The general linear planar system $\dot{\mathbf{x}} = A\mathbf{x}$ with matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is classified by its trace $\tau = a + d$ and determinant $\Delta = ad - bc$:

1. $\Delta < 0$: Saddle point (eigenvalues real with opposite signs)
2. $\Delta > 0, \tau^2 > 4\Delta$: Node (eigenvalues real, same sign)
   • Stable node if $\tau < 0$, unstable if $\tau > 0$
3. $\Delta > 0, \tau^2 < 4\Delta$: Spiral (complex eigenvalues)
   • Stable spiral if $\tau < 0$, unstable if $\tau > 0$
4. $\Delta > 0, \tau^2 = 4\Delta$: Degenerate node or star (repeated eigenvalue)
5. $\Delta > 0, \tau = 0$: Center (purely imaginary eigenvalues)

This classification in the $(\tau, \Delta)$ plane provides a complete picture of linear planar dynamics.

The classic predator-prey model is given by:

$\dot{x} = x(\alpha - \beta y), \quad \dot{y} = -y(\gamma - \delta x)$

where $x$ is prey population, $y$ is predator population, and $\alpha, \beta, \gamma, \delta > 0$ are parameters. The system has:

• A saddle at the origin $(0, 0)$
• A center at $(\gamma/\delta, \alpha/\beta)$ surrounded by closed orbits

The closed orbits represent population oscillations: prey increase when predators are scarce, predator numbers then rise, prey decline due to predation, then predators decline due to lack of food, and the cycle repeats. This model, while simplified, captures essential predator-prey dynamics.

The competitive Lotka-Volterra equations describe two species competing for resources:

$\dot{x} = x(1 - x - ay), \quad \dot{y} = y(1 - y - bx)$

where $a, b > 0$ measure interspecific competition. The system has four fixed points: $(0,0)$, $(1,0)$, $(0,1)$, and potentially a coexistence point $(x^*, y^*)$ in the interior. The stability depends on $a$ and $b$:

• If $a, b < 1$: stable coexistence at $(x^*, y^*)$
• If $a < 1 < b$: species $x$ wins ($(1,0)$ is stable)
• If $b < 1 < a$: species $y$ wins ($(0,1)$ is stable)
• If $a, b > 1$: bistability (both $(1,0)$ and $(0,1)$ stable; outcome depends on initial conditions)

This model demonstrates how competition parameters determine ecological outcomes.

The Brusselator is a model chemical reaction system:

$\dot{x} = a - (b+1)x + x^2y, \quad \dot{y} = bx - x^2y$

where $a, b > 0$ are parameters. The unique fixed point is $(a, b/a)$. Linearization yields eigenvalues with trace $\tau = a^2 - (b+1)$ and determinant $\Delta = a^2$. When $b$ crosses $b_c = 1 + a^2$, the eigenvalues cross the imaginary axis, and a Hopf bifurcation occurs: a stable limit cycle emerges from the fixed point, representing periodic chemical oscillations.

The Brusselator is a paradigm for oscillatory chemical reactions and illustrates how sustained oscillations can arise from instabilities in systems far from equilibrium.