In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...
An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the x-axis of the plot and the y-axis shows the possible long-term population values of the logistic function. The logistic map is a polynomial mapping, often cited as an archetypical example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ...
Image File history File links Burification diagram of a logistic map Released by the author (User:Ap) into the public domain. ...
The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the Feigenbaum constant. This value is traditionally known as the first Feigenbaum constant. There is a second Feigenbaum constant, also. In mathematics, a series is the sum of of the terms of a sequence of numbers. ... There are two mathematical constants called Feigenbaum constants, named after mathematician Mitchell Feigenbaum. ...
An interesting feature of this diagram is that as the periods go to infinity, r remains finite. When r is greater than (approximately) 3.57, the orbits become chaotic. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple non-linear systems. Chaos theory, in mathematics and physics, deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions (see butterfly effect). ...
The parameter k is plotted along the x-axis of the bifurcationdiagram (from 1 on the right to 4 on the left).
For small and medium k values (to the left) what the bifurcationdiagram shows is that (as we found earlier), the population reaches a single, stable population size (which increases with increasing values of k).
The bifurcationdiagram is constructed by taking a particular value for the starting population size (x=0.125 is the default value, as shown below the staircase diagram) and then, for each value of k, iterating it a number of times (70 is the default value, as shown in the left slider above the diagram).
For all these reasons, a bifurcationdiagram is a particularly powerful method for studying the attractors in the quadratic map.
Recall that a bifurcationdiagram is a plot of an asymptotic solution on the vertical axis and a control parameter on the horizontal axis.
Much of the structure in the bifurcationdiagram can only be understood by keeping track of both the stable attracting solutions and the unstable repelling solutions, as we did in constructing the orbit stability diagram (Figure 2.13).
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