Linear Systems in the Plane

Module 1: Low-Dimensional Dynamics

Reading

Compulsory: Strogatz (2024), Chapter 5, Sections 6.1-6.2, Chapter 7 and Section 8.2.
Recommended: (epstein1998introduction?).

In one dimension a trajectory can only move monotonically towards a fixed point or away from it. In two dimensions it can also rotate, and sustained oscillations become possible. The first step is to understand linear systems, because near a fixed point every smooth system looks linear (Strogatz 2024, chap. 5).

Eigenvalues from trace and determinant

Consider $\dot{\mathbf x} = A\mathbf x$ with $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Let

\[ \tau = \operatorname{tr} A = a + d, \qquad \Delta = \det A = ad - bc . \]

The eigenvalues are the roots of $\lambda^2 - \tau\lambda + \Delta = 0$:

\[ \lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}, \qquad \lambda_1 + \lambda_2 = \tau, \quad \lambda_1\lambda_2 = \Delta . \tag{1}\]

Solutions are combinations of $e^{\lambda_1 t}$ and $e^{\lambda_2 t}$, so the signs and the type (real or complex) of the eigenvalues decide the phase portrait (Strogatz 2024, sec. 5.2).

Classification

$\Delta < 0$: real eigenvalues of opposite sign. Saddle point.
$\Delta > 0$ and $\tau^2 > 4\Delta$: real eigenvalues of the same sign. Node, stable if $\tau < 0$ and unstable if $\tau > 0$.
$\Delta > 0$ and $\tau^2 < 4\Delta$: complex eigenvalues $\tau/2 \pm i\omega$. Spiral, stable if $\tau < 0$ and unstable if $\tau > 0$. The frequency is $\omega = \sqrt{4\Delta - \tau^2}/2$.
$\Delta > 0$ and $\tau = 0$: purely imaginary eigenvalues. Center, with closed orbits.
$\Delta = 0$: at least one zero eigenvalue. Non-isolated fixed points.

Figure 2: Phase portraits of four linear systems: saddle, stable node, stable spiral and center.

From linear to nonlinear

For a nonlinear system, the Jacobian at a fixed point plays the role of $A$. The linear classification carries over to the nonlinear system for saddles, nodes and spirals. Centers and the borderline cases on $\Delta = 0$ or $\tau^2 = 4\Delta$ can change type under nonlinear terms (Strogatz 2024, sec. 6.3).

Exercise

Classify the fixed point of $\dot x = -x + 2y$, $\dot y = -2x - y$.

Solution

$\tau = -2$, $\Delta = 1 + 4 = 5$, $\tau^2 - 4\Delta = 4 - 20 < 0$. The eigenvalues are $-1 \pm 2i$: a stable spiral with angular frequency 2.

print(np.linalg.eigvals([[-1, 2], [-2, -1]]))

[-1.+2.j -1.-2.j]

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman; Hall/CRC.

