Equilibrium Points (DP IB Applications & Interpretation (AI): HL): Revision Note

Equilibrium points

What is an equilibrium point?

• An equilibrium point for a system of coupled differential equations is a point $(x,y)$ at which both $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$

• If a solution trajectory starts on an equilibrium point

  • it will never move from that point

• For example, consider $\frac{dx}{dt}=2x+3y-8$ and $\frac{dy}{dt}=3x-y-1$

  • Solve $2x+3y-8=0$ and $3x-y-1=0$

  • $\left(1,2\right)$ is the equilibrium point

What are stable and unstable equilibrium points?

• An equilibrium point can be stable or unstable

• An equilibrium point is stable if

  • all solution trajectories which start close to the equilibrium point remain close to the equilibrium point

• An equilibrium point is unstable if

  • there is any solution trajectory which starts close to the equilibrium point and moves away from the equilibrium point

• A saddle point is an example of an unstable equilibrium point

  • Some solution trajectories which start close to the saddle point move towards the saddle point

  • Whereas other solution trajectories which start close to the saddle point move away from the saddle point

How can I determine the nature of an equilibrium point?

• You can use a phase portrait to determine the nature of the equilibrium point

• The origin $(0,0)$ is always an equilibrium point $\dot{\mathit{x}}=\mathit{M}\mathit{x}$, where

  • $\dot{\mathit{x}}=\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)$

  • $\mathit{M}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

  • $\mathit{x}=\left(\begin{array}{c}x\\ y\end{array}\right)$