Systems and stability · Definition

Phase plane

differential equationssystems

Before you start you need to know

It lets you see all solutions of a two-equation system at once, without solving it, by dropping time as an axis.

Definition

Coordinate plane \((x,y)\) of an autonomous system whose velocities \(f\) and \(g\) do not depend on \(t\); each solution is drawn there as a curve, its trajectory.

\[x'=f(x,y),\quad y'=g(x,y)\;\Longrightarrow\;\dfrac{dy}{dx}=\dfrac{g(x,y)}{f(x,y)}\quad(f\neq0)\]

Solving \(x'=f(x,y),\ y'=g(x,y)\) gives a pair \((x(t),y(t))\) that, as \(t\) varies, traces a curve in the \((x,y)\) plane. That curve is the trajectory; an arrow on it marks the direction in which time advances. The set of trajectories with their arrows is the phase portrait, and time no longer appears as an axis.

At each point, the vector \((f,g)\) is the velocity and is tangent to the trajectory. Since it does not depend on \(t\), exactly one trajectory passes through each non-equilibrium point and two different trajectories never cross. Where \(f\neq0\) the slope is \(dy/dx=g/f\), a first-order equation that can sometimes be solved by separation of variables.

Guiding example: for \(x'=y,\ y'=-x\) we have \(dy/dx=-x/y\), and \(x\,dx+y\,dy=0\) gives \(x^2+y^2=c\). The trajectories are circles traversed clockwise. The equilibria, where \(f=g=0\), are one-point trajectories, and the behavior of the system is read around them.

Typical mistake: confusing the trajectory with the graph of \(x\) against \(t\); the trajectory does not say when the point passes each place, only where it passes. This concept prepares the stability of equilibria, which is read directly from the phase portrait.

Think about it

In \(x'=x,\ y'=-y\), which curves are the trajectories?

We have \(dy/dx=-y/x\), so \(xy=c\): hyperbolas, with arrows moving away from the \(y\)-axis and toward the \(x\)-axis.

Exercises 10

  1. For \(x'=x+y,\ y'=x-y\), what is the slope \(dy/dx\) of the trajectory through \((2,1)\)?
  2. For \(x'=x-2y,\ y'=3x+y\), what is the velocity vector \((x',y')\) at the point \((2,1)\)?
  3. What are the equilibria of \(x'=x(1-y),\ y'=y(x-1)\)?
  4. The trajectories of \(x'=2x,\ y'=y\) are the curves:
  5. In the system \(x'=y,\ y'=-x\), which expression stays constant along each trajectory?
  6. For \(x'=y,\ y'=x\), which point lies on the same trajectory as \((1,0)\)?
  7. For \(x'=-x,\ y'=-2y\), what happens as \(t\to\infty\) to the solution starting at \((1,1)\)?
  8. In an autonomous system with smooth \(f\) and \(g\), can two different trajectories cross at a point that is not an equilibrium?
  9. For \(x'=y,\ y'=-4x\), which of these points lies on the trajectory through \((1,0)\)?
  10. In \(x'=y,\ y'=-x\), in which sense are the circular trajectories traversed?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §9.1
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §10.1
  • Steven H. Strogatz, Nonlinear Dynamics and Chaos, 2nd edition, Westview Press, 2015. §6.1