Key Concepts in Stability Analysis

Why This Matters

Stability analysis is the mathematical toolkit that lets you predict whether a system will settle down, blow up, or oscillate forever—and that question sits at the heart of differential equations. Whether you're modeling population dynamics, electrical circuits, or mechanical vibrations, you're being tested on your ability to determine what happens as time goes to infinity. The concepts here—equilibrium classification, eigenvalue analysis, linearization, and phase portraits—appear repeatedly in both computational problems and conceptual questions.

Don't just memorize definitions. Every concept in this guide connects to a core question: How do we know if a system is stable? You need to understand why eigenvalues determine stability, how linearization lets us analyze nonlinear systems, and when graphical methods reveal behavior that algebra alone might miss. Master these connections, and you'll handle everything from straightforward eigenvalue problems to FRQs asking you to sketch phase portraits and predict long-term behavior.

---

Foundations: Equilibrium and Linear Stability

Before analyzing any system, you need to identify where it might "rest" and understand the basic criteria for stability. These foundational concepts appear in nearly every stability problem you'll encounter.

Equilibrium Points

• Points where all derivatives equal zero—mathematically, where $\frac{d\mathbf{x}}{dt} = \mathbf{0}$, meaning the system has no tendency to change
• Classification depends on nearby trajectory behavior: stable equilibria attract trajectories, unstable equilibria repel them, and semi-stable points do both depending on direction
• Finding equilibria is always your first step—solve the system of equations by setting all derivatives to zero before any further analysis

Stability Criteria for Linear Systems

• Eigenvalues of the coefficient matrix determine everything—for $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$, stability depends entirely on the eigenvalues of $A$
• Negative real parts mean asymptotic stability: all trajectories eventually decay to the equilibrium as $t \to \infty$
• Any positive real part means instability—even one eigenvalue with $\text{Re}(\lambda) > 0$ causes trajectories to escape to infinity

Eigenvalues and Eigenvectors

• Eigenvalues give growth/decay rates—the real part determines whether solutions grow ($\text{Re}(\lambda) > 0$) or decay ($\text{Re}(\lambda) < 0$), while imaginary parts create oscillation
• Eigenvectors define trajectory directions: solutions move along eigenvector directions, with $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$
• Complex eigenvalues produce spirals—the imaginary part $\text{Im}(\lambda)$ determines oscillation frequency while the real part controls whether spirals wind in or out

---

Handling Nonlinearity: Linearization Techniques

Real-world systems are rarely linear, but linearization lets us apply our eigenvalue toolkit to nonlinear problems. The key insight: near an equilibrium point, a nonlinear system behaves approximately like its linear approximation.

Linearization of Nonlinear Systems

• The Jacobian matrix captures local behavior—for a system $\frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x})$, compute $J = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{pmatrix}$ evaluated at the equilibrium
• Eigenvalues of the Jacobian determine local stability: if all eigenvalues have nonzero real parts, the linearization correctly predicts stability of the nonlinear system
• Linearization fails at borderline cases—when eigenvalues are purely imaginary or zero, you need other methods (like Lyapunov theory) to determine stability

Lyapunov Stability Theory

• Determines stability without solving the system—find a scalar function $V(\mathbf{x})$ that acts like an "energy" that decreases along trajectories
• Lyapunov function requirements: $V(\mathbf{x}) > 0$ for $\mathbf{x} \neq \mathbf{0}$, $V(\mathbf{0}) = 0$, and $\frac{dV}{dt} \leq 0$ along solutions
• Especially powerful when linearization is inconclusive—works for nonlinear systems where Jacobian eigenvalues have zero real parts

---

Visualizing Dynamics: Phase Plane Methods

Phase plane analysis transforms abstract equations into geometric pictures, revealing behaviors that algebra alone might miss. These graphical techniques are essential for understanding two-dimensional systems.

Phase Plane Analysis

• Plots trajectories in the $(x_1, x_2)$ plane—each point represents a system state, and curves show how states evolve over time (time itself is implicit)
• Reveals global behavior: you can see basins of attraction, separatrices, and how different initial conditions lead to different outcomes
• Equilibrium classification becomes visual: nodes show straight-line approaches, spirals show rotating approaches, saddles show hyperbolic trajectories

Limit Cycles

• Closed loops in the phase plane—represent periodic solutions where the system repeats its behavior indefinitely
• Stable limit cycles attract nearby trajectories: perturbations decay back to the cycle, making these robust oscillators (think heartbeats, predator-prey cycles)
• Cannot exist in linear systems—limit cycles are inherently nonlinear phenomena, so their presence indicates the system cannot be fully understood through linearization alone

---

Advanced Topics: Bifurcations and Dimensional Reduction

When parameters change or systems are high-dimensional, these advanced techniques become essential. They connect stability analysis to real-world questions about how systems respond to changing conditions.

Bifurcation Theory

• Studies how equilibrium structure changes with parameters—as a parameter crosses a critical value, equilibria can appear, disappear, or exchange stability
• Common bifurcation types: saddle-node (equilibria collide and vanish), transcritical (equilibria exchange stability), pitchfork (one equilibrium splits into three), Hopf (equilibrium becomes a limit cycle)
• Predicts qualitative changes in system behavior—explains sudden transitions like population collapse or onset of oscillations

Stability of Periodic Solutions

• Floquet theory extends eigenvalue analysis to periodic orbits—instead of the Jacobian, you analyze the monodromy matrix over one period
• Floquet multipliers replace eigenvalues: if all multipliers have magnitude less than 1, the periodic orbit is stable
• Critical for oscillatory systems—determines whether rhythmic behavior persists under perturbation

Center Manifold Theory

• Reduces dimensionality near borderline equilibria—when some eigenvalues have zero real parts, the interesting dynamics occur on a lower-dimensional surface
• Separates fast and slow dynamics: stable/unstable directions decay or grow quickly, while center directions evolve slowly and determine long-term behavior
• Essential for analyzing bifurcations—most bifurcation analysis occurs on the center manifold

---

Quick Reference Table

---

Self-Check Questions

1. You compute the Jacobian at an equilibrium and find eigenvalues $\lambda = -2 \pm 3i$. What type of equilibrium is this, and is it stable? How would trajectories appear in the phase plane?

2. Compare and contrast: When would you use linearization versus Lyapunov's method to determine stability? Give a specific scenario where linearization fails but Lyapunov succeeds.

3. A system has a stable equilibrium that becomes unstable as a parameter increases, while a stable limit cycle simultaneously appears. What type of bifurcation is this, and what does it predict about the system's long-term behavior?

4. Which two concepts both involve analyzing eigenvalues but apply to different types of solutions? Explain how the mathematical setup differs between them.

5. An FRQ presents a nonlinear system and asks you to determine the stability of all equilibrium points. Outline the complete procedure you would follow, identifying which concepts from this guide you'd apply at each step.