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 and help determine stability algebraically. For nonlinear systems, we can linearize near equilibrium points and use the to understand local behavior.
Stability Types
Equilibrium Point Stability Classifications
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A simple test for asymptotic stability in some dynamical systems View original
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Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations View original
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NPG - Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate ... View original
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A simple test for asymptotic stability in some dynamical systems View original
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Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations View original
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Top images from around the web for Equilibrium Point Stability Classifications
A simple test for asymptotic stability in some dynamical systems View original
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Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations View original
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NPG - Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate ... View original
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A simple test for asymptotic stability in some dynamical systems View original
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Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations View original
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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 or in the phase plane (tr(A)<0 and det(A)>0)
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 in the phase plane (tr(A)=0 and det(A)>0)
points have solutions that diverge away from the equilibrium over time, even if starting arbitrarily close
Corresponds to a (det(A)<0), unstable node, or unstable spiral (tr(A)>0 and det(A)>0) in the phase plane
Stability Regions in Parameter Space
in parameter space show where different stability types occur based on parameter values
Regions separated by 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: tr(A)<0 and det(A)>0 with (tr(A))2−4det(A)>0
Stable spiral: tr(A)<0 and det(A)>0 with (tr(A))2−4det(A)<0
Saddle point: det(A)<0
Center: 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
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