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The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.
The bifurcation diagram is a self-similar: if we zoom in on the above-mentioned value r ≈ 3.82843 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points.
To see how this number arises, consider the real one-parameter map =.Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a 1, a 2 etc.
If μ is greater than the square root of 2, these intervals merge, and the Julia set is the whole interval from μ − μ 2 /2 to μ/2 (see bifurcation diagram). If μ is between 1 and 2 the interval [μ − μ 2 /2, μ/2] contains both periodic and non-periodic points, although all of the orbits are unstable (i.e. nearby points move away from ...
For example, by keeping b fixed at 0.3 the bifurcation diagram shows that for a = 1.25 the Hénon map has a stable periodic orbit as an attractor. Variation of 'b' showing the Bifurcation diagram. The boomerang shape is further drawn in bold at the top. Initial coordinates for each cross-section is (0, -0.2). Achieved using Python and Matplotlib.
Bifurcation diagram of the logistic map: Feigenbaum noticed in 1975 that the quotient of successive distances between bifurcation events tends to 4.6692... Work [ edit ]
In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems.
Bifurcation diagram for the Rössler attractor for varying Here, a {\displaystyle a} is fixed at 0.2, c {\displaystyle c} is fixed at 5.7 and b {\displaystyle b} changes. As shown in the accompanying diagram, as b {\displaystyle b} approaches 0 the attractor approaches infinity (note the upswing for very small values of b {\displaystyle b} ).