Bifurcation memory
Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation.
Conservative system
In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynami
Parry–Sullivan invariant
In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability o
Ushiki's theorem
In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of we
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
Kinetic scheme
In physics, chemistry and related fields, a kinetic scheme is a network of states and connections between them representing the scheme of a dynamical process. Usually a kinetic scheme represents a Mar
Lyapunov vector
In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been
System equivalence
In the systems sciences system equivalence is the behavior of a parameter or component of a system in a way similar to a parameter or component of a different system. Similarity means that mathematica
Exponential dichotomy
In the mathematical theory of dynamical systems, an exponential dichotomy is a property of an equilibrium point that extends the idea of hyperbolicity to non-autonomous systems.
Brownian dynamics
Brownian dynamics (BD) can be used to describe the motion of molecules for example in molecular simulations or in reality. It is a simplified version of Langevin dynamics and corresponds to the limit
Method of averaging
In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drif
Presymplectic form
In Geometric Mechanics a presymplectic form is a closed differential 2-form of constant rank on a manifold. However, some authors use different definitions. Recently, Hajduk and Walczak defined a pres
Hamiltonian fluid mechanics
Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.
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
Self-oscillation
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
Deterministic system
In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus a
Pugh's closing lemma
In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows: Let be a diffeomorphism of a
Black box model of power converter
The black box model of power converter also called behavior model, is a method of system identification to represent the characteristics of power converter, that is regarded as a black box. There are
Symbolic dynamics
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which correspon
Quasicircle
In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by and ,
Nonholonomic system
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differe
Stretching field
In applied mathematics, stretching fields provide the local deformation of an infinitesimal circular fluid element over a finite time interval ∆t. The logarithm of the stretching (after first dividing
Integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with suff
Differential variational inequality
In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are u
Lyapunov dimension
In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been develope
Predictive state representation
In computer science, a predictive state representation (PSR) is a way to model a state of controlled dynamical system from a history of actions taken and resulting observations. PSR captures the state
Thermodynamic operation
A thermodynamic operation is an externally imposed manipulation that affects a thermodynamic system. The change can be either in the connection or wall between a thermodynamic system and its surroundi
Geometric mechanics
Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory. Geom
Homoclinic orbit
In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the
Separatrix (mathematics)
In mathematics, a separatrix is the boundary separating two modes of behaviour in a differential equation.
Microscale and macroscale models
Microscale models form a broad class of computational models that simulate fine-scale details, in contrast with macroscale models, which amalgamate details into select categories. Microscale and macro
Injection locking
Injection locking and injection pulling are the frequency effects that can occur when a harmonic oscillator is disturbed by a second oscillator operating at a nearby frequency. When the coupling is st
Nekhoroshev estimates
The Nekhoroshev estimates are an important result in the theory of Hamiltonian systems concerning the long-time stability of solutions of integrable systems under a small perturbation of the Hamiltoni
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
Liénard equation
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Mari
Boolean delay equation
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 sem
Heteroclinic channels
Heteroclinic channels are ensembles of trajectories that can connect saddle equilibrium points in phase space. Dynamical systems and their associated phase spaces can be used to describe natural pheno
Non-autonomous system (mathematics)
In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-aut
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
Invariant manifold
In dynamical systems, a branch of mathematics, an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. Examples include the slow manifold, center ma
Equidistributed sequence
In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length
Constant-recursive sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or mor
Prime geodesic
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime
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
Time reversibility
A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed. A deterministic process is time-reversible if the
Normal form (dynamical systems)
In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior. Normal forms are often used for determining local bifurcations in a
Hitchin system
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies o
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
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
Viability theory
Viability theory is an area of mathematics that studies the evolution of dynamical systems under constraints on the system state. It was developed to formalize problems arising in the study of various
Phase reduction
Phase reduction is a method used to reduce a multi-dimensional dynamical equation describing a nonlinear limit cycle oscillator into a one-dimensional phase equation. Many phenomena in our world such
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems). In this formulation, time is not required to be a
Michael Brin Prize in Dynamical Systems
The Michael Brin Prize in Dynamical Systems, abbreviated as the Brin Prize, is awarded to mathematicians who have made outstanding advances in the field of dynamical systems and are within 14 years of
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
Analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by
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
Phase space method
In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists
Master stability function
In mathematics, the master stability function is a tool used to analyse the stability of the synchronous state in a dynamical system consisting of many identical oscillators which are coupled together
Cannon–Thurston map
In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group
Fundamental ephemeris
A fundamental ephemeris of the Solar System is a model of the objects of the system in space, with all of their positions and motions accurately represented. It is intended to be a high-precision prim
Isochron
In the mathematical theory of dynamical systems, an isochron is a set of initial conditions for the system that all lead to the same long-term behaviour.
