Phase Portraits of Two-Dimensional Systems

Phase Portraits of Linear Second-Order Systems

Phase Space as an Analytical Tool

When studying the system x' = Ax in ℝ², an explicit formula is not always necessary. Often, it is more important to understand the nature of the motion: does the system oscillate around equilibrium, tend toward it, or escape from it? For this purpose, a phase portrait is constructed—a family of trajectories in the plane (x₁, x₂).

A phase portrait is a "map" of system behavior for all initial conditions at once. A single picture replaces an infinite number of separate graphs. It was precisely phase portraits that enabled Poincaré at the end of the 19th century to lay the foundations of the qualitative theory of differential equations—a geometric approach that preceded chaos theory.

Classification of Critical Points by Eigenvalues

The behavior of trajectories near the equilibrium x* = 0 is completely determined by the eigenvalues λ₁, λ₂ of the matrix A.

Stable node: λ₁, λ₂ < 0 (real, negative). All trajectories tend to zero at exponential speed. The "fast" eigendirection (the more negative λ) dominates: trajectories are tangent to it as x → 0.

Unstable node: λ₁, λ₂ > 0. Trajectories escape from zero—a "node turned inside out."

Saddle: λ₁ < 0 < λ₂. There is a stable invariant manifold (along λ₁—trajectories converge) and an unstable one (along λ₂—trajectories diverge). A generic trajectory hyperbolically bypasses the origin. The saddle is an unstable equilibrium: a precise hit onto the stable separatrix guarantees convergence, the slightest deviation—and the trajectory escapes.

Stable focus: λ = α ± βi, α < 0. Trajectories are spirals winding towards zero. The rotation frequency ω = β, the rate of convergence is e^{αt}.

Unstable focus: α > 0. Spirals unwinding away from zero.

Center: λ = ±βi (purely imaginary). Trajectories are ellipses (or circles). Ideally conservative system—a harmonic oscillator without friction.

Stability and Traces: Rule-of-Thumb

Classification is conveniently done by the trace and determinant of matrix A:

• tr A = λ₁ + λ₂ and det A = λ₁λ₂.
• det A < 0: saddle (roots of opposite signs).
• det A > 0, (tr A)² < 4 det A: focus; (tr A)² ≥ 4 det A: node.
• tr A < 0: stable; tr A > 0: unstable; tr A = 0: center.

Example: A = [[−1, −2], [1, −3]]. tr A = −4 < 0; det A = 3 + 2 = 5 > 0. (tr A)² = 16, 4 det A = 20. (tr A)² < 4 det A → stable focus. The system tends to zero with a twist.

Eigenvalues: λ = (−4 ± √(16−20))/2 = −2 ± i. Checking: α = −2 < 0 (stable), β = 1. ✓

Physical Examples

Electrical LRC circuit: Equation: Li'' + Ri' + i/C = 0. In matrix form with x₁ = i, x₂ = i': A = [[0, 1], [−1/(LC), −R/L]]. tr A = −R/L < 0 (for R > 0); det A = 1/(LC) > 0. Type of critical point:

• Small R: focus (damped oscillations).
• Critical R = 2√(L/C): node (aperiodic return to 0).
• R > 2√(L/C): node (sluggish return to 0 without oscillations).

Pendulum (small oscillations): A = [[0, 1], [−ω₀², 0]]. tr A = 0; det A = ω₀² > 0. Center—ideal undamped oscillations. (Friction converts the center into a stable focus.)

Quantitative Characteristics of Stability

For engineers, not just the type of critical point is important, but also quantitative characteristics:

Damping ratio ζ = −α/√(α² + β²). At ζ = 1—critical damping (boundary between focus and node). For ζ < 1—weak damping (focus); ζ > 1—strong (node).

Natural frequency: ω₀ = √(det A). Frequency of damped oscillations: ωd = β = ω₀√(1 − ζ²).

Settling time: τ = 1/|α|—characteristic damping time. Over 5τ the amplitude decreases by e⁵ ≈ 150 times.

All these characteristics are standard in control theory (PID controllers), mechanical system design, and electrical circuit analysis.

Hurwitz Criterion and Stability of Control Systems

For an n-th order system with characteristic polynomial p(λ) = λⁿ + a₁λⁿ⁻¹ + ... + aₙ, the Hurwitz criterion gives necessary and sufficient stability conditions through the signs of Hurwitz determinants. For n = 2: a₁ > 0 and a₂ > 0. For n = 3: a₁ > 0, a₃ > 0 and a₁a₂ > a₃. In PID controller theory the selection of coefficients (proportional, integral, differential) reduces to ensuring the stability of the closed system. The root locus method is a visualization of the movement of roots of the characteristic equation when the feedback coefficient K changes. The stability boundary is the value of K at which at least one root crosses the imaginary axis.

Question for thought: On the (tr A, det A) diagram draw the boundaries of different types of critical point regions. Which type of critical point "disappears" when the symmetry (antisymmetry) condition of the matrix is violated?