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

Equilibrium Points

What is an equilibrium point?

• For a system of coupled differential equations, an equilibrium point is a point at which both and 

  • Because both derivatives are zero, the rates of change of both and are zero

  • This means that and will not change, and therefore that if the system is ever at the point then it will remain at that point forever

• An equilibrium point can be stable or unstable

  • An equilibrium point is stable if for all points close to the equilibrium point the solution trajectories move back towards the equilibrium point

    • This means that if the system is perturbed away from the equilibrium point, it will tend to move back towards the state of equilibrium

  • If an equilibrium point is not stable, then it is unstable

    • If a system is perturbed away from an unstable equilibrium point, it will tend to continue moving further and further away from the state of equilibrium

• For a system that can be represented in the matrix form , where , , and , the origin is always an equilibrium point

  • Considering the nature of the phase portrait for a particular system will tell us what sort of equilibrium point the origin is

  • If both eigenvalues of the matrix  are real and negative, then the origin is a stable equilibrium point

    • This sort of equilibrium point is sometimes known as a sink

  • If both eigenvalues of the matrix  are real and positive, then the origin is an unstable equilibrium point

    • This sort of equilibrium point is sometimes known as a source

  • If both eigenvalues of the matrix  are real, with one positive and one negative, then the origin is an unstable equilibrium point

    • This sort of equilibrium point is known as a saddle point (you will be expected to identify saddle points if they occur in an AI HL exam question)

  • If both eigenvalues of the matrix  are imaginary, then the origin is an unstable equilibrium point

    • Recall that for all points other than the origin, the solution trajectories here all ‘orbit’ around the origin along circular or elliptical paths

  • If both eigenvalues of the matrix  are complex with a negative real part, then the origin is an stable equilibrium point

    • All solution trajectories here spiral in towards the origin

  • If both eigenvalues of the matrix  are complex with a positive real part, then the origin is an unstable equilibrium point

    • All solution trajectories here spiral away from the origin