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Chua's Circuit
Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos". Chaotic criteria An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria before it can display chaotic behaviour. It must contain: # one or more nonlinear elements, # one or more locally active resistors, # three or more energy-storage elements. Chua's circuit is the simplest electronic circuit meeting these criteria. As shown in the top figure, the energy storage elements are two capacitors (label ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Circuit
A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit ''F(x)'' when a linear combination of signals ''ax1(t) + bx2(t)'' is applied to it is equal to the linear combination of the outputs due to the signals ''x1(t)'' and ''x2(t)'' applied separately: :F(ax_1 + bx_2) = aF(x_1) + bF(x_2)\, It is called a linear circuit because the output voltage and current of such a circuit are linear functions of its input voltage and current. This kind of linearity is not the same as that of straight-line graphs. In the common case of a circuit in which the components' values are constant and don't change with time, an alternate definition of linearity is that when a sinusoidal input voltage or current of frequency ''f'' is applied, any steady-state output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency ''f''. A linear circuit with constant comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diode
A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other. A diode vacuum tube or thermionic diode is a vacuum tube with two electrodes, a heated cathode and a plate, in which electrons can flow in only one direction, from cathode to plate. A semiconductor diode, the most commonly used type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals. Semiconductor diodes were the first semiconductor electronic devices. The discovery of asymmetric electrical conduction across the contact between a crystalline mineral and a metal was made by German physicist Ferdinand Braun in 1874. Today, most diodes are made of silicon, but other semiconducting materials such as gallium arsenide and germanium are also used. Among many uses, diodes are found in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Memristor
A memristor (; a portmanteau of ''memory resistor'') is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which comprises also the resistor, capacitor and inductor. Chua and Kang later generalized the concept to memristive systems. Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably ReRAM. The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated. As a fundamental electrical component Chua in his 1971 paper identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiscroll Attractor
In the mathematics of dynamical systems, the double-scroll attractor (sometimes known as Chua's attractor) is a strange attractor observed from a physical electronic chaotic circuit (generally, Chua's circuit) with a single nonlinear resistor (see Chua's diode). The double-scroll system is often described by a system of three nonlinear ordinary differential equations and a 3-segment piecewise-linear equation (see Chua's equations). This makes the system easily simulated numerically and easily manifested physically due to Chua's circuits' simple design. Using a Chua's circuit, this shape is viewed on an oscilloscope using the X, Y, and Z output signals of the circuit. This chaotic attractor is known as the double scroll because of its shape in three-dimensional space, which is similar to two saturn-like rings connected by swirling lines. The attractor was first observed in simulations, then realized physically after Leon Chua invented the autonomous chaotic circuit which became ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaotic 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 manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Attractor
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Entropy
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer-assisted Proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program. Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using automated reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear
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 because 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. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirchhoff's Circuit Laws
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. These laws can be applied in time and frequency domains and form the basis for network analysis. Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits. Kirchhoff's current law This law, also called Kirchhoff's first law, or Kirchhoff's junction rule, states that, for any node (junction) in an electrical circuit, the sum of currents ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
From Periodicity To Chaotic Order
From may refer to: * From, a preposition * From (SQL), computing language keyword * From: (email message header), field showing the sender of an email * FromSoftware, a Japanese video game company * Full range of motion, the travel in a range of motion * Isak From (born 1967), Swedish politician * Martin Severin From (1825–1895), Danish chess master * Sigfred From Sigfred From (12 December 1925 – April 1998), was a Danish chess player. Biography From the begin of 1960s to the begin of 1970s Sigfred From was one of Danish leading chess players. He regularly played in Danish Chess Championships. Her best ... (1925–1998), Danish chess master * ''From'' (TV series), a sci-fi-horror series that debuted on Epix in 2022 {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
