∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 3 Theory

Systems of ODEs, Eigenvalues & 3D Phase Portraits

Spiral sinks, saddle nodes, and the 3D Lorenz chaotic attractor

Orbit around a live 3D dynamical phase portrait (including coupled spiral trajectories and the Lorenz Strange Attractor) integrated in real time.

1. Eigenvalue Phase Classification & 3D Attractors

For a linear system x' = Ax, the eigenvalues λ = α ± iβ of matrix A dictate the phase geometry: real eigenvalues of opposite sign produce a Saddle; complex eigenvalues with α < 0 produce an inward Spiral Sink.

det⁡(A−λI)=λ2−(tr A)λ+det⁡A=0\det(A - \lambda I) = \lambda^2 - (\text{tr } A)\lambda + \det A = 0

Quick Reference & Cheat Sheet

Trace-Determinant Plane
det⁡A<0  ⟹  Saddle,det⁡A>0,  tr A<0  ⟹  Stable Sink\det A < 0 \implies \text{Saddle}, \quad \det A > 0,\; \text{tr } A < 0 \implies \text{Stable Sink}
Classifies stability directly from matrix trace and determinant.