Introduction to Nonlinear Dynamics

Chaos in schizophrenia associations, reality or metaphor?

Many of the so‐called paradoxes of orthodox quantum mechanics can be shown to have parallel, more logical interpretations in the realm of nonlinear dynamics and chaos theory. Among these are violations of Bell‐type inequalities, which in comparing “classical” mechanics with quantum mechanics implicitly compare uncorrelated and correlated statistics. During the past decade research in the field of nonextensive thermodynamics (including Tsallis entropy) has demonstrated the existence of many statistical correlations in classical, nonlinear systems. When such correlations exist, the conventional classical upper limit on statistical correlations in Bell‐type experiments can easily be raised to overlap with quantum mechanical predictions involving correlated states such as the Bell singlet state, a favorite for deriving Bell inequalities. Thus, arguments based on experimental violations of Bell‐type inequalities, which rule out the existence of “local reality,” become moot. Perhaps quantum mechanics does have a deterministic, ontological basis, albeit one based in nonlinear dynamics and chaos theory. If so, deterministic chaos could provide Einstein’s longed‐for fundamental determinism, but because chaotic systems must be interpreted statistically, this also fits in quite well with the ideas of Bohr — Einstein and Bohr both could have been correct! It should be emphasized that the concept of nonlinear dynamics and chaos underpinning quantum mechanics does not involve hidden variables, nor does the fact that chaos is deterministic interlope on the existence of free will.

Measurement of features from the chaos theory or as popularly known, the concept of nonlinear dynamics, as indicatives of several pathological conditions and cognition states using the electroencephalography (EEG) signal is very popular. In this paper, the analysis of scalp EEG signals of normal subjects and brain tumour patients using the nonlinear dynamic features has been presented. The nonlinear dynamic features that represent the dimensional and waveform complexities of the signal being analyzed have been considered. The statistical analysis of the selected nonlinear dynamic features has been presented. The results show that the nonlinear dynamic features significantly discriminate the brain tumour group from the normal group.

Nonlinearity and Chaos in Molecular Vibrations

The Earth system shows the tendency to change in nonlinear and sometimes abrupt ways; small changes in external forcing can lead to large and perhaps irreversible changes in outcome. The prospect of crossing important ‘tipping points’ and realizing their impacts poses unique challenges to decision makers within society, hoping to avoid damaging anthropogenic influence on Earth systems. Abrupt and nonlinear changes are by their very nature highly uncertain and difficult to predict, and so hard to avoid or adapt to. After briefly introducing key concepts in nonlinear dynamics, we summarize paleoclimate evidence for past abrupt and nonlinear changes in major Earth systems such as, ocean and atmospheric circulation patterns, sea ice and terrestrial ice sheets, atmospheric composition, and the terrestrial biosphere. For each of these systems we then review observational, theoretical, and modeling evidence for potential future abrupt changes, and associated impacts. We outline the extra challenges that are faced in predicting abrupt or nonlinear as opposed to more gradual climate change, and in providing a risk analysis for their impacts on Earth and societal systems. We examine the potential for early warning systems of abrupt change, and discuss differences in attitude to risk which may dictate societal response to low probability–high impact events. Finally, we outline the promising directions of research needed to better quantify the risk of abrupt and nonlinear climate change. WIREs Clim Change 2011 2 663–686 DOI: 10.1002/wcc.130This article is categorized under: Assessing Impacts of Climate Change > Evaluating Future Impacts of Climate Change

