Stability and Classification

This chapter is the payoff of our eigenvalue work. We have learned how to find eigenvalues and eigenvectors, and how to use them to write explicit solutions. Now we ask: without computing the full solution, what can the eigenvalues alone tell us about the behavior of a system?

The answer is: everything that matters qualitatively. The eigenvalues determine whether solutions approach equilibrium or flee from it, whether they spiral or move monotonically, whether the system is stable or unstable. This classification gives us immediate geometric insight into any linear system.

The Central Question: What Happens Near Equilibrium?

Consider a linear system in the form:

The origin $\mathbf{x} = 0$ is always an equilibrium point: if the system starts at rest at the origin, it stays there. The question is: what happens to solutions that start near the origin? Do they return to equilibrium? Move away? Orbit forever?

This question has profound physical significance. A ball at the bottom of a valley is in stable equilibrium; if nudged, it rolls back. A ball balanced on top of a hill is in unstable equilibrium; the slightest push sends it tumbling away. A frictionless pendulum at rest is in a different kind of equilibrium: displaced slightly, it oscillates forever, neither returning to rest nor running away.

Eigenvalues in the Complex Plane

The key insight is that the eigenvalues of $A$, viewed as points in the complex plane, encode all this information. Every 2x2 matrix has two eigenvalues (counting multiplicity), which are either both real or a complex conjugate pair.

Where these eigenvalues sit in the complex plane determines the system's behavior:

• Real part determines stability: Eigenvalues with negative real parts cause solutions to decay. Positive real parts cause growth.
• Imaginary part determines oscillation: Nonzero imaginary parts produce rotation, leading to spiraling or orbiting behavior.

Drag the eigenvalues around the complex plane and watch the classification change. The green region (negative real part) corresponds to stable behavior. Complex eigenvalues (off the real axis) produce oscillation. Real eigenvalues (on the horizontal axis) produce monotonic approach or escape.

The Six Cases

For a 2x2 system with matrix $A$, the eigenvalues fall into one of six fundamental cases. Each produces a distinct phase portrait geometry.

Case 1: Stable Node (Both Eigenvalues Negative Real)

When both eigenvalues $\lambda_1$ and $\lambda_2$ are negative real numbers, all solutions decay exponentially toward the origin. The general solution has the form:

Since both exponentials decay as $t \to \infty$, every trajectory approaches the origin. The origin is a stable node (also called a sink).

If $|\lambda_1| > |\lambda_2|$, the component along $\mathbf{v}_1$ decays faster. Eventually trajectories become tangent to $\mathbf{v}_2$, the direction of slower decay.

Case 2: Unstable Node (Both Eigenvalues Positive Real)

When both eigenvalues are positive real, both exponentials grow. Every solution (except the trivial one at the origin) escapes to infinity. The origin is an unstable node (also called a source).

Run time backward, and trajectories approach the origin. Run time forward, and they flee.

Case 3: Saddle Point (One Positive, One Negative Real Eigenvalue)

When $\lambda_1 > 0 > \lambda_2$, one eigenvalue is positive and one is negative. This creates a saddle point, the only unstable case with real eigenvalues that has interesting geometry.

Solutions along the eigenvector $\mathbf{v}_2$ (corresponding to $\lambda_2 < 0$) approach the origin. These form the stable manifold. Solutions along $\mathbf{v}_1$ (corresponding to $\lambda_1 > 0$) escape from the origin. These form the unstable manifold.

Generic trajectories follow a hyperbolic path: approaching along the stable direction, passing near the origin, then fleeing along the unstable direction. The saddle is unstable because almost all solutions eventually escape.

Case 4: Stable Spiral (Complex Eigenvalues with Negative Real Part)

When the eigenvalues are complex conjugates $\alpha \pm \beta i$ with $\alpha < 0$, solutions spiral inward toward the origin. The real part $\alpha$ controls the rate of decay, while the imaginary part $\beta$ controls the rotation speed.

The solution involves terms like $e^{\alpha t} \cos(\beta t)$ and $e^{\alpha t} \sin(\beta t)$: oscillation modulated by exponential decay. This is the behavior of a damped oscillator like a swinging pendulum with friction.

Case 5: Unstable Spiral (Complex Eigenvalues with Positive Real Part)

When $\alpha > 0$ but the eigenvalues are still complex, solutions spiral outward from the origin. The oscillation continues, but now it grows rather than decays. This is an unstable oscillator, like an electrical circuit with positive feedback.

Case 6: Center (Pure Imaginary Eigenvalues)

When $\alpha = 0$ and the eigenvalues are pure imaginary $\pm \beta i$, solutions oscillate forever without decay or growth. Every trajectory is a closed ellipse around the origin. This is a center.

The center represents a frictionless oscillator. Displaced from equilibrium, the system oscillates at a fixed amplitude forever. It is stable in the sense that solutions stay bounded, but not asymptotically stable: they do not return to equilibrium.

Click through the different equilibrium types to see their phase portraits. Notice how the eigenvalue structure determines the trajectory geometry in each case.

