state space model by using eigenfunction expansion method and then Grünwald–Letnikov approximation is applied to solve the system numerically. The necessary data for response surface analysis are read from the obtained numerical solution. Finally, second-order polynomial response surface mathematical model for the experimental design is presented and the optimum control parameters are predicted from this response surface model. The proposed optimization method is compared with the technique of minimization of integral square error by means of settling time and the results are discussed.

We examined the Van der Pol system with external forcing and a memory possessing fractional damping term. Calculating the basins of attraction we showed broad spectrum of nonlinear behaviour connected with sensitivity to the initial conditions. To quantify dynamical response of the system we propose the statistical 0–1 test. The results have been confirmed by bifurcation diagrams, phase portraits and Poincare sections.

Fractional calculus is currently gaining more and more popularity in the control engineering world. Several tuning algorithms for fractional order controllers have been proposed so far. This chapter describes a simple tuning rule for fractional order PI controllers for single-input–single-output processes and an extension of this method to the multivariable case. The implementation of a fractional order PI on an FPGA target for controlling the DC motor speed, as well as the implementation of a multivariable fractional order PI controller for a time delay system is presented. Experimental results are given to show the efficiency and robustness of the tuning algorithm.

This paper presents the application of model-based predictive control (MPC) in combination with a sensor for the measurement of analgesia (pain relief) in an unconscious patient in order to control the level of anesthesia. The MPC strategy uses fractional-order impedance models (FOIMs) to model the diffusion process that occurs in the human body when an analgesic drug is taken up. Based on this control strategy an early dawn concept of the pain sensor is developed. The grand challenges that coincide with this development include identification of the patient model, validation of the pain sensor, and validation of the effect of the analgesic drug.

Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects.In all models with local interactions, these ordered phases disappear in the strong segregation regime (low temperature). It is expected, however, that these phases should persist in the case of long-range interactions, which can’t be correctly described by a Ginzburg–Landau type model with only a finite number of spatial derivatives of the order parameter.An alternative approach is to study the dynamics of the phase transition or pattern formation. While, in the usual process of Ostwald ripening, succession of doubling of the domain size leads to a total segregation, or macro-segregation, Misbah and Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern which then remains in a micro-structured state or super-crystal. We show that this is the case for a modified Cahn–Hilliard dynamics due to Oono which includes a nonlocal term and which is particularly well suited to describe systems with a modulated phase.

Main problems of simulation and mathematical modeling of high-frequency signals for analog Costas loop and for analog phase-locked loop (PLL) are considered. Two approachers which allow to solve these problems are considered. In the first approach, nonlinear models of classical PLL and classical Costas loop are considered. In the second approach, engineering solutions for this problems are described. Nonlinear differential equations are derived for both approaches.

The present study examines static complexity of assembly supply chains (ASCs). While static complexity describes the structure of the supply chain, the number and the variety of its components, and interactions between relevant units; the dynamic complexity of supply chains involves the aspects of time and randomness. The aim is to come up with a methodological framework for conceptual modeling of ASC structures. Models of such ASC structures are divided into classes on the basis of the numbers of initial suppliers. Subsequently, we propose to apply different indices for measuring a structural complexity of ASC structures based on specific demand conditions. Special attention is also paid here to so-called Vertex Degree Index. It is a complexity measure originating from information theory and is based on the Shannon entropy. Finally, we outline a reference model for defining levels of parameterized complexity of ASC structures.

In this chapter, we briefly recall the theory of non-commutative tomography in a pedagogical way. We then consider its applications to signal analysis. The advantages and drawbacks of these techniques to finite samples of data are discussed. Then the method is applied, first to signals originating from reflectometry measurements in magnetized fusion plasmas, and then to data obtained from the advection of tracers in a two-dimensional time-dependent flow generated by three point vortices. In the first case, we show that the tomogram allows to pick a base to represent our signal which has the advantage of isolating the reflection coming from the plasma and then to improve the estimation of the density profile. In the second case, we show how, with a “tricky transformation” the method allows us to detect Lévy flights and extract some of their properties.

