Dynamical systems

mathematical model which describes the time dependence of a point in a geometrical space

thumb|right|Electric displacement field of a ferroelectric material as the [[electric field is first decreased, then increased. The curves form a hysteresis loop.]] 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 changed in the past. Such a system is called hysteretic. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variab

classical mechanics problem of three massive point particles interacting via Newtonian gravity; special case of the 𝑛‐body problem for 𝑛=3

formulation of classical mechanics in terms of phase space and Hamiltonian function

to determine the motion of two point particles that interact only with each other

system in which the change of the output is not proportional to the change of the input

complex motion of a massive astronomical body

250px|thumb|A metastable state of weaker bond (1), a transitional "saddle" configuration (2) and a stable state of stronger bond (3).

formalism of mechanics based on the least action principle

discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling

principle

abstract space whose coordinates are the dynamic variables of the studied system

System of ordinary differential equations first studied by Edward Lorenz

idealized concept of a pendulum

In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, where the capacity of the final form to do thermodynamic work is less than that of the initial form. For example, transfer of energy as heat is dissipative because it is a transfer of energy other than by thermodynamic work or by transfer of matter, and spreads previously concentrated energy. Following the second law of thermodynamics, in conduction and ra

constant solution to a differential equation

parameters that describe the configuration of the system relative to some reference configuration

mathematical quantity

pendulum with another pendulum attached to its end

mathematical constants

in analytical mechanics, the work of a force on a particle along a virtual displacement

mathematical model of a system based on the use of a linear operator

method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs; uses statistical methods to build mathematical models of dynamical systems from measured data

area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations

property of a dynamical system where solutions near an equilibrium point remain so

non-conservative oscillator with non-linear damping

In mathematics, linearization (British English: linearisation) 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 interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.

closed trajectory in a 2d phase space of a dynamical system such that that another trajectory spirals into it as time approaches ±∞

Displacement in analytical mechanics

chaotic dynamical system introduced by Michel Hénon

mathematical or physical system whose future states are not affected by random chance

the rate of separation of infinitesimally close trajectories

mathematical operation of composing a function with itself repeatedly

mathematical equations

characteristic timescale on which a dynamical system is chaotic

geometric representation

scientific educational toy

the set of values that a process can take in a dynamical system

property of certain dynamical systems

thumb|300px|Schematic representation of a self-oscillation as a positive feedback loop. The oscillator V produces a feedback signal B. The controller at R uses this signal to modulate the external power S that acts on the oscillator. If the power is modulated in phase with the oscillator's velocity, a negative damping is established and the oscillation grows until limited by nonlinearities.

type of map used in mathematics, particularly dynamical systems

time behavior

dynamical system that exhibits both continuous and discrete dynamic behavior

concept in mathematics

(of a dynamical system) function of the dependent variables that is a constant (in other words, conserved) along each trajectory of the system

branch of ordinary differential equations

invariant of homeomorphisms of the circle

harmonic oscillator whose parameters oscillate in time

mathematical description of the behavior of bouncing balls

On topology of algebraic curves and surfaces

dynamical system abstract an ideal game of billiards, with elastic collisions off boundaries

Type of mathematical system

frameworks for modeling variables that evolve over time

mathematical problem

change of state over time, especially in physics

system where the output depends only on past and current inputs

branch of mathematics about iteration of complex-valued functions

mathematical formalization of the motion of particles in a fluid

node-link graph representing overlaps between sequences of symbols