Butterfly effect in popular culture
The butterfly effect describes a phenomenon in chaos theory whereby a minor change in circumstances can cause a large change in outcome. The scientific concept is attributed to Edward Lorenz, a mathem
Complexity
Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and
Synchronization of chaos
Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled. Because of the exponential divergence of the nearby trajectories of chaotic systems, h
Fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical da
Feigenbaum function
In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:
* the solution to the Feigenbaum-Cvit
Optical chaos
In the field of photonics, optical chaos is chaos generated by laser instabilities using different schemes in semiconductor and fiber lasers. Optical chaos is observed in many non-linear optical syste
Nonlinear Dynamics (journal)
Nonlinear Dynamics, An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems is a monthly peer-reviewed scientific journal covering all nonlinear dynamic phenomena associated wi
Uncertainty exponent
In mathematics, the uncertainty exponent is a method of measuring the fractal dimension of a . In a chaotic scattering system, the invariant set of the system is usually not directly accessible becaus
Chaotic scattering
Chaotic scattering is a branch of chaos theory dealing with scattering systems displaying a strong sensitivity to initial conditions. In a classical scattering system there will be one or more impact
Complexor
The word complexor was coined by Marcial Losada derived from the words "complex order", to refer to chaotic attractors that are strange and thus have fractal structure (in contrast to fixed point or l
Lagrangian coherent structure
Lagrangian coherent structures (LCSs) are distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest. The type of
Pomeau–Manneville scenario
In the theory of dynamical systems (or turbulent flow), the Pomeau–Manneville scenario is the transition to chaos (turbulence) due to intermittency. Named after Yves Pomeau and .
Chaotic bubble
Chaotic bubbles within physics and mathematics, occur in cases when there are any dynamic processes that generate bubbles that are nonlinear. Many exhibit mathematically chaotic patterns consistent wi
Transfer operator
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In
Butterfly effect
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a la
Chaos communications
Chaos communications is an application of chaos theory which is aimed to provide security in the transmission of information performed through telecommunications technologies. By secure communications
Feigenbaum constants
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after t
Chaotic mixing
In chaos theory and fluid dynamics, chaotic mixing is a processby which flow tracers develop into complex fractals under the actionof a fluid flow.The flow is characterized by an exponential growth of
Fine-tuning
In theoretical physics, fine-tuning is the process in which parameters of a model must be adjusted very precisely in order to fit with certain observations. This had led to the discovery that the fund
Information fluctuation complexity
Information fluctuation complexity is an information-theoretic quantity defined as the fluctuation of information about entropy. It is derivable from fluctuations in the predominance of order and chao
Malkus waterwheel
The Malkus waterwheel, also referred to as the Lorenz waterwheel or chaotic waterwheel, is a mechanical model that exhibits chaotic dynamics. Its motion is governed by the Lorenz equations. While clas
Quantum ergodicity
In quantum chaos, a branch of mathematical physics, quantum ergodicity is a property of the quantization of classical mechanical systems that are chaotic in the sense of exponential sensitivity to ini
Relativistic chaos
In physics, relativistic chaos is the application of chaos theory to dynamical systems described primarily by general relativity, and also special relativity. One of the earlier references on the topi
The Chemical Basis of Morphogenesis
"The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from
Singularity (system theory)
In the study of unstable systems, James Clerk Maxwell in 1873 was the first to use the term singularity in its most general sense: that in which it refers to contexts in which arbitrarily small change
Bus bunching
In public transport, bus bunching, clumping, convoying, piggybacking or platooning is a phenomenon whereby two or more transit vehicles (such as buses or trains) that were scheduled at regular interva
Arnold–Beltrami–Childress flow
The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional incompressible velocity field which is an exact solution of Euler's equation. Its repre
For Want of a Nail
"For Want of a Nail" is a proverb, having numerous variations over several centuries, reminding that seemingly unimportant acts or omissions can have grave and unforeseen consequences.
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
Chaos machine
In mathematics, a chaos machine is a class of algorithms constructed on the base of chaos theory (mainly deterministic chaos) to produce pseudo-random oracle. It represents the idea of creating a univ
The Collapse of Chaos
The Collapse of Chaos: Discovering Simplicity in a Complex World (1994) is a book about complexity theory and the nature of scientific explanation written by biologist Jack Cohen and mathematician Ian
Multiscale turbulence
Multiscale turbulence is a class of turbulent flows in which the chaotic motion of the fluid is forced at different length and/or time scales. This is usually achieved by immersing in a moving fluid a
Pyragas method
In the mathematics of chaotic dynamical systems, in the Pyragas method of stabilizing a periodic orbit, an appropriate continuous controlling signal is injected into the system, whose intensity is nea
Turing pattern
The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and
Supersymmetric theory of stochastic dynamics
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicabili
Self-organized criticality control
In applied physics, the concept of controlling self-organized criticality refers to the control of processes by which a self-organized system dissipates energy. The objective of the control is to redu
Correlation sum
In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close: where is the number of consider
Catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation
Laminar–turbulent transition
In fluid dynamics, the process of a laminar flow becoming turbulent is known as laminar–turbulent transition. The main parameter characterizing transition is the Reynolds number. Transition is often d
Recurrence plot
In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment in time, the times at which the state of a dynamical system returns to the previous state at ,i.e.
Chaos theory
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
Stability of the Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational eff
Eden's conjecture
In the mathematics of dynamical systems, Eden's conjecture states that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic o
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
Control of chaos
In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic
Recurrence quantification analysis
Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a
Chaotic cryptology
Chaotic cryptology is the application of the mathematical chaos theory to the practice of the cryptography, the study or techniques used to privately and securely transmit information with the presenc
Correlation dimension
In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For exam
Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "Wha
Poincaré plot
A Poincaré plot, named after Henri Poincaré, is a type of recurrence plot used to quantify self-similarity in processes, usually periodic functions. It is also known as a return map. Poincaré plots ca
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in
Self-organized criticality
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance charac
Emergence
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the pa
Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a f
Chaotic hysteresis
A nonlinear dynamical system exhibits chaotic hysteresis if it simultaneously exhibits chaotic dynamics (chaos theory) and hysteresis. As the latter involves the persistence of a state, such as magnet
Chaos: Making a New Science
Chaos: Making a New Science is a debut non-fiction book by James Gleick that initially introduced the principles and early development of the chaos theory to the public. It was a finalist for the Nati
Chirikov criterion
The Chirikov criterion or Chirikov resonance-overlap criterionwas established by the Russian physicist Boris Chirikov.Back in 1959, he published a seminal article,where he introduced the very first ph
Oscillon
In physics, an oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in granular media result from vertically vibrating a plate with a layer of uniform p
Fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal patter
Edge of chaos
The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engender
Correlation integral
In chaos theory, the correlation integral is the mean probability that the states at two different times are close: where is the number of considered states , is a threshold distance, a norm (e.g. Euc
Chaos game
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively crea