Stability: Asymptotic versus Lyapunov

We must distinguish two notions of stability:

Asymptotically stable: Solutions starting near the equilibrium actually converge to it as $t \to \infty$. This requires all eigenvalues to have strictly negative real parts.

Lyapunov stable (or marginally stable): Solutions starting near the equilibrium stay bounded forever but may not converge. Centers are Lyapunov stable but not asymptotically stable.

Unstable equilibria are neither: solutions can escape to infinity.

Start a solution from a perturbed initial condition and watch its fate. For a stable spiral, it returns to equilibrium. For an unstable spiral, it escapes. For a center, it orbits at constant distance forever.

The Stability Criterion

The criterion is simple: an equilibrium is asymptotically stable if and only if all eigenvalues have strictly negative real parts.

For a 2x2 matrix, we can state this in terms of the trace and determinant:

Asymptotic stability requires:

• $\det(A) > 0$ (both eigenvalues have the same sign or are complex conjugates)
• $\text{trace}(A) < 0$ (the sum of eigenvalues is negative)

If $\det(A) < 0$, the eigenvalues have opposite signs, giving a saddle point (unstable).

The Trace-Determinant Plane

Rather than computing eigenvalues directly, we can classify equilibria using the trace $\tau = \text{trace}(A)$ and determinant $\Delta = \det(A)$. These are easier to compute and give a complete picture.

The discriminant of the characteristic equation is:

• If $\tau^2 - 4\Delta > 0$: eigenvalues are real and distinct
• If $\tau^2 - 4\Delta < 0$: eigenvalues are complex conjugates
• If $\tau^2 - 4\Delta = 0$: repeated eigenvalue

This diagram partitions all 2x2 systems into regions with the same qualitative behavior. The parabola $\tau^2 = 4\Delta$ separates nodes (outside) from spirals (inside). The horizontal axis $\Delta = 0$ separates saddles (below) from other cases (above). The vertical axis $\tau = 0$ separates stable (left) from unstable (right).

Classification Summary

Linearization of Nonlinear Systems

Most real-world systems are nonlinear. The pendulum equation $\theta'' + \sin\theta = 0$ involves $\sin\theta$, not $\theta$. How can our linear theory help?

The key insight is linearization: near an equilibrium point, any smooth nonlinear system behaves approximately like a linear system. We replace the nonlinear system with its linear approximation and classify the equilibrium using eigenvalues.

For a system $\mathbf{x}' = \mathbf{f}(\mathbf{x})$ with equilibrium at $\mathbf{x}_0$ (where $\mathbf{f}(\mathbf{x}_0) = 0$), the linearization is:

where $J$ is the Jacobian matrix of $\mathbf{f}$ evaluated at $\mathbf{x}_0$:

The eigenvalues of $J$ determine the local stability of the nonlinear system, with one caveat: if the linearization predicts a center (pure imaginary eigenvalues), the nonlinear terms can tip the balance either way, and further analysis is needed.

Compare trajectories of a nonlinear pendulum system with its linearization at the origin. Near the equilibrium, they are nearly identical. Far from equilibrium, the nonlinear effects become significant, and additional equilibria appear that the linear model cannot capture.

Physical Interpretation

The classification has deep physical meaning:

Nodes represent overdamped systems. A door with a strong closer swings shut without oscillating. An RC circuit with large resistance discharges monotonically.

Spirals represent underdamped systems. A lightly damped spring oscillates many times before settling. An RLC circuit with small resistance rings.

Centers represent conservative systems with no damping. A frictionless pendulum swings forever. An LC circuit oscillates at constant amplitude.

Saddles represent systems with competing tendencies. A ball on a saddle-shaped surface rolls toward the valley along one axis but away from it along the perpendicular axis. Such equilibria are always unstable.

Robustness and Structural Stability

Nodes, spirals, and saddles are structurally stable: small perturbations to the matrix $A$ do not change their qualitative type. A stable node remains a stable node if you slightly adjust the coefficients.

Centers are structurally unstable. Adding the slightest friction to a frictionless pendulum turns the center into a stable spiral. Removing just a bit of damping from an underdamped system could push it toward a center. This is why centers are rare in physical systems: any tiny imperfection destroys them.

This robustness principle explains why we see spirals and nodes everywhere in nature but almost never see perfect centers: real systems always have some dissipation.

Key Takeaways

• The eigenvalues of $A$ completely determine the qualitative behavior of $\mathbf{x}' = A\mathbf{x}$ near the origin
• Negative real parts mean decay and stability; positive real parts mean growth and instability
• Nonzero imaginary parts mean oscillation (spirals or centers)
• Saddle points have one positive and one negative real eigenvalue; they attract along one direction and repel along another
• Asymptotically stable means solutions converge to equilibrium; Lyapunov stable means solutions stay bounded but may not converge
• The trace-determinant plane provides a complete classification without computing eigenvalues explicitly
• Linearization extends this classification to nonlinear systems near their equilibria
• Most cases are structurally stable; centers are the exception and require careful analysis