Concept
Bifurcation diagram
Related concepts (11)
Attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.
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 solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
Feigenbaum constants
In 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.
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 one having double the period of the original. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.
Sharkovskii's theorem
In 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.
Rössler attractor
The Rössler attractor ˈrɒslər is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine. Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.
Logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written where xn is a number between zero and one, which represents the ratio of existing population to the maximum possible population.
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, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.
Periodic point
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Given a mapping f from a set X into itself, a point x in X is called periodic point if there exists an n so that where f_n is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x.
Mandelbrot set
The Mandelbrot set (ˈmændəlbroʊt,_-brɒt) is a two dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at , i.e., for which the sequence , , etc., remains bounded in absolute value. This set was first defined and drawn by Robert W.