Quantum Mechanics, Nonlinear Dynamics, and Correlated Statistical Mechanics

Chaos and Clinical Theory Douglas W. Heinrichs (bio) In considering the specific issues raised by these three very thoughtful commentaries, it is helpful to reflect on the status of a theory or model for a specifically clinical discipline—what is it trying to accomplish and how might it proceed to do so? Kellert (2005) sees my proposal in terms of "borrowed knowledge"—the metaphorical application of a theory established in one field of inquiry to another field—and views my application of chaos theory as a "nonliteral use of knowledge from the natural sciences" (p. 239). He likens it to attempts to explore consequences of relativity theory or quantum mechanics for areas other than those for which they were developed. He notes the rhetorical functions—including appeal to the disciplinary prestige of natural science—of such metaphorical borrowings. Although clearly there are significant metaphorical aspects at work here, there are at least two factors that distinguish this model from, say, the metaphorical application of quantum mechanics to a field other than subatomic physics. First, quantum mechanics purports to say something specific about the behavior of the subatomic world. Its development is wed to that specific content area. On the other hand, nonlinear dynamics, including chaos theory, is a formal mathematical model intending to characterize the behavior of any system of elements that possess certain formal relational properties. It does not start out as field or discipline specific. In fact it is not uniquely tied to the natural sciences at all, beyond the fact that it may be easier to obtain the sort of detailed repeated measures needed to demonstrate chaotic behavior in physical systems. So it is not just a question of whether chaos theory is a useful intuitive device for thinking freshly about clinical psychiatry. Rather, it is the quite literal and nonmetaphorical question as to whether some of the observable aspects of persons that are relevant to their psychiatric care behave chaotically, and if so what are the implications. Now it is quite true that direct, mathematically rigorous demonstrations of chaos in clinically relevant data sets is currently limited. In fact, one hoped for consequence of presenting this model is to encourage just such studies, and I suggest several promising directions for doing so. However, there are good reasons for thinking the presence of such chaotic behavior to be likely, and this relates to the notion of a clinical theory. One of the consequences of what Globus (2005) calls "triumphalist biological psychiatry" (p. 229) is the frequent assumption that the model or paradigm of psychiatry—an applied clinical practice, a techne—is synonymous with the paradigm of neuroscience—a basic science or episteme. This is clearly not so for many reasons, including the fact that the former has to incorporate the comprehensive treatment of discrete individuals in a way that the latter does not. Yet the respective theories are related more than metaphorically. [End Page 243] The paradigms of neuroscience and those of other relevant basic sciences, such as psychology, substantially inform the clinical paradigm and the clinical model must be at least compatible with those of the relevant basic sciences. Thus, it is reasonable to assume that if chaotic processes are demonstrable in the nervous system, for which significant evidence already exists, there is good reason to expect them to be reflected in clinically relevant domains. The clinical paradigm should make room for it and its consequences, rather than thinking within an outmoded neuroscience model that makes no room for the implications of nonlinear dynamics. And when this generates a picture more compatible than the old model with our usual sense of what it is to be human, the odds that we are on to something is further enhanced. Of course until such time that enough clinical research based on this model is completed, considerable speculation—indeed aided by metaphorical devices—is involved in generating a concrete sense of how the model would be applied to specific clinical problems, such as the treatment of depressed persons. One clear task of the clinical paradigm is the methodologic integration of interventions growing out of different informing basic sciences. Because chaotic modeling can be applied across various content domains and...

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.

Chaos theory moots that consequences can be predicted in a system of deterministic thinking. Moreover, in general, chaotic systems can be accepted as random as they are predicted. Of course, the prediction depends on how the dimensions are clear. As chaos theory is applied for social sciences, many related studies were done on business management and to its functions. However both the studies on chaos theory and human resources management are satisfactory in numbers, studies on the relationship between these two are scarce. This paper provides a review of literature that underlines the basic principles of chaos and complexity theory, and the functions of human recourses management. In addition to this common practices of human resource management functions will be discussed in the context of chaos and complexity theory. During the text, the question of how HRM can be explained with chaos theory will be answered. It is aimed to fill the gap in the area with showing the connections between two levels of reality. Then, the study was designed to serve the daily business life with is discussions.

The article starts with a self‐contained brief overview of fundamental topics on nonlinear dynamics, chaos, and bifurcations, reaching out to the contemporary subjects of controlling chaos, controlling bifurcations, and anticontrolling chaos.The section on nonlinear dynamics includes basic concepts of nonlinear dynamical systems, equilibria, limit sets, attractors, periodic orbit, limit cycles, Poincaré maps, homoclinic and heteroclinic orbits, Lyapunov stabilities, and the orbital stability of dynamical systems.In the section on chaos theory, the concept of chaos and its dynamical features are first introduced, followed by some measures and testing methods of chaos. Fractals and chaos in control systems are briefly described. In parallel, the section on bifurcations introduces several typical types of bifurcations and the period‐doubling bifurcations route to chaos. Similarly, bifurcations in control systems are briefly discussed.The above prepared the readers to step forward into the new realm of research on controlling chaos and controlling bifurcations, presented in the following two sections. Why chaos control and why bifurcation control are first motivated, by both theoretical challenges and real‐world examples. Then, some effective chaos control and bifurcation control methodologies are introduced, respectively, with distinctions from conventional control theory and techniques for nonchaotic especially linear systems.The last section is novel and unique‐anticontrolling chaos in the sense of creating chaos or retaining chaos, by means of control, when chaos is beneficial. This topic has a great potential in nonconventional applications of control theory and practice, with a corpus of opportunities in future technology and engineering.

This chapter presents an introduction to fundamental concepts in nonlinear dynamics, which are essential for modeling and testing the mechanical behavior of MEMS devices. Because nonlinear dynamics is a wide topic, the focus here is on subjects that are most common and relevant to MEMS. Basic analytical techniques to analyze and tackle nonlinear systems are presented. Several applications on MEMS devices demonstrating phenomena at the microscale level are discussed. The aim of this chapter is to give MEMS researchers fundamental background on the subject. This should lead to proper understanding of the behavior of microsystems and enable the development of modeling methodologies that can capture the essence of the static and dynamic aspects of MEMS devices behaving nonlinearly.

The article discusses the application of the theory of dynamic chaos to the study of social phenomena. Appeal to the origins of the creation of the theory of dynamic chaos in natural science (A. Poincaré, I. Prigogine, E. Lorenz, and others) revealed nonlinear dynamic systems in the natural environment (turbulent flows, atmosphere, biological populations, etc.). The category of “chaos” is now firmly established in the arsenal of the social sciences and humanities, although only recently it referred exclusively to natural science knowledge (the theory of chaos in mathematics, physics, biology, etc.). In synergetics, for the first time, the description of self-organization processes as a mutual transition of order and chaos was proposed by I.R. Prigogine.But in the social sciences such systems are society, its economic, political and other spheres, which have the properties of non-closure, instability and non-linear development. In Ukrainian philosophical thought, one of the first works in which the problem of the development of nonlinear self-developing systems was highlighted was the work of I.S. Dobronravova (1991). Scientific monograph I.V. Yershova-Babenko (1992) also had a significant impact on the development of studies of complex non-linear systems, since for the first time the system of the human psyche was considered as a non-linear self-organized system. The psycho-synergetic model of social reality is based on the fact that social reality is a psychomeric environment, i.e. a complex nonlinear system consisting of other complex nonlinear integrity, which are determined by phase transitions between different states of chaos and order. The application of chaos theory is also possible at the micro and macro levels of social research, which is presented by Ukrainian researchers in synergetics (I. S. Dobronravova, L. Finkel) and in psychosynergetics (I.V. Yershova-Babenko), L. Bevzenko and others.

The popular subject 'chaos theory' has captured the imagination of a wide variety of scientists and engineers. CFD has always been faced with nonlinear systems and it is natural to assume that nonlinear dynamics will play a role at sometime in such work. This paper will attempt to introduce some of the concepts and analysis procedures associated with nonlinear dynamics theory. In particular, results from computations of an airfoil at high angle of attack which exhibits a sequence of bifurcations for single frequency unsteady shedding through period doublings cascading into low dimensional chaos are used to present and demonstrate various aspects of nonlinear dynamics in CFD.

Chaos in schizophrenia associations, reality or metaphor?

