Integrated Exercises: Phase Planes, Linearization, and Stability

date2026-07-16document_iddoc_080c17d5eb0ba4147ba5834559003990description固有方向と Jordan 構造から相図を読み、線型化と安定性の定義を使い分ける統合演習である。prerequisites対角化・Jordan 形と連立系 / 線型化と固有値判定の入口 / 相平面と安定性type問題演習content_typeexercisestatusactive

mathdifferential-equationsexercisesystemsstability

data/lecture/math/differential-equations/linearization-and-eigenvalue-stability.lecture.n.md data/lecture/math/differential-equations/phase-planes-and-stability.lecture.n.md

1Exercise Strategy

These exercises use eigenvalues, eigendirections, Jordan structure, and time orientation to describe phase portraits. Distinguish the complete criterion for an exact linear system from the local linearization theorem for a nonlinear system.

2Problem 1

For

x^{'} = (\begin{matrix} 1 & 1 \\ 0 & - 2 \end{matrix}) x,

find the eigenvalues and eigendirections, and describe the stable direction, unstable direction, and time orientation of the orbits.

2.1Sample Answer

○

The eigenvalues are $1$ and $- 2$ . Corresponding eigenvectors are $v_{u} = (1, 0)^{T}$ and $v_{s} = (1, - 3)^{T}$ . The general solution is

x (t) = c_{u} e^{t} v_{u} + c_{s} e^{- 2 t} v_{s} .

On $span (v_{s})$ , solutions approach the origin as time increases; on $span (v_{u})$ , they depart from it. When $c_{u} c_{s} \neq 0$ , a general orbit is asymptotic to the stable direction in backward time and to the unstable direction in forward time.