Vibration Energy Harvesting has received a lot of attention in recent years, because of the ubiquitous existence of vibrations in a variety of environments. In real-world device implementation, however, several problems are encountered particularly when the harvesters are intended to power miniaturized systems at micro and/or nano scale; in these cases, to store the harvested energy can pose significant problems due to the very low level of voltages involved, thereby conflicting with the threshold of blocking diodes. Investigations on this specific subject have lead us to the development of a family of devices which exploits the synergetic use of piezoelectric materials, flexible beams, magnets and mechanical stoppers together with some concepts of nonlinear dynamics used to accurately model and understand the device behaviors. Here we present an excursion that begins with the genesis of these ideas and leads to a family of devices able to capture mechanical energy, convert it into electrical energy, and store this energy regardless of the voltage level. The switching mechanism with the mechanical stopper is used to overcome the diode threshold. Few building blocks (Piezoelectric cantilevers, magnets and stoppers) have been identified that, once suitably arranged and used, can lead to novel devices operating as detectors and/or energy harvesters. Beyond energy harvesting, devices able to multiply voltages and rectify signals will be presented, these devices can perform, even at very low voltages because do not use diode. A review of these devices together with working principles, models and experimental characterization results is reported in this review paper.

La Dynamique Posturale Non-Linéaire. Partie I : bases et fondements

Geometrical concepts and computational techniques of nonlinear dynamics

The issues of the formation of inhomogeneities and cracks as well as the effects of their spontaneous healing are considered on the well-known principles and concepts of nonlinear dynamics and quantum technologies in order to fully describe the picture of possible processes of the genesis of heterogeneities and their self-organization. At the same time, the process of spontaneous overgrowth of micro-inhomogeneities in solid-state/metal products, such defects in which occur under different operating conditions, is discussed on the example of thermal power plants of various types with cyclic processes and vibrations. The mechanism of such healing is determined by dynamic processes of unsteady diffusion with certain temporal characteristics evaluated within the framework of the corresponding model concepts. A model of diffusion-limited aggregation of defects/particles implemented by the cellular automaton method from the Neumann neighborhood, a model of random and ballistic deposition, as well as a percolation approach are considered. The process of modification and development of a 3D fractured structure based on the Griffiths load theory for dislocations is described, taking into account the gradual sequential transition from a stable state to a growth stage. As a result of the action of such dynamic loads in a solidstate object, a rapid change in the parameters of microcracking is observed, which can be represented as local micro-explosions, viz. the growth and fusion of natural microcracks into larger ones; the emergence of new microcracks; the disclosure of large microcracks with the formation of defects of the next hierarchical level. In the overview aspect, the universal modes of operation of a nonlinear dynamic system, well-known in mathematics and physics and suitable for analyzing the stability and sustainability of thermal power plants, are considered. The ongoing processes are associated with different types and strategies for the development of heterogeneities, such as: collapse and stagnation; stable periodicity; parameters “at odds”; chaotic development within certain limits; disruption/sudden crisis; a sharp leap and a breakthrough in development. The proposed approaches can be useful in improving the real operational condition of chambers with working matter in power plants when they operate in various operating modes.

Atmosphere-terrestrial ecosystem interactions: implications for coupled modeling

The concepts of classical nonlinear dynamics are employed to interpret the spectroscopic properties of the molecular highly excited vibration. These concepts include the Morse oscillator, the pendulum dynamics, chaos and the overlapping of resonances leading to chaos. The relations of resonance, constant of motion and the basic dynamical unit to the pendulum dynamics are stressed. An algebraic Hamiltonian in the coset space is employed for the dynamical analysis from which the dynamical potential can be easily obtained. The dynamical potential is closely related to the classical fixed points in which the quantized levels are embedded in various quantal environments. Localized modes are easily identified in various systems which share similar dynamical potentials. The dissociation of DCO radical is finally interpreted by these concepts from the classical nonlinear dynamics.

Nonlinear dynamics are pivotal in understanding complex systems, where traditional linear models fail to capture the intricacies of real-world phenomena. This paper explores the mathematical foundations of nonlinear dynamics and their applications across various complex systems. We begin with an introduction to the core concepts of nonlinear dynamics, including chaos theory, bifurcation theory, and fractals. We then discuss the mathematical tools used to analyze nonlinear systems, such as differential equations and attractor reconstruction. The paper further investigates practical applications in fields such as engineering, biology, and economics. By emphasizing both theoretical and applied aspects, this work aims to provide a comprehensive overview of how nonlinear dynamics can be harnessed to model and predict the behavior of complex systems.