Years ago, many researchers proposed that chaos theory could provide a kind of universal theory of the qualitative behavior of all dynamical systems. Thus it could provide a solid mathematical foundation for unifying efforts to address the three really fundamental questions of basic science: (1) what is life?; (2) what is mind or intelligence?; (3) what are the underlying physical laws of the universe? Chaos theory is essentially a recent extension of a larger field of mathematics which it is part of: nonlinear system dynamics (NSD). It mainly addresses systems governed by Ordinary Differential Equations (ODE) or their discrete-time cousins. Because the vast bulk of models now used in computational neuroscience are ODE models, it is only natural that neuroscience has developed great interest in the use of NSD as a tool for analyzing and designing neural network models. Walter Freeman, in particular, has played a seminal role in describing how chaos theory can be used in computational neuroscience, in his classical models of the olfactory system, described in articles in Scientific American and many scientific journals and IJCNN proceedings. His colleague Morris Hirsch in mathematics has helped provide a solid foundation for this work. This work has helped to inspire many efforts throughout the world to use chaos as a design principle in computer systems and in artificial neural network (ANN) design. Recently, however, several issues or questions have begun to emerge, some at Berkeley and some in other places. The author discusses these questions and their implications.

Background: From a mathematical perspective, dynamics can be thought of as linear or non-linear. Linear equations work quite well for a number of problems in the physical sciences. For instance, they are very useful if one wants to predict the orbit of the planets or understand the effects of wind resistance and gravity on the trajectory of a missile. To explore systemic change, non-linear dynamics uses non-linear equations. Non-linear equations are not additive; therefore, they are often difficult to solve. Non-linear systems tend to settle down over time. This settling down, or convergence, tends to result in one of four typical patterns. These patterns, when graphed in diagrams that show periodic changes in behaviour, are called attractors. The trajectories of these attractors typically converge on a discrete point, a simple oscillating cycle, a quasiperiodic cycle, or chaotic cycle. A practical translation of this is the fractal, an object consisting of a pattern that when magnified, reveals repetitive levels of detail so that a similar structure exists on all scales. This property is known as self-similarity. Aims: To examine a novel view-point by examining the possible applications of chaos theory in the study of mental health. Method: For this paper, we studied the available research correlating chaos theory and mental health, especially schizophrenia and bipolar disorder. Results and Discussion: The papers we reviewed show dynamic and chaotic models for both schizophrenia and bipolar disorder. Examples have been found of chaotic responses to initial stimuli in schizophrenia where overstimulation or sensory overload may be paradoxically associated with negative symptoms, fixity, or even catatonia. Meanwhile, regarding bipolar disorder, the attractor hypothesis has been analysed. It is thought that healthy persons have a single strange attractor regarding their mood shifts, while in patients suffering from bipolar disorder, this attractor is destroyed and replaced by two attractors with chaotic transients between them. The second pillar of chaos theory has also been analysed regarding mental health by using fractal analysis to more accurately determine MRI changes in mental patients. Conclusion: This analysis is still in the opening phases and more research is needed to further cement chaos theory as a valid approach in psychiatry, especially since initial results seem promising.

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

For the last decade I have been demonstrating that many of the so-called paradoxes generated by the Copenhagen interpretation of quantum mechanics have less puzzling analogs in nonlinear dynamics and chaos theory. This raises questions about the possibility of nonlinearities inherent in the foundations of quantum theory. Since most of us do not think intuitively with nonlinear logic, I take this opportunity to dwell on several peculiarities of nonlinear dynamics and chaos: nonlinear logic itself and the possible connection of infinite nonlinear regressions with free will. Superficially, nonlinear dynamics can be every bit as counterintuitive as quantum theory; yet, its apparent paradoxes are more amenable to logical analysis. As a result, using nonlinear dynamics to resolve quantum paradoxes winds up being more straightforward than many of the alternative interpretations currently being formulated to replace the orthodox interpretation. Chaos theory could be a candidate for bridging the gap between the determinism so dear to Einstein and the statistical interpretation of the Copenhagen School: For deterministic chaos is indeed deterministic; however, intrinsic limitations on precision in measuring initial conditions necessitate analyzing it statistically. Einstein and Bohr both could have been correct in their debates.

In the literature it is sometimes claimed that chaos theory, non-linear dynamics, and the theory of fractals have major implications for philosophy of medicine, especially for our analysis of the...

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.