In this chapter, a simple nonlinear controller is applied to investigate the generalized projective synchronization for two gyroscopes with different dynamical behaviors. The projective synchronization conditions are developed through the theory of discontinuous dynamical systems. The synchronization invariant domain from the synchronization conditions is presented. The parameter maps are obtained for a better understanding of the synchronicity of two gyroscopes. Finally, the partial and full generalized projective synchronizations of two nonlinear coupled gyroscope systems are carried out to verify the effectiveness of the scheme. The scaling factors in such synchronization are observed through numerical simulations.

This paper introduces the concept of fractional order models for characterizing viscoelasticity in the lungs. A technique to detect and analyse these nonlinear, low-frequency contributions in the lung tissue is presented, along with some experimental data. The measurements are performed using the forced oscillation technique and a non-invasive lung function testing procedure which takes only 40 s, while the patient is breathing at rest. The index introduced to quantify the nonlinear contributions in the lungs in healthy is then employed in a theoretical analysis to show that the values are changing in case of disease. The results indicate that the proposed method and index are useful for clinical classification of viscoelastic properties in the lungs.

There are many mathematical models of drilling systems Despite, huge efforts in constructing models that would allow for precise analysis, drilling systems, still experience breakdowns. Due to complexity of systems, engineers mostly use numerical analysis, which may lead to unreliable results. Nowadays, advances in computer engineering allow for simulations of complex dynamical systems in order to obtain information on the behavior of their trajectories. However, this simple approach based on construction of trajectories using numerical integration of differential equations describing dynamical systems turned out to be quite limited for investigation of stability and oscillations of these systems. This issue is very crucial in applied research; for example, as stated in Lauvdal et al. (Proceedings of the IEEE control and decision conference, 1997) the following phrase: “Since stability in simulations does not imply stability of the physical control system (an example is the crash of the YF22) stronger theoretical understanding is required”. In this work, firstly a mathematical model of a drilling system developed by a group of scientists from the University of Eindhoven will be considered. Then a mathematical model of a drilling system with perfectly rigid drill-string actuated by induction motor will be analytically and numerically studied. A modification of the first two models will be considered and it will be shown that even in such simple models of drilling systems complex effects such as hidden oscillations may appear, which are hard to find by standard computational procedures.

The paper studies a second order nonlinear differential equation whose right hand side is a piecewise linear function. It shows the coexistence of a countable set of periodic solutions and an uncountable set of bounded non-periodic solutions. The result can be also used to explain the chaos on some smooth nonlinear dynamical systems.

We examine the dynamical response and the power output of a vibration energy harvesting electromechanical system with kinematic ambient excitation and impact. Due to the stopper nonlinearities the examined system exhibits multiple solutions. We characterize their properties and stability by the voltage output and the corresponding basins of attraction.

This chapter discusses the dynamics of a mass–damper–spring system with two rigid constraints and impact interactions. Impacting chatter and stuck phenomena are investigated for the mass with constraints and the corresponding conditions for such phenomena are determined. Analytical predictions are presented for the system to give a more precise and complete demonstration of the phenomena in the system. Finally, an analytical parameter map is given to show how the system changes for varying parameters. From these conditions, numerical simulations are performed to demonstrate these phenomena in the system.

The formations of transitional zones in shock wave, governed by Burgers’ equation, are studied from viewpoint of saddle-node bifurcations. First, the inviscid Burgers’ equation is studied in detail, the solution of the system with a certain smooth initial condition is obtained, and the solution in vector form is reduced into a Map in order to investigate the stability and bifurcation in the system. It is proved that there exists a thin spatial zone where a saddle-node bifurcation occurs in finite time, and the velocity of the fluid behaves as jumping, namely, the characteristic of shock wave. Further, the period-doubling bifurcation is captured, that means there exist multiple states as time increases, and the complicated spatio-temporal pattern is formatted. In addition to above, the viscous Burgers’ equation is further studied to extend to dissipative systems. By traveling wave transformation, the governing equation is reduced into an ordinary differential equation. More, the instability or bifurcation condition is obtained, and it is proved that there are three singular points in the system as the bifurcation condition is satisfied. The results show that the discontinuity resulting from saddle-node bifurcations is removed with the introduction of viscosity, and another kind of velocity change with strong gradient is obtained. However, the change of velocity is continuous with sharp slopes. As a conclusion, it can be drawn that all results can provide a fundamental understanding of the nonlinear phenomena relevant to shock wave and other complicated nonlinear phenomena, from viewpoint of nonlinear dynamics.

