Category: Nonlinear systems

Bifurcation memory
Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation.
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
Kuramoto model
The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本 由紀, Kuramoto Yoshiki), is a mathematical model used to describing synchronization. More specifically, it is a model
Superstatistics is a branch of statistical mechanics or statistical physics devoted to the study of non-linear and non-equilibrium systems. It is characterized by using the superposition of multiple d
Period-doubling bifurcation
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new
Excitable medium
An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time
Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external
Resonant interaction
In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wa
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solut
Additive state decomposition
Additive state decomposition occurs when a system is decomposed into two or more subsystems with the same dimension as that of the original system. A commonly used decomposition in the control field i
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
Quadratic integrate and fire
The quadratic integrate and fire (QIF) model is a biological neuron model and a type of integrate-and-fire neuron which describes action potentials in neurons. In contrast to physiologically accurate
Autowave reverberator
In the theory of autowave phenomena an autowave reverberator is an autowave vortex in a two-dimensional active medium. A reverberator appears a result of a rupture in the front of a plane autowave. Su
Variable structure system
A variable structure system, or VSS, is a discontinuous nonlinear system of the form where is the state vector, is the time variable, and is a piecewise continuous function. Due to the piecewise conti
SETAR (model)
In statistics, Self-Exciting Threshold AutoRegressive (SETAR) models are typically applied to time series data as an extension of autoregressive models, in order to allow for higher degree of flexibil
Convergent cross mapping
Convergent cross mapping (CCM) is a statistical test for a cause-and-effect relationship between two variables that, like the Granger causality test, seeks to resolve the problem that correlation does
Nonlinear system
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
Bailout embedding
In the theory of dynamical systems, a bailout embedding is a system defined as Here the function k(x) < 0 on a set of unwanted orbits; otherwise k(x) > 0. The trajectories of the full system of a bail
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
Nonlinear acoustics
Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dy
Hysteretic model
Hysteretic models are mathematical models capable of simulating the complex nonlinear behavior characterizing mechanical systems and materials used in different fields of engineering, such as aerospac
Fermi–Pasta–Ulam–Tsingou problem
In physics, the Fermi–Pasta–Ulam–Tsingou problem or formerly the Fermi–Pasta–Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactl
In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar
Neural accommodation
Neural accommodation or neuronal accommodation occurs when a neuron or muscle cell is depolarised by slowly rising current (ramp depolarisation) in vitro. The Hodgkin–Huxley model also shows accommoda
Nonlinear system identification
System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include a
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 i
FitzHugh–Nagumo model
The FitzHugh–Nagumo model (FHN), named after (1922–2007) who suggested the system in 1961 and J. Nagumo et al. who created the equivalent circuit the following year, describes a prototype of an excita
Parametric array
A parametric array, in the field of acoustics, is a nonlinear transduction mechanism that generates narrow, nearly side lobe-free beams of low frequency sound, through the mixing and interaction of hi
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
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field change
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
Dispersive partial differential equation
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength
Empirical dynamic modeling
Empirical dynamic modeling (EDM) is a framework for analysis and prediction of nonlinear dynamical systems. Applications include population dynamics, ecosystem service, medicine, neuroscience, dynamic
Thomas' cyclically symmetric attractor
In the dynamical systems theory, Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas. It has a simple form which is cyclically symmetric in the x,y, and
STAR model
In statistics, Smooth Transition Autoregressive (STAR) models are typically applied to time series data as an extension of autoregressive models, in order to allow for higher degree of flexibility in
Three-wave equation
In nonlinear systems, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude waves in a variety of non-linear media, inc
Coupled map lattice
A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the cha
Dynamic fluid film equations
Fluid films, such as soap films, are commonly encountered in everyday experience. A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right. Altern
Abelian sandpile model
The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized cr
Spiral wave
Spiral waves are travelling waves that rotate outward from a center in a spiral. They are a feature of many excitable media. Spiral waves have been observed in various biological systems including sys
Compartmental neuron models
Compartmental modelling of dendrites deals with multi-compartment modelling of the dendrites, to make the understanding of the electrical behavior of complex dendrites easier. Basically, compartmental
Hindmarsh–Rose model
The Hindmarsh–Rose model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron. The relevant variable is the
Theta model
The theta model, or Ermentrout–Kopell canonical model, is a biological neuron model originally developed to model neurons in the animal Aplysia, and later used in various fields of computational neuro
Social complexity
In sociology, social complexity is a conceptual framework used in the analysis of society. Contemporary definitions of complexity in the sciences are found in relation to systems theory, in which a ph
Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy
Wave turbulence
In continuum mechanics, wave turbulence is a set of nonlinear waves deviated far from thermal equilibrium. Such a state is usually accompanied by dissipation. It is either or requires an external sour
Crisis (dynamical systems)
In applied mathematics and astrodynamics, in the theory of dynamical systems, a crisis is the sudden appearance or disappearance of a strange attractor as the parameters of a dynamical system are vari
Exponential integrate-and-fire
Exponential integrate-and-fire models are compact and computationally efficient nonlinear spiking neuron models with one or two variables. The exponential integrate-and-fire model was first proposed a
Hodgkin–Huxley model
The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equati