Advanced 1-d flows

Get inspired:

Stability and instability are amongst the most fundamental concepts of dynamical systems theory. Though you will use some abstract mathematics to study issues of stability, never forget that for differential equations that model real systems in the world, stability has very concrete, practical implications. For instance, in my own research, I use partial differential equations to model locust swarms, which have caused up to billions of dollars in crop losses during some years. Under some conditions, a state in which the locusts are dispersed in the environment is stable. But under other conditions, this state becomes unstable, and is replaced by a state in which the locust aggregate and eat voraciously. Thus, the difference between stability and instability is the difference between this and this.

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

  • Analyze linear stability of fixed points.
  • Interpret linear stability geometrically.
  • Compute the characteristic time scale near a fixed point.
  • Distinguish between local and global existence of solutions.
  • Determine if \dot{x}=f(x) has local existence.
  • Discuss all qualitatively different possibly behaviors of \dot{x}=f(x).
  • Analyze \dot{x}=f(x) via a potential.

Before class, please:

In class, we will:

After class, please: