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ZixuanZhang
ZixuanZhang
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Stability of Fixed Points

A fixed point is stable if small deviations converge to as , and unstable if they diverge from .

Definition 5.15 (Stability of Fixed Points)

A fixed point is

  • stable if whenever deviates slightly from , converges to as .
  • unstable if whenever deviates slightly from , diverges from as .

Linearization

Near a fixed point of ,

If , higher-order terms are needed. For an autonomous system with , the fixed point is stable if and unstable if .

Related

Stated in