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Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.
Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or in time. Systems that operate with all parts in synchrony are said to be synchronous or in sync—and those that are not are asynchronous.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Helioseismology, a term coined by Douglas Gough, is the study of the structure and dynamics of the Sun through its oscillations. These are principally caused by sound waves that are continuously driven and damped by convection near the Sun's surface. It is similar to geoseismology, or asteroseismology, which are respectively the studies of the Earth or stars through their oscillations. While the Sun's oscillations were first detected in the early 1960s, it was only in the mid-1970s that it was realized that the oscillations propagated throughout the Sun and could allow scientists to study the Sun's deep interior. The modern field is separated into global helioseismology, which studies the Sun's resonant modes directly, and local helioseismology, which studies the propagation of the component waves near the Sun's surface.
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.
In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation:
In the field of computational neuroscience, the theory of metastability refers to the human brain’s ability to integrate several functional parts and to produce neural oscillations in a cooperative and coordinated manner, providing the basis for conscious activity.
Neural oscillations, or brainwaves, are rhythmic or repetitive patterns of neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either by mechanisms within individual neurons or by interactions between neurons. In individual neurons, oscillations can appear either as oscillations in membrane potential or as rhythmic patterns of action potentials, which then produce oscillatory activation of post-synaptic neurons. At the level of neural ensembles, synchronized activity of large numbers of neurons can give rise to macroscopic oscillations, which can be observed in an electroencephalogram. Oscillatory activity in groups of neurons generally arises from feedback connections between the neurons that result in the synchronization of their firing patterns. The interaction between neurons can give rise to oscillations at a different frequency than the firing frequency of individual neurons. A well-known example of macroscopic neural oscillations is alpha activity.
A Boolean Delay Equation (BDE) is an evolution rule for the state of dynamical variables whose values may be represented by a finite discrete numbers os states, such as 0 and 1. As a novel type of semi-discrete dynamical systems, Boolean delay equations (BDEs) are models with Boolean-valued variables that evolve in continuous time. Since at the present time, most phenomena are too complex to be modeled by partial differential equations (as continuous infinite-dimensional systems), BDEs are intended as a (heuristic) first step on the challenging road to further understanding and modeling them. For instance, one can mention complex problems in fluid dynamics, climate dynamics, solid-earth geophysics, and many problems elsewhere in natural sciences where much of the discourse is still conceptual.
René Thomas (14 May 1928 - 9 January 2017 was a Belgian scientist. His research included DNA biochemistry and biophysics, genetics, mathematical biology, and finally dynamical systems. He devoted his life to the deciphering of key logical principles at the basis of the behaviour of biological systems, and more generally to the generation of complex dynamical behaviour. He was professor and laboratory head at the Université Libre de Bruxelles, and taught and inspired several generations of researchers.
The Atlantic Multidecadal Oscillation (AMO), also known as Atlantic Multidecadal Variability (AMV), is the theorized variability of the sea surface temperature (SST) of the North Atlantic Ocean on the timescale of several decades.
A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.
The repressilator is a genetic regulatory network consisting of at least one feedback loop with at least three genes, each expressing a protein that represses the next gene in the loop. In biological research, repressilators have been used to build cellular models and understand cell function. There are both artificial and naturally-occurring repressilators. Recently, the naturally-occurring repressilator clock gene circuit in Arabidopsis thaliana and mammalian systems have been studied.
In neurophysiology, several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e., to reproduce the measurements of current and voltage exactly. The renowned Hodgkin–Huxley model of the axon from the Loligo squid exemplifies such models. Although qualitatively correct, the H-H model does not describe every type of excitable membrane accurately, since it considers only two ions, each with only one type of voltage-sensitive channel. However, other ions such as calcium may be important and there is a great diversity of channels for all ions. As an example, the cardiac action potential illustrates how differently shaped action potentials can be generated on membranes with voltage-sensitive calcium channels and different types of sodium/potassium channels. The second type of mathematical model is a simplification of the first type; the goal is not to reproduce the experimental data, but to understand qualitatively the role of action potentials in neural circuits. For such a purpose, detailed physiological models may be unnecessarily complicated and may obscure the "forest for the trees". The FitzHugh–Nagumo model is typical of this class, which is often studied for its entrainment behavior. Entrainment is commonly observed in nature, for example in the synchronized lighting of fireflies, which is coordinated by a burst of action potentials; entrainment can also be observed in individual neurons. Both types of models may be used to understand the behavior of small biological neural networks, such as the central pattern generators responsible for some automatic reflex actions. Such networks can generate a complex temporal pattern of action potentials that is used to coordinate muscular contractions, such as those involved in breathing or fast swimming to escape a predator.