--- title: "Linear Systems in the Plane" subtitle: "Module 1: Low-Dimensional Dynamics" format: html --- ::: {.callout-note} ## Reading - **Compulsory:** @strogatz2024nonlinear, Chapter 5, Sections 6.1-6.2, Chapter 7 and Section 8.2. - **Recommended:** @epstein1998introduction. ::: In one dimension a trajectory can only move monotonically towards a fixed point or away from it. In two dimensions it can also rotate, and sustained oscillations become possible. The first step is to understand linear systems, because near a fixed point every smooth system looks linear [@strogatz2024nonlinear, Chapter 5]. ## Eigenvalues from trace and determinant Consider $\dot{\mathbf x} = A\mathbf x$ with $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Let $$ \tau = \operatorname{tr} A = a + d, \qquad \Delta = \det A = ad - bc . $$ The eigenvalues are the roots of $\lambda^2 - \tau\lambda + \Delta = 0$: $$ \lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}, \qquad \lambda_1 + \lambda_2 = \tau, \quad \lambda_1\lambda_2 = \Delta . $$ {#eq-eigen} Solutions are combinations of $e^{\lambda_1 t}$ and $e^{\lambda_2 t}$, so the signs and the type (real or complex) of the eigenvalues decide the phase portrait [@strogatz2024nonlinear, Section 5.2]. ## Classification - $\Delta < 0$: real eigenvalues of opposite sign. **Saddle point**. - $\Delta > 0$ and $\tau^2 > 4\Delta$: real eigenvalues of the same sign. **Node**, stable if $\tau < 0$ and unstable if $\tau > 0$. - $\Delta > 0$ and $\tau^2 < 4\Delta$: complex eigenvalues $\tau/2 \pm i\omega$. **Spiral**, stable if $\tau < 0$ and unstable if $\tau > 0$. The frequency is $\omega = \sqrt{4\Delta - \tau^2}/2$. - $\Delta > 0$ and $\tau = 0$: purely imaginary eigenvalues. **Center**, with closed orbits. - $\Delta = 0$: at least one zero eigenvalue. Non-isolated fixed points. ```{python} #| echo: false #| label: fig-trace-det #| fig-cap: "Classification of fixed points of linear systems in the (trace, determinant) plane. The parabola is tau^2 = 4 Delta." #| fig-width: 6 #| fig-height: 4.5 import matplotlib.pyplot as plt import numpy as np tau = np.linspace(-4, 4, 400) fig, ax = plt.subplots(figsize=(6, 4.5)) ax.plot(tau, tau**2 / 4, "k") ax.axhline(0, color="k") ax.plot([0, 0], [0, 4.5], "k", lw=2) ax.fill_between(tau, -2, 0, color="tab:gray", alpha=0.2) ax.text(-1.0, -1.2, "saddles") ax.text(-3.6, 1.2, "stable\nnodes") ax.text(2.5, 1.2, "unstable\nnodes") ax.text(-1.9, 3.3, "stable\nspirals") ax.text(0.5, 3.3, "unstable\nspirals") ax.text(0.08, 4.1, "centers", rotation=90, fontsize=8) ax.set_xlim(-4, 4) ax.set_ylim(-2, 4.5) ax.set_xlabel("trace τ") ax.set_ylabel("determinant Δ") plt.tight_layout() plt.show() ``` ```{python} #| echo: false #| label: fig-portraits #| fig-cap: "Phase portraits of four linear systems: saddle, stable node, stable spiral and center." #| fig-width: 9 #| fig-height: 2.6 examples = { "saddle": [[1, 0], [0, -1]], "stable node": [[-2, 0], [0, -0.5]], "stable spiral": [[-0.3, -1], [1, -0.3]], "center": [[0, -1], [1, 0]], } x, y = np.meshgrid(np.linspace(-2, 2, 20), np.linspace(-2, 2, 20)) fig, axs = plt.subplots(1, 4, figsize=(9, 2.6)) for ax, (name, A) in zip(axs, examples.items()): A = np.array(A) ax.streamplot(x, y, A[0, 0] * x + A[0, 1] * y, A[1, 0] * x + A[1, 1] * y, density=0.7, linewidth=0.7, arrowsize=0.7) ax.plot(0, 0, "ko", ms=4) ax.set_title(name, fontsize=9) ax.set_xticks([]) ax.set_yticks([]) plt.tight_layout() plt.show() ``` ## From linear to nonlinear For a nonlinear system, the Jacobian at a fixed point plays the role of $A$. The linear classification carries over to the nonlinear system for saddles, nodes and spirals. Centers and the borderline cases on $\Delta = 0$ or $\tau^2 = 4\Delta$ can change type under nonlinear terms [@strogatz2024nonlinear, Section 6.3]. ::: {.callout-note} ## Exercise Classify the fixed point of $\dot x = -x + 2y$, $\dot y = -2x - y$. ::: ::: {.callout-tip collapse="true"} ## Solution $\tau = -2$, $\Delta = 1 + 4 = 5$, $\tau^2 - 4\Delta = 4 - 20 < 0$. The eigenvalues are $-1 \pm 2i$: a stable spiral with angular frequency 2. ```{python} print(np.linalg.eigvals([[-1, 2], [-2, -1]])) ``` ::: ## Next