Lyapunov exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitat
Homoclinic connection
In dynamical systems, a branch of mathematics, a structure formed from the stable manifold and unstable manifold of a fixed point.
Community matrix
In mathematical biology, the community matrix is the linearization of the Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equili
Bouncing ball
The physics of a bouncing ball concerns the physical behaviour of bouncing balls, particularly its motion before, during, and after impact against the surface of another body. Several aspects of a bou
Linear flow on the torus
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus which is represented by the following
Outer billiards
Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or
Contact dynamics
Contact dynamics deals with the motion of multibody systems subjected to unilateral contacts and friction. Such systems are omnipresent in many multibody dynamics applications. Consider for example
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Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of
Fermi–Pustyl'nikov model
The Fermi–Pustyl'nikov model, named after Enrico Fermi and , is a model of the Fermi acceleration mechanism. A point mass falls with a constant acceleration vertically on the infinitely heavy horizont
Painlevé conjecture
In physics, the Painlevé conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4. The theorem was proven for n ≥ 5 in 1988
Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory,
Numerical model of the Solar System
A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model estab
Bogdanov map
In dynamical systems theory, the Bogdanov map is a chaotic 2D map related to the Bogdanov–Takens bifurcation. It is given by the transformation: The Bogdanov map is named after Rifkat Bogdanov.
Dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system
Non-smooth mechanics
Non-smooth mechanics is a modeling approach in mechanics which does not require the time evolutions of the positions and of the velocities to be smooth functions anymore. Due to possible impacts, the
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 .
Quasi-invariant measure
In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An i
Tau function (integrable systems)
Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in h
Eulerian coherent structure
In applied mathematics, objective Eulerian coherent structures (OECSs) are the instantaneously most influential surfaces or curves that exert a major influence on nearby trajectories in a dynamical sy
Quasiperiodic motion
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies
Parasitic oscillation
Parasitic oscillation is an undesirable electronic oscillation (cyclic variation in output voltage or current) in an electronic or digital device. It is often caused by feedback in an amplifying devic
Action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of os
Inertial manifold
In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems. Inertial manifolds are finite-dimensional, smooth, invariant manifolds t
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
Denjoy's theorem on rotation number
In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. D
Rotation number
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
Suspension (dynamical systems)
Suspension is a construction passing from a map to a flow. Namely, let be a metric space, be a continuous map and be a function (roof function or ceiling function) bounded away from 0. Consider the qu
Prefix order
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix
Arithmetic dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the c
Andronov–Pontryagin criterion
The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.
