Abstract

Citation

Researchers should cite this work as follows:

• Swim, E. (2022). 1-009-Bifurcation-TechniqueNarrative. SIMIODE, QUBES Educational Resources. doi:10.25334/BTYP-JP32

  BibTex | EndNote

Article Context

Description

Bifurcation analysis is a useful mathematical tool applied in a wide variety of application areas such as neural network models for human sensory systems, quantum motion in molecular systems, density dependent models of logistic growth in ecology,, etc. In these situations, solutions to initial value problems are often sensitive to changes in the value of one or more parameter within the model.

In particular, it is possible that an equilibrium solution may suddenly change from being asymptotically stable to unstable as parameter values are varied. For example, in quantum mechanics a system of particles may change phase at absolute zero temperature based upon changes in quantum momentum or some other parameter.

Whenever quantum momentum is high enough, the system is typically disordered, while at lower momentum values a topologically ordered state of matter may be observed. Unlike thermal phase transitions  between solid and liquid states where changes in pressure at a fixed positive temperature may produce mesophases such as liquid crystal, a quantum change of phase occurs suddenly for a single value of one or more parameters.

Frequently, these changes are modeled using differential equations for local quantities such as magnetization where a singularity in a derivative indicates the instantaneous change in topological order .

In the setting of ordinary differential equations in one real variable, models for various biological systems include a similar behavior. Species of animals where reproductive rates depend highly on food availability at a time prior to when eggs hatch have population dynamics modeled by a delay modification to the logistic equation