Stellar pulsations are caused by expansions and contractions in the outer layers as a star seeks to maintain equilibrium. These fluctuations in stellar radius cause corresponding changes in the luminosity of the star. Astronomers are able to deduce this mechanism by measuring the spectrum and observing the Doppler effect. Many intrinsic variable stars that pulsate with large amplitudes, such as the classical Cepheids, RR Lyrae stars and large-amplitude Delta Scuti stars show regular light curves.
Phase resetting in neurons is a behavior observed in different biological oscillators and plays a role in creating neural synchronization as well as different processes within the body. Phase resetting in neurons is when the dynamical behavior of an oscillation is shifted. This occurs when a stimulus perturbs the phase within an oscillatory cycle and a change in period occurs. The periods of these oscillations can vary depending on the biological system, with examples such as: (1) neural responses can change within a millisecond to quickly relay information; (2) In cardiac and respiratory changes that occur throughout the day, could be within seconds; (3) circadian rhythms may vary throughout a series of days; (4) rhythms such as hibernation may have periods that are measured in years. This activity pattern of neurons is a phenomenon seen in various neural circuits throughout the body and is seen in single neuron models and within clusters of neurons. Many of these models utilize phase response (resetting) curves where the oscillation of a neuron is perturbed and the effect the perturbation has on the phase cycle of a neuron is measured.
Andreas Karl Engel is a German neuroscientist. He is the director of the Department of Neurophysiology and Pathophysiology at the University Medical Center Hamburg-Eppendorf (UKE).
In mathematics and mathematical biology, the Mackey–Glass equations, named after Michael Mackey and Leon Glass, refer to a family of delay differential equations whose behaviour manages to mimic both healthy and pathological behaviour in certain biological contexts, controlled by the equation's parameters. Originally, they were used to model the variation in the relative quantity of mature cells in the blood. The equations are defined as:
BH Virginis is a binary star system in the equatorial constellation of Virgo. With a typical apparent visual magnitude of 9.6, it is too faint to be visible to the naked eye. Based on parallax measurements, it is located at a distance of approximately 488 light years from the Sun. The system is drifting closer with a net radial velocity of −23 km/s.
References
- 1 2 Chandra, F. A.; Buzi, G.; Doyle, J. C. (2011). "Glycolytic Oscillations and Limits on Robust Efficiency". Science . 333 (6039): 187–192. Bibcode:2011Sci...333..187C. CiteSeerX 10.1.1.368.4950 . doi:10.1126/science.1200705. PMID 21737735. S2CID 10836848.
- ↑ Goldbeter, A.; Berridge, M. J. (1996). "Oscillatory enzymes: simple periodic behaviour in an allosteric model for glycolytic oscillations". Biochemical Oscillations and Cellular Rhythms. pp. 31–88. doi:10.1017/CBO9780511608193.005. ISBN 9780511608193.
- ↑ Duysens, L. N. M.; Amesz, J. (1957). "Fluorescence spectrophotometry of reduced phosphopyridine nucleotide in intact cells in the near-ultraviolet and visible region". Biochimica et Biophysica Acta. 24 (1): 19–26. doi:10.1016/0006-3002(57)90141-5. hdl: 1874/15621 . PMID 13426197.
- ↑ Letellier, C. (2013). "Chaos in Biology and Biomedicine". Chaos in Nature. World Scientific Series on Nonlinear Science Series A. Vol. 81. pp. 277–322. doi:10.1142/9789814374439_0013. ISBN 978-981-4374-42-2. S2CID 88603020.
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