We study the dynamics of a milling process of a composite material basing on the experimental time series of cutting force components measured in the feeding direction. By using the recurrence plots we observe the differences in the response of the system depending on the feeding direction with respect to composite fibers orientation. This effect has been found after decomposition on the Huang experimental modes. Showing the results of recurrences in particular experimental modes we advocate to use this quantity to analyze the stability of the cutting of composites. The difference between different cases was also noticed using Fourier transform and statistical parameters such as RMS and kurtosis, but for these methods the necessary time interval of the examined time series has to be much longer, while recurrence approach is designed for shorter time series.

In recent years a number of bracing devices have been proposed, analyzed, and applied to real cases, since in engineering applications the construction of frames equipped with braces is a widespread practice. In the present contribution, a nonlinear bracing system is introduced and applied to the case of shear-type moment-resistant frames. The frame is considered here as the primary structure and is assumed to have linear elastic behavior and the bracing system is considered as a secondary, additional structure. The bracing system is made of two chains, each of them constructed as the assemblage of two axial elements (springs) undergoing axial force, only. The springs that are assumed to have linear elastic behavior are connected to each other in the chain and to the frame through hinges. The global behavior of the system is nonlinear, since the restoring force of the bracing system is a piecewise-defined function. In order to asses the performance of the whole nonlinear system, its behavior is compared with that of the linear primary structure alone, through a suitable concise descriptor.

Analytical method is presented for the determination of free vibration characteristics of high speed viscoelastic rotating disks. In the development of this analytical solution, two-dimensional elastodynamic theory is employed and the viscoelastic material for the medium is allowed by assuming complex elastic moduli. The general governing equations of motion are derived and a solution for a single rotating disk with different boundary conditions is developed for a wide range of rotating speeds and any radius ratios, such as those for solid disks or thin rings. The proposed solution is used to investigate the influences of hysteretic material damping on dimensionless natural frequencies and modal loss factors for the rotating disks. Furthermore, the solution is expanded to consider the effect of adding disk segment with different material on the inner or outer sides of a disk on the natural frequencies and critical speeds of the equivalent single disk. The dimensionless results for these cases are presented for a wide range of rotational speeds.

In the present study we consider, on one hand, a differentiated Stackelberg model, and, on the other hand, a differentiated Cournot model, when one of the firms engages in an R & D process that gives an endogenous cost-reducing innovation. The aim of this study is two fold. The first is to study the licensing of the cost-reduction in the Stackelberg model. The second is to do a direct comparison between Stackelberg model and Cournot model. We analyse the implications of these types of licensing contracts over the R & D effort, the profits of the firms, the consumer surplus and the social welfare. By using comparative static analysis, we conclude that the degree of the differentiation of the goods assumes a great importance in the results.

We analyse the relationship between the privatization of a public firm and government preferences for tax revenue, by considering a (sequential) Stackelberg duopoly with the public firm as the leader. We assume that the government payoff is given by a weighted sum of tax revenue and the sum of consumer and producer surplus. We get that if the government puts a sufficiently larger weight on tax revenue than on the sum of both surpluses, it will not privatize the public firm. In contrast, if the government puts a moderately larger weight on tax revenue than on the sum of both surpluses, it will privatize the public firm. Furthermore, we compare our results with the ones previously published by an other author obtained in a (simultaneous) Cournot duopoly.