Bifurcations of dynamical systems

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As a parameter is varied in a dynamical system, new types of solutions may be created or destroyed, and/or the stability of various states might change. These qualitative changes in the behavior of the differential equation are referred to as bifurcations. All sorts of dynamical systems have bifurcations of practical importance. For instance, you can see a brief discussion of why bifurcations are important to switching of the cell cycle, which is pretty fundamental to our existence! Personally, my favorite bifurcations are those that lead to the formation of patterns (much of my research is on pattern forming bifurcations in partial differential equations). For instance, check out this movie, in which the parameters being varied are the amplitude and frequency used to shake a container that has corn starch solution inside. Another type of pattern-forming bifurcation is thought to be responsible for the appearance of animal coat patterns.

By the end of this lesson, you should be able to:

  • Explain what a bifurcation is.
  • Connect information about fixed points and stability with bifurcation diagrams.
  • Describe the qualitative change associated with transcritical and pitchfork bifurcations.
  • Determine whether an ordinary differential equation has saddle-node, trasncritical, and/or pitchfork bifurcation, and if so, where.
  • Distinguish between subcritical and supercritical pitchfork bifurcations.
  • Explain bistability, jumps, and hysteresis in context of a subcritical pitchfork bifurcation.

Before class, please:

In class, we will:

After class, please: