Phase Plane Analysis and Nonhomogeneous Systems

MATH 341 — Notes 11

University of Scranton

2026-06-16

Goals

What We Will Cover

  1. Table 4.1 and Theorem 4.41 — eigenvalue classification and stability criterion \(\text{tr}\,A<0\), \(\det A>0\)

  2. Theorem 4.40 — the trace–determinant plane; reading off the phase portrait type at a glance

  3. Drawing phase diagrams — nullclines, direction fields, eigenvectors

  4. Applied examples — crop–soil, glucose–insulin, circuits, chemical reactors

  5. Fundamental matrix \(\Phi(t)\) for homogeneous systems

  6. Variation of parameters for nonhomogeneous systems \(\mathbf{x}'=A\mathbf{x}+\mathbf{f}(t)\)

  7. Undetermined coefficients for nonhomogeneous systems

Note

Sections 4.5 and 4.6 of (Logan 2015).

Tip

New vocabulary: trace–determinant plane, stability theorem, nullcline, fundamental matrix, variation of parameters (systems), undetermined coefficients (systems).

Section 4.5

Phase Plane Analysis

Classification Summary (Table 4.1)

Eigenvalues of \(A\) determine everything about the phase portrait of \(\mathbf{x}' = A\mathbf{x}\):

Eigenvalues Portrait Stability
\(\lambda_1,\lambda_2>0\), real, unequal Unstable node Unstable
\(\lambda_1,\lambda_2<0\), real, unequal Stable node Asymp. stable
\(\lambda_1<0<\lambda_2\) Saddle Unstable
\(\lambda=\pm bi\) Center Stable
\(\lambda=a\pm bi\), \(a>0\) Unstable spiral Unstable
\(\lambda=a\pm bi\), \(a<0\) Stable spiral Asymp. stable
\(\lambda_1=\lambda_2>0\) Unstable node Unstable
\(\lambda_1=\lambda_2<0\) Stable node Asymp. stable

But we never need to compute eigenvalues to check stability:

Theorem 4.41 — The Stability Theorem

Theorem 4.41

The origin \((0,0)\) of \(\mathbf{x}'=A\mathbf{x}\) is asymptotically stable if, and only if, \[\text{tr}\,A < 0 \quad \text{and} \quad \det A > 0.\]

Why this works:

  • \(\det A > 0\): eigenvalues are either both-negative-real, or complex-conjugates (no zero eigenvalue, no opposite signs)
  • \(\text{tr}\,A = \lambda_1+\lambda_2 < 0\): real parts are negative

Tip

In applications: compute \(\text{tr}\,A\) and \(\det A\) in terms of parameters, then check their signs. This immediately settles stability without solving the characteristic equation.

Theorem 4.40 — The Trace–Determinant Plane

The parabola \((\text{tr}\,A)^2 = 4\det A\) is the discriminant boundary separating real eigenvalues (below/on) from complex eigenvalues (above).

Theorem 4.40

Plot the point \((\text{tr}\,A,\, \det A)\). The region it falls in determines the portrait:

Region Portrait
\(\det A < 0\) Saddle
\(\det A > 0\), \(\text{tr}\,A = 0\) Center
\(\det A > 0\), \(\text{tr}\,A > 0\), on/below parabola Unstable node
\(\det A > 0\), \(\text{tr}\,A > 0\), above parabola Unstable spiral
\(\det A > 0\), \(\text{tr}\,A < 0\), on/below parabola Stable node
\(\det A > 0\), \(\text{tr}\,A < 0\), above parabola Stable spiral

The Trace–Determinant Plane — Diagram

Figure 1

Using the Trace–Determinant Plane

Example. Classify the origin for each system:

System \(\text{tr}\,A\) \(\det A\) Discriminant Type
\(\begin{pmatrix}-2&-3\\3&-2\end{pmatrix}\) \(-4\) \(13\) \(16-52=-36<0\) Stable spiral
\(\begin{pmatrix}1&1\\4&1\end{pmatrix}\) \(2\) \(-3\) — Saddle
\(\begin{pmatrix}-2&0\\0&-5\end{pmatrix}\) \(-7\) \(10\) \(49-40=9>0\) Stable node

Tip

Workflow: (1) Compute \(\text{tr}\,A\) and \(\det A\). (2) Check \(\det A\): if \(<0\), it’s a saddle. (3) If \(\det A>0\), check \(\text{tr}\,A\) for stability. (4) Compare \((\text{tr}\,A)^2\) to \(4\det A\) for node vs. spiral.

How to Draw a Phase Diagram

(a) Real unequal eigenvalues: Draw linear orbits along eigenvectors (arrows toward origin for \(\lambda<0\), away for \(\lambda>0\)). Fill between with direction field. Add nullclines.

(b) Purely imaginary eigenvalues (\(\lambda=\pm bi\)): Orbits are closed ellipses. Compute one direction field vector to determine CW or CCW. Add nullclines.

(c) Complex eigenvalues (\(\lambda=a\pm bi\)): Orbits are spirals; sign of \(a\) determines in vs. out. Compute one direction field vector for orientation.

(d) \(\det A = 0\): Line of equilibria along \(ax+by=0\). Orbits are parallel lines \(y=mx+C\).

Applied Examples

Example 4.42 — Crop–Soil Model

Pesticide in crops (\(x\)) and soil (\(y\)): \[x'=-\beta x+\alpha y, \quad y'=\beta x-(\alpha+\gamma)y, \quad \alpha,\beta,\gamma>0.\]

Nullclines: \(x'=0\) on \(y=(\beta/\alpha)x\); \(\;y'=0\) on \(y=\beta/(\alpha+\gamma)\,x\).

The \(x\)-nullcline has greater slope than the \(y\)-nullcline.

Stability: \(\text{tr}\,A=-(\alpha+\beta+\gamma)<0\) and \(\det A=\beta\gamma>0\).

\(\implies\) Asymptotically stable by Theorem 4.41.

The pesticide eventually disappears from the system. \(\square\)

Note

We reached this conclusion without solving the system — just from \(\text{tr}\,A\) and \(\det A\). This is the power of the geometric approach.

Example 4.42 — Phase Portrait

Figure 2

Example 4.44 — Glucose–Insulin Dynamics

\[x'=-gx-ry,\quad y'=sx-dy, \quad A=\begin{pmatrix}-g&-r\\s&-d\end{pmatrix}, \quad g,r,s,d>0.\]

\(\text{tr}\,A=-(g+d)<0\) and \(\det A=gd+rs>0\) \(\implies\) always asymptotically stable.

Eigenvalues: \(\lambda=\tfrac{1}{2}\!\left(-(g+d)\pm\sqrt{(g-d)^2-4rs}\right)\).

Condition Eigenvalue type Behavior
\((g-d)^2>4rs\) Real, negative Stable node — monotone return
\((g-d)^2<4rs\) Complex, \(\text{Re}<0\) Stable spiral — decaying oscillations (hypoglycemic overshoot)

For \(g=2.9\), \(r=4.3\), \(s=0.21\), \(d=0.78\): \((g-d)^2-4rs \approx -3.22 < 0\) \(\implies\) stable spiral.

Figure 3

Example 4.45 — Electrical Circuit

Two-loop circuit (capacitor \(C\), resistor \(R\), inductor \(L\)):

\[I_2'=\frac{1}{RC}(-I_2+I_3),\quad I_3'=-\frac{R}{L}I_2. \qquad A=\begin{pmatrix}-\frac{1}{RC} & \frac{1}{RC}\\[4pt]-\frac{R}{L} & 0\end{pmatrix}.\]

\(\text{tr}\,A=-1/(RC)<0\) and \(\det A=1/(LC)>0\) \(\implies\) always asymptotically stable.

\((\text{tr}\,A)^2-4\det A = \dfrac{1}{(RC)^2}-\dfrac{4}{LC}\):

Condition Type
\(L/(R^2C)>4\) Node (overdamped — no oscillations)
\(L/(R^2C)<4\) Spiral (underdamped — decaying oscillations)

Note

The case \(L/(R^2C)<4\) models a resonant RLC circuit — currents oscillate while decaying toward zero.

Example 4.46 — Coupled Chemical Reactors

Two tanks, concentrations \(C_1(t)\), \(C_2(t)\) with \(c_\text{in}=0\):

\[C_1'=-\frac{q_1}{V}C_1+\frac{q_2}{V}C_2, \qquad C_2'=\frac{q_1}{V}C_1-\frac{q_2+q}{V}C_2.\]

\[A=\begin{pmatrix}-q_1/V & q_2/V \\ q_1/V & -(q_2+q)/V\end{pmatrix}.\]

\(\text{tr}\,A = -(q_1+q_2+q)/V < 0\) and \(\det A = q_1 q/V^2 > 0\).

\(\implies\) Asymptotically stable. Both concentrations \(\to 0\) as \(t\to\infty\). \(\square\)

Section 4.6

Nonhomogeneous Systems

Setup

We now study the nonhomogeneous system

\[\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t). \tag{4.43}\]

where \(\mathbf{f}(t)=(f_1(t),f_2(t))^T\) is a prescribed forcing function.

Structure of the general solution (same as for scalar equations):

\[\mathbf{x}(t) = \underbrace{\mathbf{x}_h(t)}_{\text{homogeneous}} + \underbrace{\mathbf{x}_p(t)}_{\text{particular}}.\]

Two methods for finding \(\mathbf{x}_p\):

  1. Variation of parameters — works for any \(\mathbf{f}(t)\)
  2. Undetermined coefficients — efficient when \(\mathbf{f}(t)\) is a polynomial, exponential, sine/cosine, or their products

The Fundamental Matrix

Definition

A fundamental matrix \(\Phi(t)\) is a \(2\times 2\) matrix whose columns are two independent solutions of the homogeneous system \(\mathbf{x}'=A\mathbf{x}\): \[\Phi(t)=\bigl(\mathbf{x}_1(t)\;\big|\;\mathbf{x}_2(t)\bigr).\]

Key properties:

  • \(\Phi'(t)=A\Phi(t)\) (satisfies the matrix ODE)
  • General solution of homogeneous system: \(\mathbf{x}_h(t)=\Phi(t)\mathbf{c}\) for arbitrary \(\mathbf{c}=(c_1,c_2)^T\)
  • \(\Phi(t)\) is invertible (columns are independent, so \(\det\Phi\neq 0\))

Variation of Parameters — Derivation

Assume a particular solution \(\mathbf{x}(t)=\Phi(t)\mathbf{c}(t)\) (vary the constants).

Differentiate: \[\mathbf{x}'(t)=\Phi(t)\mathbf{c}'(t)+\Phi'(t)\mathbf{c}(t)=\Phi(t)\mathbf{c}'(t)+A\Phi(t)\mathbf{c}(t).\]

Set equal to \(A\mathbf{x}+\mathbf{f}(t)\): \[\Phi(t)\mathbf{c}'(t)=\mathbf{f}(t)\implies \mathbf{c}'(t)=\Phi(t)^{-1}\mathbf{f}(t).\]

\(\Phi(t)^{-1}\) Is Time-Varying

\(\Phi(t)^{-1}\) is not a constant matrix — it is the matrix inverse evaluated at each \(t\) separately, using the time-varying entries of \(\Phi(t)\). For a \(2\times 2\) fundamental matrix this means applying the inverse formula \[\Phi(t)^{-1} = \frac{1}{\det\Phi(t)}\begin{pmatrix}d(t)&-b(t)\\-c(t)&a(t)\end{pmatrix}\] at every \(t\), where \(\det\Phi(t)\) is the Wronskian \(W(t)\). This is distinct from the constant-matrix inverse \(A^{-1}\) from Section 4.2.

Integrate:

Variation of Parameters Formula (4.45)

\[\mathbf{x}(t)=\underbrace{\Phi(t)\mathbf{k}}_{\text{general homogeneous}}+\underbrace{\Phi(t)\int_0^t\Phi(s)^{-1}\mathbf{f}(s)\,ds}_{\text{particular solution}}.\]

Example 4.47 — Variation of Parameters

\[\mathbf{x}'=\begin{pmatrix}4&3\\-1&0\end{pmatrix}\mathbf{x}+\begin{pmatrix}0\\t\end{pmatrix}.\]

Eigenpairs: \(\lambda_1=1\), \(\mathbf{v}_1=(1,-1)^T\); \(\;\lambda_2=3\), \(\mathbf{v}_2=(-3,1)^T\).

\[\Phi(t)=\begin{pmatrix}e^t&-3e^{3t}\\-e^t&e^{3t}\end{pmatrix},\qquad \Phi^{-1}(t)=-\frac{1}{2}\begin{pmatrix}e^{-t}&3e^{-t}\\e^{-3t}&e^{-3t}\end{pmatrix}.\]

Note

\(\Phi^{-1}(t)\) is computed by applying the \(2\times 2\) inverse formula to \(\Phi(t)\) at each \(t\): \(\det\Phi(t) = e^t\cdot e^{3t} - (-3e^{3t})(-e^t) = e^{4t}-3e^{4t} = -2e^{4t}\), so \(\Phi^{-1}(t) = \tfrac{1}{-2e^{4t}}\bigl[\begin{smallmatrix}e^{3t}&3e^{3t}\\e^t&e^t\end{smallmatrix}\bigr]\), which simplifies to the expression above.

After evaluating the integrals by parts, formula (4.45) gives:

\[\mathbf{x}(t)=\begin{pmatrix}k_1e^t-3k_2e^{3t}\\-k_1e^t+k_2e^{3t}\end{pmatrix} +\begin{pmatrix}t+\tfrac{4}{3}\\[4pt]-\tfrac{4}{3}t-\tfrac{13}{9}\end{pmatrix}.\;\square\]

Tip

The particular solution has the same polynomial degree as \(\mathbf{f}(t)\) — a degree-1 forcing gives a degree-1 particular solution.

Undetermined Coefficients for Systems

Strategy

Guess \(\mathbf{x}_p\) to match the form of \(\mathbf{f}(t)\). If a term appears in one component of \(\mathbf{f}\), it must appear in all components of the guess.

\(\mathbf{f}(t)\) contains Guess for \(\mathbf{x}_p\)
Polynomial degree \(n\) \(\mathbf{x}_p=\mathbf{a}_n t^n+\cdots+\mathbf{a}_0\)
\(e^{\mu t}\) (\(\mu\neq\) eigenvalue) \(\mathbf{x}_p=\mathbf{a}e^{\mu t}\)
\(\cos\omega t\) or \(\sin\omega t\) \(\mathbf{x}_p=\mathbf{a}\cos\omega t+\mathbf{b}\sin\omega t\)

Rules are identical to the scalar case from Chapter 2.

Example 4.49 — Undetermined Coefficients

Revisit Example 4.47. \(\mathbf{f}(t)=(0,t)^T\) is linear, so guess

\[\mathbf{x}_p=\begin{pmatrix}at+b\\ct+d\end{pmatrix}.\]

Substitute \(\mathbf{x}_p'=A\mathbf{x}_p+\mathbf{f}\): \[\begin{pmatrix}a\\c\end{pmatrix}=\begin{pmatrix}4&3\\-1&0\end{pmatrix}\begin{pmatrix}at+b\\ct+d\end{pmatrix}+\begin{pmatrix}0\\t\end{pmatrix}.\]

Comparing coefficients of \(t\) and constants:

\[a=1,\quad b=-c=\frac{4}{3},\quad d=-\frac{13}{9}.\]

\[\mathbf{x}_p=\begin{pmatrix}t+\tfrac{4}{3}\\[4pt]-\tfrac{4}{3}t-\tfrac{13}{9}\end{pmatrix}.\;\square\]

Agrees exactly with the variation of parameters result — much less work!

Example 4.48 — Guessing for Higher-Degree Forcing

If the forcing is \(\mathbf{f}(t)=\begin{pmatrix}1\\t^2+2\end{pmatrix}\), the highest degree is 2, so we guess

\[\mathbf{x}_p(t)=\begin{pmatrix}a_1t^2+b_1t+c_1\\a_2t^2+b_2t+c_2\end{pmatrix}.\]

Both components are degree 2, regardless of which component of \(\mathbf{f}\) has which degree. Substituting and comparing coefficients determines the six unknowns \(a_1,b_1,c_1,a_2,b_2,c_2\). \(\square\)

Tip

When \(\mathbf{f}(t)\) is complicated or involves resonance (forcing frequency = eigenvalue), variation of parameters is always available as a fallback. For simple \(\mathbf{f}\), undetermined coefficients is faster.

Summary

Key Takeaways

  • The trace–determinant plane (Figure 4.13, Theorem 4.40) gives a complete map: locate \((\text{tr}\,A,\,\det A)\) to read off the portrait type without computing eigenvalues. The parabola \((\text{tr}\,A)^2 = 4\det A\) separates nodes from spirals.

  • Theorem 4.41: the origin is asymptotically stable \(\iff\) \(\text{tr}\,A<0\) and \(\det A>0\). In applied problems this reduces stability analysis to checking two simple inequalities.

  • Nullclines (\(x'=0\) and \(y'=0\)) are powerful tools for sketching phase portraits. On the \(x\)-nullcline the vector field is vertical; on the \(y\)-nullcline it is horizontal. Together with one computed direction-field vector per region, the qualitative behavior is completely determined.

  • For the nonhomogeneous system \(\mathbf{x}'=A\mathbf{x}+\mathbf{f}(t)\): the fundamental matrix \(\Phi(t)\) (columns = independent homogeneous solutions) is the key object. The variation of parameters formula (4.45) always works; undetermined coefficients is faster when \(\mathbf{f}\) is a polynomial, exponential, or sinusoid.

  • In all applied models (crop–soil, glucose–insulin, circuits, reactors): stability follows immediately from \(\text{tr}\,A\) and \(\det A\), without ever solving the characteristic equation. The geometric approach is often all that is needed.

Tip

Looking Ahead: These techniques extend to nonlinear systems in Chapter 5. Near an equilibrium \(\mathbf{x}^*\) the linearization (Jacobian) governs local behavior — the same trace–determinant classification applies to the Jacobian.

Note

Next: Nonlinear Systems and Linearization — Logan §5.1–5.2.

References

Logan, J David. 2015. A First Course in Differential Equations, Third Edition.