Shadowing lemma
In the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit (whi
Elastic pendulum
In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring s
Cliodynamics
Cliodynamics (/ˌkliːoʊdaɪˈnæmɪks/) is a transdisciplinary area of research that integrates cultural evolution, economic history/cliometrics, macrosociology, the mathematical modeling of historical pro
Recurrence period density entropy
Recurrence period density entropy (RPDE) is a method, in the fields of dynamical systems, stochastic processes, and time series analysis, for determining the periodicity, or repetitiveness of a signal
Strejc method
The Strejc system identification method allows the estimate of the transfer function of a non-periodic, black box-type system based on its step response and is widely used in all branches of industria
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
Three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion acco
Dragon king theory
Dragon king (DK) is a double metaphor for an event that is both extremely large in size or impact (a "king") and born of unique origins (a "dragon") relative to its peers (other events from the same s
Monogenic system
In classical mechanics, a physical system is termed a monogenic system if the force acting on the system can be modelled in a particular, especially convenient mathematical form. The systems that are
Orbit portrait
In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps. In simple words one can say that it is :
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Stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions emb
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.
Composition operator
In mathematics, the composition operator with symbol is a linear operator defined by the rule where denotes function composition. The study of composition operators is covered by AMS category 47B33.
Fermi–Ulam model
The Fermi–Ulam model (FUM) is a dynamical system that was introduced by Polish mathematician Stanislaw Ulam in 1961. FUM is a variant of Enrico Fermi's primary work on acceleration of cosmic rays, nam
Jet Propulsion Laboratory Development Ephemeris
Jet Propulsion Laboratory Development Ephemeris (abbreved JPL DE(number), or simply DE(number)) designates one of a series of mathematical models of the Solar System produced at the Jet Propulsion Lab
Heteroclinic network
In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic network
Lyapunov time
In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse o
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked lo
Melnikov distance
In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
Morse–Smale system
In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbol
Poincaré map
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynami
Linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the non
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordina
Cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homoge
Historical dynamics
Historical dynamics broadly includes the scientific modeling of history. This might also be termed computer modeling of history, historical simulation, or simulation of history - allowing for an exten
Rayleigh–Ritz method
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz
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
Dynamical neuroscience
The dynamical systems approach to neuroscience is a branch of mathematical biology that utilizes nonlinear dynamics to understand and model the nervous system and its functions. In a dynamical system,
Matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous poin
Superslow process
Superslow processes are processes in which values change so little that their capture is very difficult because of their smallness in comparison with the measurement error.
Spectral submanifold
In dynamical systems, a spectral submanifold (SSM) is the unique smoothest invariant manifold serving as the nonlinear extension of a spectral subspace of a linear dynamical system under the addition
Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-
Equilibrium point
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
Double pendulum
In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system
Generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of th
Positive-definite function
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Rational dependence
In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A colle
Hybrid bond graph
A hybrid bond graph is a graphical description of a physical dynamic system with discontinuities (i.e., a hybrid dynamical system). Similar toa regular bond graph, it is an energy-based technique. How
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian m
Linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interes
Cobweb plot
A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logis
Cocycle
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likew
Marginal stability
In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if
Lattès map
In mathematics, a Lattès map is a rational map f = ΘLΘ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus to the complex sphere and L is an affine map z → az + b
Phase space crystals
Phase space crystal is the state of a physical system that displays discrete symmetry in phase space instead of real space. For a single-particle system, the phase space crystal state refers to the ei
Conley–Zehnder theorem
In mathematics, the Conley–Zehnder theorem, named after Charles C. Conley and Eduard Zehnder, provides a lower bound for the number of fixed points of Hamiltonian diffeomorphisms of standard symplecti
Interplanetary Transport Network
The Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes partic
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
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
Free motion equation
A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a refer
Rational difference equation
A rational difference equation is a nonlinear difference equation of the form where the initial conditions are such that the denominator never vanishes for any n.
