Topological conjugacyIn mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that and are iterated functions, and there exists a homeomorphism such that so that and are topologically conjugate.
Chaos theoryChaos 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 conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.
Nonlinear systemIn mathematics and science, a nonlinear system (or a non-linear 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 since 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.
Dyadic transformationThe dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) (where is the set of sequences from ) produced by the rule Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function The name bit shift map arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
Tent mapIn mathematics, the tent map with parameter μ is the real-valued function fμ defined by the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ the unit interval [0, 1] into itself, thus defining a discrete-time dynamical system on it (equivalently, a recurrence relation). In particular, iterating a point x0 in [0, 1] gives rise to a sequence : where μ is a positive real constant.
PredictabilityPredictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Causal determinism has a strong relationship with predictability. Perfect predictability implies strict determinism, but lack of predictability does not necessarily imply lack of determinism. Limitations on predictability could be caused by factors such as a lack of information or excessive complexity. In experimental physics, there are always observational errors determining variables such as positions and velocities.
Feigenbaum constantsIn mathematics, specifically bifurcation theory, the Feigenbaum constants ˈfaɪɡənˌbaʊm are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.
Butterfly effectIn 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 later state. The term is closely associated with the work of mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier.
Recurrence plotIn 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., when the phase space trajectory visits roughly the same area in the phase space as at time . In other words, it is a plot of showing on a horizontal axis and on a vertical axis, where is the state of the system (or its phase space trajectory). Natural processes can have a distinct recurrent behaviour, e.g.
Sharkovskii's theoremIn mathematics, Sharkovskii's theorem (also spelled Sharkovsky's theorem, Sharkovskiy's theorem, Šarkovskii's theorem or Sarkovskii's theorem), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. For some interval , suppose that is a continuous function.
