Bifurcation (mathematics)

A bifurcation or branching is a qualitative change of state in nonlinear systems under the influence of a parameter . The concept of bifurcation was introduced by Henri Poincaré . ${\ displaystyle \ mu}$

Nonlinear systems, the behavior of which depends on a parameter, can suddenly change their behavior when the parameter changes. For example, a system that was previously aiming for a limit value can now jump back and forth between two values, i.e. have two accumulation points. This is called a bifurcation . Certain systems can experience an infinite number of bifurcations with a finite change in the parameter and thus have an infinite number of accumulation points. The behavior of such systems changes to deterministic chaotic behavior when the parameter is changed . An example of this is the logistic mapping . ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$

definition

A dynamic system can be described by a function that determines the development of the system status over time . This function is now dependent on a parameter , which is expressed by the notation . If the system has a qualitatively different behavior for parameter values below a certain critical value than for values above , then one speaks of the system experiencing a bifurcation in the parameter . The parameter value is then referred to as the bifurcation point. ${\ displaystyle F (x)}$ ${\ displaystyle x}$ ${\ displaystyle \ mu}$ ${\ displaystyle F (x, \ mu)}$ ${\ displaystyle \ mu _ {c}}$ ${\ displaystyle \ mu _ {c}}$ ${\ displaystyle \ mu _ {c}}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu _ {c}}$

What a “qualitative change” is can be formally described with the term topological equivalence or topological conjugation : As long as the systems and are topologically equivalent to each other for two parameter values and , there is no qualitative change in the above sense. ${\ displaystyle \ mu _ {1}}$ ${\ displaystyle \ mu _ {2}}$ ${\ displaystyle F (x, \ mu _ {1})}$ ${\ displaystyle F (x, \ mu _ {2})}$

The change at the bifurcation point consists in most cases either in a change in the number of attractors such as fixed points or periodic orbits, or in a change in the stability of these objects.

Bifurcation diagram

Figure 1: Bifurcation diagram of the logistic equation

Bifurcations can be represented graphically in bifurcation diagrams. In the case of a one-dimensional system, the fixed points of the system are plotted against the parameter . The number and position of these points is displayed for each parameter value. In addition, you can have stable and unstable fixed points such. B. can be distinguished by different colors. In a multi-variable system, similar diagrams can be drawn by looking at only a subspace of phase space , such as a Poincaré cut . ${\ displaystyle \ mu}$

The best-known bifurcation diagram is the fig tree diagram shown in Figure 1 , which is derived from the logistic equation and depicts a period doubling bifurcation. It can be seen that with small parameter values there is only one stable fixed point, which merges into an orbit of two alternating accumulation points at the first bifurcation point. This orbit then doubles its period each time at further bifurcation points (i.e. only comes back to the same point after 2, 4, 8 etc. passes) until it changes into a chaotic state at a parameter value of around 3.57 , wherever at all no more period is recognizable. All these transitions can be clearly illustrated with the help of the bifurcation diagram.

example

A typical example of a bifurcation is the buckling of a rod under pressure.

Imagine a vertically standing, massless rod with a weight at the top, clamped in the ground . The angular deviation of the bar from the vertical corresponds to the variable x. ${\ displaystyle \ mu}$

As long as the weight remains small enough, the system is in a stable equilibrium position , i.e. H. For small deviations, the rod automatically realigns itself to the vertical ( ). If the weight is continuously increased, the vertical equilibrium position becomes unstable at a certain weight (the buckling load or also branching load ). At the same time (for a plane system) two new (stable) equilibrium positions arise (by bending the rod to the left or right.) The transition of the system from one (stable) to three (one unstable, two stable) equilibrium positions is the bifurcation, which in in this case is a pitchfork bifurcation . ${\ displaystyle x = 0}$ ${\ displaystyle x = 0}$ ${\ displaystyle \ mu}$

Types

Literature and Sources

Hassan K. Khalil: Nonlinear Systems. 3. Edition. Prentice Hall, 2002, ISBN 0-13-067389-7 .
RI Leine, H. Nijmeijer: Dynamics and Bifurcations in Non-Smooth Mechanical Systems. In: Lecture Notes in Applied and Computational Mechanics. Vol. 18, Springer-Verlag, Berlin / Heidelberg / New York 2004, ISBN 3-540-21987-0 .
Steven H. Strogatz : Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Perseus Books Group, ISBN 0-7382-0453-6 .

Web links

Wiktionary: Bifurcation - explanations of meanings, word origins, synonyms, translations

World of Bifurcation (English)
Bifurcations and Two Dimensional Flows by Elmer G. Wiens (English)