Metastability
In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy.A ball resting in a hollow on a slope is a simpl
Stable manifold theorem
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given
Intermittency
In dynamical systems, intermittency is the irregular alternation of phases of apparently periodic and chaotic dynamics (Pomeau–Manneville dynamics), or different forms of chaotic dynamics (crisis-indu
Closed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the
Pendulum (mechanics)
A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position
Differential inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form where F is a multivalued map, i.e. F(t, x) is a set rather than a single point
Linear dynamical system
Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exa
Artin–Mazur zeta function
In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. I
Convergence group
In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian
Hilbert's sixteenth problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original probl
Fatou conjecture
In mathematics, the Fatou conjecture, named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
Graph dynamical system
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis
Arnold diffusion
In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the
Heteroclinic orbit
In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If
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
Combinatorics and dynamical systems
The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about n
Numerical continuation
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, The parameter is usually a real scalar, and the solution an n-vector. For a fixe
Iterated function
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times
Conserved quantity
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conse
Sequential dynamical system
Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provi
D'Alembert's principle
D'Alembert's principle is another way of formulating Newton's second law of motion.The principle has been defined as "the negative of the product of mass times acceleration. If this force is added to
Two-body problem
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with
Slow manifold
In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to
Data-driven control system
Data-driven control systems are a broad family of control systems, in which the identification of the process model and/or the design of the controller are based entirely on experimental data collecte
Fractional-order system
In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer o
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
De Bruijn graph
In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mn vertices, consisting of all possible length-n sequences
Superintegrable Hamiltonian system
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold: (i) There exist independent integrals of mot
False nearest neighbor algorithm
Within abstract algebra, the false nearest neighbor algorithm is an algorithm for estimating the embedding dimension. The concept was proposed by Kennel et al. (1992). The main idea is to examine how
Lagrange stability
Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange. For any point in the state space, in a real continuous dynamical system , where is , th
Dynamical systems theory
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 eq
Amplitude death
In the theory of dynamical systems, amplitude death is complete cessation of oscillations. The system can be in a state of either periodic motion or chaotic motion before it goes to amplitude death. A
Orbit (dynamics)
In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phas
Volume entropy
The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely r
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
Parametric oscillator
A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency o
LaSalle's invariance principle
LaSalle's invariance principle (also known as the invariance principle, Barbashin-Krasovskii-LaSalle principle, or Krasovskii-LaSalle principle) is a criterion for the asymptotic stability of an auton
Iterated filtering
Iterated filtering algorithms are a tool for maximum likelihood inference on partially observed dynamical systems. Stochastic perturbations to the unknown parameters are used to explore the parameter
Perturbation (astronomy)
In astronomy, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fou
Shift space
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and symbolic dyn
Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its de
Orbit modeling
Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational a
Expansive homeomorphism
In mathematics, the notion of expansivity formalizes the notion of points moving away from one another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definit
Dynamical billiards
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the bounda
Universality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universali
Time-scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite
State space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intell
Krylov–Bogoliubov averaging method
The Krylov–Bogolyubov averaging method (Krylov–Bogolyubov method of averaging) is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics. The method is based o
Discrete time and continuous time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Multibody system
Multibody system is the study of the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.
Center manifold
In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was rea
Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbol
Multibody simulation
Multibody simulation (MBS) is a method of numerical simulation in which multibody systems are composed of various rigid or elastic bodies. Connections between the bodies can be modeled with kinematic
Translation surface
In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a
Floquet theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form with a piecewise continuous periodi
Pentagram map
In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shorte
Equichordal point problem
In Euclidean plane geometry, the equichordal point problem is the question whether a closed planar convex body can have two equichordal points. The problem was originally posed in 1916 by Fujiwara and
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
Exponential stability
In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative
Heteroclinic cycle
In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclini
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
Sommerfeld effect
In mechanics, Sommerfeld effect is a phenomenon arising from feedback in the energy exchange between vibrating systems: for example, when for the rocking table, under given conditions, energy transmit
Markov partition
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition,
Newtonian limit
In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly
Tire model
In vehicle dynamics, a tire model is a type of multibody simulation used to simulate the behavior of tires. In current vehicle simulator models, the tire model is the weakest and most difficult part t
Linear recurrence with constant coefficients
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equatio
Rulkov map
The Rulkov map is a two-dimensional iterated map used to model a biological neuron. It was proposed by Nikolai F. Rulkov in 2001. The use of this map to study neural networks has computational advanta
Virtual displacement
In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) shows how the mechanical system's trajectory can hypothetically (hence the ter
Brjuno number
In mathematics, a Brjuno number is a special type of irrational number.
