Phase Planes and Stability

1Introduction

This lecture explains how to represent two-dimensional systems in the phase plane and classify trajectories near equilibria by eigenvalues and eigendirections.

The phase plane uses the state $$(x,y)$$(x,y), rather than time, as its coordinates. The image traced by a solution $$\left(x\right(t),y(t\left)\right)$$(x(t),y(t)), equipped with the direction of increasing time, is an orbit. The vector field that assigns the velocity $$\mathit{F}(x,y)$$\boldsymbol{F}(x,y) to each point, together with its family of orbits, is a phase portrait.

2Standard Form and Stability Terms

We study either

or a nonlinear system $$x^{\prime }=\mathit{F}\left(\mathit{x}\right)$$\boldsymbol{x}'=\boldsymbol{F}(\boldsymbol{x}). For a nonlinear system, first find each equilibrium and then linearize with the Jacobian at that equilibrium.

An equilibrium is Lyapunov stable when every trajectory starting sufficiently near it remains near it. It is locally attractive when every trajectory in some neighborhood converges to it. Both properties together give local asymptotic stability. If these conditions and convergence hold for every initial value with global forward existence, the equilibrium is globally asymptotically stable.

3Classification of Real Two-Dimensional Linear Systems

The following table classifies the exact linear system $$x^{\prime }=A\mathit{x}$$\boldsymbol{x}'=A\boldsymbol{x}. A node approaches or departs along real eigendirections; a saddle has both approaching and departing directions; a focus rotates while approaching or departing; a center is surrounded by closed orbits.

For repeated real eigenvalues, inspect the eigenspace dimension and hence the Jordan blocks. Both $$-I$$-I and $$\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)$$\begin{pmatrix}-1&1\\0&-1\end{pmatrix} are globally asymptotically stable, but only the first has every direction as an eigendirection; the second contains a $$t{e}^{-t}$$te^{-t} term. Away from the imaginary axis, Jordan structure changes orbit geometry without changing decay or growth. On the imaginary axis, block size also changes Lyapunov stability.

If zero is an eigenvalue, then $$\ker A\ne \left\{0\right\}$$\ker A\ne\{\boldsymbol{0}\}; the origin is nonisolated, and all of $$\ker A$$\ker A consists of equilibria. A Jordan block of size at least two for zero produces a direction with polynomial growth in $$t$$t. Thus zero-eigenvalue cases are degenerate boundary cases, not centers covered by the pure-imaginary row.

4Classification Procedure

1. Solve $$\mathit{F}\left(\mathit{x}\right)=0$$\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0} for the equilibria.
2. Use the coefficient matrix $$A$$A for a linear system or the Jacobian $$D\mathit{F}$$D\boldsymbol{F} at each equilibrium for a nonlinear system.
3. Determine stability from real parts. For real eigenvalues, inspect eigendirections and Jordan structure; for complex eigenvalues, determine rotation direction from the vector field.
4. Add arrows in the direction of increasing time and verify approach or departure against the definitions.
5. For a nonlinear system with no positive real part but with a zero real part, do not infer stability from linearization alone. For the exact linear system, Jordan structure gives the complete criterion.

5Example 1: Node and Saddle

For

the eigenvalues are $$-1,-2$$-1,-2, so the origin is a stable node. For

the eigenvalues are $$1,-1$$1,-1, so the origin is a saddle. Although solutions on the stable eigendirection approach the origin, arbitrarily close initial values depart along the unstable direction; hence the saddle is Lyapunov unstable.

6Example 2: Stable Focus and Rotation Direction

For

the eigenvalues are $$-1\pm 2i$$-1\pm2i. The negative real part gives exponential radial decay, and the nonzero imaginary part gives rotation. At $$(1,0)$$(1,0) the velocity is $$(-1,2)$$(-1,2), whose upward component shows counterclockwise rotation. The origin is therefore a counterclockwise stable focus. In general the sign of the imaginary part alone does not determine orientation without reference to the matrix or vector field.

7Example 3: Nonlinear Linearization

For

the equilibria are $$(0,0)$$(0,0) and $$(1,0)$$(1,0), and

At $$(0,0)$$(0,0) the eigenvalues are $$1,-1$$1,-1, so the nonlinear equilibrium is unstable. At $$(1,0)$$(1,0) they are $$-1,-1$$-1,-1, so the nonlinear equilibrium is locally asymptotically stable. These are local conclusions obtained separately at each equilibrium; they do not make the nonlinear system globally linear.

8Scope

For nonlinear systems, linearization contains only local information. When no eigenvalue has positive real part and at least one has zero real part, nonlinear terms may create either stability or instability. A linearized center therefore need not imply a nonlinear center. For the exact linear system, by contrast, the eigenvalues together with the Jordan blocks on the imaginary axis provide a complete stability criterion.

Up to this point, we have studied trajectories of autonomous systems in state space. The next lecture shifts to forcing responses on the time axis and treats linear equations with step and impulse inputs. It begins a different strand of ordinary-differential-equation analysis rather than serving as a direct continuation of phase-plane theory.

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