3.4 Stability criteria and classification

Linear systems stability is all about equilibrium points. We classify them as asymptotically stable, Lyapunov stable, or unstable based on how nearby solutions behave over time. These classifications correspond to different phase plane behaviors like stable nodes, centers, or saddle points.

We use various techniques to analyze stability. The trace-determinant plane and Routh-Hurwitz criterion help determine stability algebraically. For nonlinear systems, we can linearize near equilibrium points and use the Hartman-Grobman theorem to understand local behavior.

Stability Types

Equilibrium Point Stability Classifications

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• Asymptotic stability occurs when an equilibrium point attracts all nearby solutions as time approaches infinity
  • Solutions starting close to the equilibrium point converge to it over time
  • Corresponds to a stable node or stable spiral in the phase plane ($\text{tr}(A) < 0$ and $\det(A) > 0$)
• Lyapunov stability describes an equilibrium point where solutions starting nearby remain close for all future time
  • Solutions may not converge to the equilibrium point, but they stay within a small neighborhood
  • Corresponds to a center in the phase plane ($\text{tr}(A) = 0$ and $\det(A) > 0$)
• Unstable equilibrium points have solutions that diverge away from the equilibrium over time, even if starting arbitrarily close
  • Corresponds to a saddle point ($\det(A) < 0$), unstable node, or unstable spiral ($\text{tr}(A) > 0$ and $\det(A) > 0$) in the phase plane

Stability Regions in Parameter Space

• Stability regions in parameter space show where different stability types occur based on parameter values
  • Regions separated by bifurcation curves where the stability type changes
  • Helps visualize how parameters influence the stability of equilibrium points
• Stable regions correspond to parameter values yielding asymptotically stable or Lyapunov stable equilibria
• Unstable regions correspond to parameter values resulting in unstable equilibria

Stability Analysis Techniques

Graphical and Algebraic Criteria

• Trace-determinant plane classifies stability based on the trace and determinant of the Jacobian matrix $A$
  • Stable node: $\text{tr}(A) < 0$ and $\det(A) > 0$ with $(\text{tr}(A))^2 - 4\det(A) > 0$
  • Stable spiral: $\text{tr}(A) < 0$ and $\det(A) > 0$ with $(\text{tr}(A))^2 - 4\det(A) < 0$
  • Saddle point: $\det(A) < 0$
  • Center: $\text{tr}(A) = 0$ and $\det(A) > 0$
• Routh-Hurwitz criterion determines stability from coefficients of the characteristic polynomial
  • Constructs Routh array from coefficients and checks signs of entries in the first column
  • All positive entries imply asymptotic stability, sign changes indicate instability

Local Stability and Topological Equivalence

• Linearization approximates a nonlinear system near an equilibrium point using a linear system
  • Stability of the linearized system corresponds to local stability of the nonlinear system
  • Analyze the Jacobian matrix evaluated at the equilibrium point
• Hartman-Grobman theorem states that a nonlinear system is locally topologically equivalent to its linearization near a hyperbolic equilibrium point
  • Hyperbolic equilibrium: Jacobian matrix has no eigenvalues with zero real part
  • Topological equivalence means the phase portraits are qualitatively the same (homeomorphic)
  • Allows using the linearized system to determine stability and local behavior