Nonlinear dynamic analysis serves an increasingly important role in process systems engineering research. Understanding the nonlinear dynamics from the mathematical model of a process helps to find the boundaries of all achievable process conditions and identify the system instabilities. The information on such boundaries is beneficial for optimizing the design and formulating a control structure. However, a systematic approach to analyzing nonlinear dynamics of chemical processes considering such boundaries in a quantifiable and adaptable way is yet to exist in the literature. The primary aim of this work is to formulate theoretical concepts for dynamic operability, as well as develop the practical implementation methods for the analysis of dynamic performance in chemical processes. Process operability is a powerful tool for analyzing the relationships between the input variables, the output variables, and the disturbances via the geometric computation of variable sets. The operability sets are described by unions of polyhedra, which can be translated to sets of inequality constraints, so the results of the operability analysis can be used for process optimization and advanced process control. Nonetheless, existing process operability approaches in the literature are currently limited for steady-state processes and a generalized definition of dynamic operability that retains the core principles of steady-state operability as a controllability measure. A unified dynamic operability concept is proposed in this dissertation with two different adaptations to represent the complex relationships between the design, control structure, and control law of a given process. The existing operability mapping methods discretize the input space by partitioning the ranges of each input variable evenly, and all possible input combinations are simulated to achieve the output sets. The procedure is repeated for each value in the expected disturbance set to find the output regions that are guaranteed to be achieved regardless of the disturbance scenario. However, for dynamic systems, the same set of manipulated inputs can take different values at different time intervals, so the number of possible input combinations, which is also the number of simulations required, increases exponentially with the number of time intervals. This tractability challenge motivates the development of novel dynamic operability mapping approaches. A linear time-invariant dynamic system is first considered to tackle the dynamic mapping of achievable output sets. For a linear system, the achievable output set (AOS) at a fixed predicted time is the smallest convex hull that contains all the images of the extreme points of the available input set

Evaluation of squeezed film air damping is critical in the design and control of dynamic MEMS devices. The published squeezed film air damping models are generally derived from the analytical solutions of Reynolds equation or its other modified forms under the supposition of trivial pressure boundary conditions on the peripheral borders. These treatments ignoring the border effect can not give faithful result for structure with smaller air venting gap or the double-gimbaled structure in which the inner frame and outer one affect the air venting. In this paper, we use Green’s function to solve the nonlinear Reynolds equation with inhomogeneous boundary conditions. For two typical normal motion cases of parallel plate, the analytical models of squeeze film damping force with border effect are established. The viscous and inertial losses with real values and image values acoustic impedance are all included in the model. These models reduced the time consumption while giving satisfactory result. Without multifield coupling analysis, the estimation of the dynamic behavior of MEMS device is also allowed, and the simulation of the system performance is more convenient.

Because the vocal folds are nonlinear systems with many degrees of freedom, bifurcations and chaos should be expected for certain parameter configurations. Indeed, bifurcations and chaos appear in some of the simplest models of vocal fold vibration. They are also seen in more complex models, in models that incorporate left–right asymmetries, and in biomechanical simulations of vocal fold oscillations. Furthermore, an acoustical analysis of many types of rough voice (e.g., creaky voice, vocal fry, and newborn infant cries) reveals an intimate relationship between voice mechanics and bifurcations and chaos. Likely physiological mechanisms of the observed nonlinear phenomena are presented. Basic concepts and analysis techniques from nonlinear dynamics are reviewed and compared with traditional voice analysis schemes. Finally, it is shown how much of the terminology in the literature describing rough voice can be unified within the framework of nonlinear dynamics. [This research was supported by Grant. No. P60 DC00976 from the National Institute on Deafness and Other Communication Disorders.]