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Limit Cycle
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). Definition We consider a two-dimensional dynamical system of the form x'(t)=V(x(t)) where V : \R^2 \to \R^2 is a smooth function. A ''trajectory'' of this system is some smooth function x(t) with values in \mathbb^2 which satisfies this differential equation. Such a trajectory is called ''closed'' (or ''periodic'') if it is not constant but returns to its starting point, i.e. if there exists some t_0>0 such that x(t + t_0) = x(t) for all t \in \R. An orbit (dynamics), orbit is the ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Cycle Poincare Map
Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 song by Calvin Harris from ''Ready for the Weekend'' (album) * "Limits", a 2019 song by Paenda; see Austria in the Eurovision Song Contest 2019 * ''Limits'' (collection), a collection of short stories and essays by Larry Niven * The Limit, a Dutch band * "The Limit", an episode from ''Adventure Time'' * "The Limit", an episode from ''The Amazing World of Gumball'' * " The Limit is Just Me", a documentary film about world's longest triathlon. Mathematics * Limit (mathematics), the value that a function or sequence "approaches" as the input or index approaches some value ** Limit of a function ***(ε,_δ)-definition of limit, formal definition of the mathematical notion of limit ** Limit of a sequence ** One-sided limit, either of the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Bendixson Theorem
In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an open subset of the plane, every non-empty compact ''ω''-limit set of an orbit, which contains only finitely many fixed points, is either * a fixed point, * a periodic orbit, or * a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these. Moreover, there is at most one orbit connecting different fixed points in the same direction. However, there could be countably many homoclinic orbits connecting one fixed point. Discussion A weaker version of the theorem was originally conceived by , although he lacked a complete proof which was later given by . Continuous dynamical systems that are defined on two-dimensional manifolds other than the plane (or cylinder or two-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an ''n''-dimensional vector. The attractor is a region in ''n''-dimensional space. In physical systems, the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Pol Oscillator
In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-Conservative force, conservative, oscillating system with non-linear damping. It evolves in time according to the second-order differential equation - \mu(1-x^2) + x = 0, where is the position coordinate system, coordinate—which is a function (mathematics), function of the time —and is a scalar (mathematics), scalar parameter indicating the nonlinearity and the strength of the damping. History The Van der Pol oscillator was originally proposed by the Dutch electrical engineering, electrical engineer and physicist Balthasar van der Pol while he was working at Philips. Van der Pol found stable oscillations, which he subsequently called relaxation oscillator, relaxation-oscillations and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes. When these circuits are driven near the limit cycle, they become entrainment (phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrical Circuit
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, current sources, Electrical resistance and conductance, resistances, inductances, capacitances). An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Thus all circuits are networks, but not all networks are circuits (although networks without a closed loop are often referred to as "open circuits"). A resistive network is a network containing only resistors and ideal current and voltage sources. Network analysis (electrical circuits), Analysis of resistive networks is less complicated than analysis of networks containing capacitors and inductors. If the sources are constant (Direct current, DC) sources, the result is a DC network. The effective resistance and current dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cancer Cell
Cancer cells are cells that divide continually, forming solid tumors or flooding the blood or lymph with abnormal cells. Cell division is a normal process used by the body for growth and repair. A parent cell divides to form two daughter cells, and these daughter cells are used to build new tissue or to replace cells that have died because of aging or damage. Healthy cells stop dividing when there is no longer a need for more daughter cells, but cancer cells continue to produce copies. They are also able to spread from one part of the body to another in a process known as metastasis. Classification There are different categories of cancer cell, defined according to the cell type from which they originate. * Carcinoma, the majority of cancer cells are epithelial in origin, beginning in a tissue that lines the inner or outer surfaces of the body. * Leukaemia, originate in the tissues responsible for producing new blood cells, most commonly in the bone marrow. * Lymphoma and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Migration
