Bifurcation theory

A bifurcation or branching is a qualitative change of state in nonlinear systems under the influence of a parameter μ. The concept of bifurcation was introduced by Henri Poincaré.

Nonlinear systems whose behavior depends on a parameter, can suddenly change their behavior with a change of the parameter. For example, a system that zustrebte above a threshold, then jump between two values ​​back and forth, so have two accumulation points. This is called a bifurcation. Certain systems can learn infinitely many bifurcations under finite change of the parameter μ, and thus have an infinite amount of cumulative points. The behavior of such systems is transformed by changing the parameter μ thus deterministic chaotic behavior. An example of this is the logistic map.

Definition of a bifurcation

A dynamic system can be described by a function that determines the time evolution of the system state. This function is now on a parameter μ depends on what one expresses by the notation. Now, if the system parameter values ​​below a certain critical value has a qualitatively different behavior for values ​​above, then one speaks the fact that the system undergoes a bifurcation in the parameter. The parameter value is then referred to as bifurcation.

What a "qualitative change", one can formally describe the notion of topological equivalence or topological conjugation: as long as two are for parameter values ​​and the systems and topologically equivalent to each other, is no qualitative change in the above sense before.

The change in the bifurcation, in most cases, either a change in the number of fixed points or attractors as periodic orbits, or a change in the stability of these properties.

Bifurcation diagram

Bifurcations can be represented graphically in Bifurkationsdiagrammen. In a one-dimensional system while the fixed points of the system are plotted against the parameter. As the number and location of these points is displayed for each parameter value. Additionally, you can stable and unstable fixed points eg distinguished by different color. In a system with multiple variables can be similar diagrams are characterized by considering only a subspace of the phase space, such as by a Poincaré section.

The best-known bifurcation diagram is the fig tree diagram to the right, which is derived from the logistic equation and a Periodenverdoppelungsbifuraktion maps. It can be seen that for small parameter values, only one stable fixed point exists, which merges into an orbit of two alternating cluster points at the first bifurcation point. This orbit then doubled to further bifurcation points every time his period ( only comes after 2, 4, 8, etc. runs back to the same point ) until it merges with a parameter value of about 3.57 in a chaotic state where no longer period is visible. All these transitions can be well illustrated using the Bifurkationsdiagrammes.

Example

A typical example of a bifurcation, the buckling of a bar under pressure.

One in the bottom clamped, vertical, massless rod Imagine having a weight μ at the top. The angular deviation of the rod from the vertical corresponds to the variable x.

As long as the weight is small enough, x = 0 is a stable equilibrium position of the system, ie for small deviations of the rod is directed again independently in the vertical (x = 0) from. If the weight μ increased continuously, so at a certain weight ( the buckling load or buckling load ) the vertical position of equilibrium is unstable. At the same result ( for a planar system) ( by the rod bends to the left or right. ) Two new (stable ) equilibrium, the transition of the system from a ( stable ) to three ( two stable unstable ) equilibrium is the bifurcation, which in this case is a pitchfork bifurcation.

( See also: Elastic theory )

Types of bifurcations

Literature and sources

  • Hassan K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, 2002, ISBN 0-13-067389-7
  • Leash, R.I. & Nijmeijer, H. Dynamics and Bifurcations in Non- Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics Vol 18, Berlin Heidelberg New York, Springer -Verlag, 2004, ISBN 3-540-21987-0
  • Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, ISBN 0-738-20453-6
Topological conjugacy Orbit (disambiguation) Elastica theory Complex system
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