Hybrid system
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both flow (described by a differential equation) and jump (described by a state ma
Vague torus
In classical mechanics, a vague torus is a region in phase space that is characterized by approximate constants of motion, as opposed to an actual torus defined by exact constants of motion.The concep
Micromagnetics
Micromagnetics is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the ma
Valentin Afraimovich
Valentin Afraimovich (Russian: Валентин Сендерович Афраймович, 2 April 1945, Kirov, Kirov Oblast, USSR – 21 February 2018, Nizhny Novgorod, Russia) was a Soviet, Russian and Mexican mathematician. He
Lagrangian system
In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of
Line field
In mathematics, a line field on a manifold is a formation of a line being tangent to a manifold at each point, i.e. a section of the line bundle over the manifold. Line fields are of particular intere
Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechan
Behavioral modeling
The behavioral approach to systems theory and control theory was initiated in the late-1970s by J. C. Willems as a result of resolving inconsistencies present in classical approaches based on state-sp
Koenigs function
In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representatio
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
Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map where θ is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the bo
Distribution function (physics)
In molecular kinetic theory in physics, a system's distribution function is a function of seven variables, , which gives the number of particles per unit volume in single-particle phase space. It is t
Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poin
Projected dynamical system
Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and appli
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
Gauss–Kuzmin–Wirsing operator
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss ma
Dissipation factor
In physics, the dissipation factor (DF) is a measure of loss-rate of energy of a mode of oscillation (mechanical, electrical, or electromechanical) in a dissipative system. It is the reciprocal of qua
Parry–Daniels map
In mathematics, the Parry–Daniels map is a function studied in the context of dynamical systems. Typical questions concern the existence of an invariant or ergodic measure for the map. It is named aft
Relativistic Lagrangian mechanics
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.
System identification
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design of experiments
Hyperbolic set
In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, o
Wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the syst
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
Quasiperiodicity
Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a
Illumination problem
Illumination problems are a class of mathematical problems that study the illumination of rooms with mirrored walls by point light sources.
Bony–Brezis theorem
In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant
Euler's Disk
Euler's Disk, invented between 1987 and 1990 by Joseph Bendik, is a trademark for a scientific educational toy. It is used to illustrate and study the dynamic system of a spinning and rolling disk on
Virtual work
In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves a
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
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-Fren
Langevin dynamics
In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by French physicist Paul Langevin. The approach is character
Bost–Connes system
In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number fie
Normally hyperbolic invariant manifold
A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described heuristically as follows: For a manifold t
Poincaré–Birkhoff theorem
In symplectic topology and dynamical systems, Poincaré–Birkhoff theorem (also known as Poincaré–Birkhoff fixed point theorem and Poincaré's last geometric theorem) states that every area-preserving, o
Rotational–vibrational coupling
In physics, rotational–vibrational coupling occurs when the rotation frequency of a system is close to or identical to a natural internal vibration frequency. The animation on the right shows ideal mo
Hysteresivity
Hysteresivity derives from “hysteresis”, meaning “lag”. It is the tendency to react slowly to an outside force, or to not return completely to its original state. Whereas the area within a hysteresis
Phase portrait
A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point. Phase portrai
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
Markov odometer
In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem
Trapping region
In applied mathematics, a trapping region of a dynamical system is a region such that every trajectory that starts within the trapping region will move to the region's interior and remain there as the
Kaplan–Yorke conjecture
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest , let j be t
Lefschetz zeta function
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map , the zeta-function is defined as the formal se
Misiurewicz point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical po
Causal system
In control theory, a causal system (also known as a physical or nonanticipative system) is a system where the output depends on past andcurrent inputs but not future inputs—i.e., the output depends on
Autonomous system (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable