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Everything about Linear stability

State of linear equations

In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation. If there exists an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem". Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Linear stabilityA 5-minute orientation
  2. Skim a recent paper: Stability of conformable linear infinite-dimensional systemsInfluential work since 2022
  3. Pick up a classic book: Stability and Stabilization of Linear Systems with Saturating ActuatorsThe most-reprinted book on the subject
  4. Read the landmark paper: Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control SystemsCited 5,278 times
  5. Go sideways: Hopf bifurcationA closely related topic
  6. Go sideways: Vakhitov–Kolokolov stability criterionA closely related topic

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