Cell migration is a central process in the development and maintenance of multicellular organisms. Tissue formation during embryogenesis, embryonic development, wound healing and immune system, immune responses all require the orchestrated movement of cells in particular directions to specific locations. Cells often migrate in response to specific external signals, including chemotaxis, chemical signals and mechanotaxis, mechanical signals. Errors during this process have serious consequences, including intellectual disability, cardiovascular disease, vascular disease, tumor, tumor formation and metastasis. An understanding of the mechanism by which cells migrate may lead to the development of novel therapeutic strategies for controlling, for example, invasive tumour cells. Due to the highly viscous environment (low Reynolds number), cells need to continuously produce forces in order to move. Cells achieve active movement by very different mechanisms. Many less complex prokaryotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circadian Rhythm
A circadian rhythm (), or circadian cycle, is a natural oscillation that repeats roughly every 24 hours. Circadian rhythms can refer to any process that originates within an organism (i.e., Endogeny (biology), endogenous) and responds to the environment (is Entrainment (chronobiology), entrained by the environment). Circadian rhythms are regulated by a circadian clock whose primary function is to rhythmically co-ordinate biological processes so they occur at the correct time to maximize the fitness of an individual. Circadian rhythms have been widely observed in animals, plants, fungi and cyanobacteria and there is evidence that they evolved independently in each of these kingdoms of life. The term ''circadian'' comes from the Latin ', meaning "around", and ', meaning "day". Processes with 24-hour cycles are more generally called diurnal rhythms; diurnal rhythms should not be called circadian rhythms unless they can be confirmed as endogenous, and not environmental. Although ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glycolysis
Glycolysis is the metabolic pathway that converts glucose () into pyruvic acid, pyruvate and, in most organisms, occurs in the liquid part of cells (the cytosol). The Thermodynamic free energy, free energy released in this process is used to form the high-energy molecules adenosine triphosphate (ATP) and NADH, reduced nicotinamide adenine dinucleotide (NADH). Glycolysis is a sequence of ten reactions catalyzed by enzymes. The wide occurrence of glycolysis in other species indicates that it is an ancient metabolic pathway. Indeed, the reactions that make up glycolysis and its parallel pathway, the pentose phosphate pathway, can occur in the Great Oxygenation Event, oxygen-free conditions of the Archean oceans, also in the absence of enzymes, catalyzed by metal ions, meaning this is a plausible prebiotic pathway for abiogenesis. The most common type of glycolysis is the ''Embden–Meyerhof–Parnas (EMP) pathway'', which was discovered by Gustav Embden, Otto Meyerhof, and Jakub Kar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuron
A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell that fires electric signals called action potentials across a neural network (biology), neural network in the nervous system. They are located in the nervous system and help to receive and conduct impulses. Neurons communicate with other cells via synapses, which are specialized connections that commonly use minute amounts of chemical neurotransmitters to pass the electric signal from the presynaptic neuron to the target cell through the synaptic gap. Neurons are the main components of nervous tissue in all Animalia, animals except sponges and placozoans. Plants and fungi do not have nerve cells. Molecular evidence suggests that the ability to generate electric signals first appeared in evolution some 700 to 800 million years ago, during the Tonian period. Predecessors of neurons were the peptidergic secretory cells. They eventually ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action Potential
An action potential (also known as a nerve impulse or "spike" when in a neuron) is a series of quick changes in voltage across a cell membrane. An action potential occurs when the membrane potential of a specific Cell (biology), cell rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of Membrane potential#Cell excitability, excitable cells, which include animal cells like neurons and myocyte, muscle cells, as well as some plant cells. Certain endocrine cells such as pancreatic beta cells, and certain cells of the anterior pituitary gland are also excitable cells. In neurons, action potentials play a central role in cell–cell interaction, cell–cell communication by providing for—or with regard to saltatory conduction, assisting—the propagation of signals along the neuron's axon toward axon terminal, synaptic boutons situated at the ends of an axon; these signals can then connect wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
