Phase Plane

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Phase Plane

Definition: Linear Systems of DEs

A n×n linear system of DEs has the form

x1(t)=a11x1+a12x2++a1nxn+f1x2(t)=a21x1+a22x2++a2nxn+f2xn(t)=an1x1+an2x2++annxn+fn

Definition: Nonlinear Systems of DEs

A n×n nonlinear system of DEs is not in the above linear form.

Definition: 2D Phase Plane

  1. The phase plane is the x1x2-plane, i.e., R2, for the system of DEs: xx(t)=AA(t)xx(t)+ff(t).

Nullclines in 2D Phase Planes

Definition: Nullclines x1-nullclines are sets of points in a phase plane such that dx1dt=0. x2-nullclines are sets of points in a phase plane such that dx2dt=0.

Equilibrium Solutions in 2D Phase Planes

Definition: Equilibrium Solutions

Let AA be the [nonsingular, i.e., det(AA)00] 2×2 coefficient matrix of a system of DEs.

Solutions xx for which AxAx=00 are critical points and correspond to equilibrium solutions, a.k.a., fixed points, to the system of DEs.

Equilibrium points occur at the intersection(s) of the x1-nullclines with the x2-nullclines.

Example 01:

Find the nullclines and equilibrium solutions for

x1(t)=x2x121+x12+x22x2(t)=x1x221+x12+x22

Solution:

  1. Find x1 nullclines: x1(t)=x2x121+x12+x22=set0x2=x12

  2. Find x2 nullclines: x2(t)=x1x221+x12+x22=set0x1=x22

  3. Find equilibrium solutions, solve for x1 and x2 using the above two equations, i.e., \newline substitute x2=x12 into x1=x22: x1=(x12)2x14+x1=0x1(x13+1)=0x1=0,1. Thus, the equilibrium points are (0,0) and (1,1).

Here is the graph of the nullclines in the phase plane for the above system of DEs.

Phase Portraits

Definition: Trajectory

A trajectory is an implicit solution to an IVP of a system of DEs.

Definition: Phase portrait

A phase portrait is a plot of several solutions, i.e., trajectories, to the system of DEs.

Here is the graph of the nullclines with several solution curves, i.e., trajectories, for the above system of DEs. Note the behavior of the solutions near the equilibrium solutions.

Here is the graph of the nullclines with several solution curves, i.e., trajectories, in the phase plane for the above system of DEs.

Classification of Equilibrium Points

Unstable Saddle

Asymptotically Stable Node

Unstable Node

Stable Center

Asymptotically Stable Spiral

Unstable Spiral

Equilibrium Solutions in 2×2 Linear Systems

Let AA be the 2×2 coefficient matrix of a linear system of DEs with eigenvalues λ1 and λ2.

Unstable Saddle

The eigenvalues of AA are of different signs, e.g., λ1<0<λ2

Asymptotically Stable Node

The eigenvalues of AA are distinct and negative, e.g., λ1<λ2<0

Unstable Node

The eigenvalues of AA are distinct and positive, e.g., λ2>λ1>0

Stable Center

The eigenvalues of AA are complex with α=0, i.e., λ1,2=±iβ with β>0

Asymptotically Stable Spiral

The eigenvalues of AA are complex, i.e., λ1,2=α±iβ with α<0 and β>0

Unstable Spiral

The eigenvalues of AA are complex, i.e., λ1,2=α±iβ with α>0 and β<0

Defective Node

The eigenvalues of AA are a non-zero, repeated real number with one linearly independent eigenvector. The defective node is unstable if λ>0 and asymptotically stable if λ<0

Star Node

The eigenvalues of AA are a non-zero, repeated real number with two linearly independent eigenvectors. The defective node is unstable if λ>0 and asymptotically stable if λ>0

Example 02:

Consider the system: x1(t)=x1(3x1+x2)x2(t)=6x2

Solution:

NOTE: This system of DEs is non-linear since the first equation has x12.

NOTE: The phase portrait of the system is:

NOTE:

x1 nullclines: x1(t)=x1(3x1+x2)=set0x1=x23 and x1=0. x2 nullclines: x2(t)=6x2=set0x2=0.

NOTE: By solving for all solutions to the three equations [nullclines]: x1=x23, x1=0, and x2=0, the only equilibrium point is (0,0). The equilibrium point is an unstable saddle.

Investigation 01

For each system of DEs below, answer the following questions:

  1. x1(t)=3x1+x2x2(t)=4x2

  2. x1(t)=5x1+x2x2(t)=2x1+5x2

  3. x1(t)=6x1+5x2x2(t)=7x12x2

  4. x1(t)=x1+2x2x2(t)=2x1x2

  5. x1(t)=x1x2x2(t)=x1+3x2

  6. x1(t)=x1x2x2(t)=x1+3x2

  7. x1(t)=x1x26x1x2(t)=x1x22x1+x22

  8. x1(t)=x12x1x2x2(t)=x1x23x1+2

  9. x1(t)=(x1x2)(1x1x2)x2(t)=x1(2+x2)

  10. x1(t)=3x12x2,x1(0)=3x2(t)=2x12x2,x2(0)=12

Investigation 02

For each system of DEs below, answer the following questions:

  1. x1(t)=3x1+x2x3x2(t)=x1+2x2x3x3(t)=3x1+3x2x3

  2. x1(t)=x1+x2,x1(0)=2x2(t)=x1+2x2+x3,x2(0)=2x3(t)=3x2x3,x3(0)=1

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Last Modified: Thursday, 15 October 2020 6:42 EDT