10.2 Stability Analysis

Stability analysis is a crucial tool for understanding the behavior of differential equations. It helps us determine whether solutions will stay close to equilibrium points or diverge over time, giving insights into long-term system behavior.

We'll explore techniques like linearization, Lyapunov functions, and stable/unstable manifolds. These methods allow us to analyze complex nonlinear systems without solving them explicitly, revealing key properties of their dynamics and equilibrium points.

Linearization and Jacobian Matrix

Approximating Nonlinear Systems

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• Linearization approximates a nonlinear system near an equilibrium point by a linear system
• Involves computing the Jacobian matrix, which contains partial derivatives of the system at the equilibrium point
• Allows for studying the local behavior of nonlinear systems using linear techniques
• Provides insights into stability, oscillations, and other dynamic properties near the equilibrium

Eigenvalues and Eigenvectors

• The Jacobian matrix is used to determine the eigenvalues and eigenvectors of the linearized system
• Eigenvalues characterize the stability of the equilibrium point
  • Negative real parts indicate stability
  • Positive real parts indicate instability
  • Complex conjugate pairs with negative real parts suggest oscillations
• Eigenvectors represent the directions along which the system evolves near the equilibrium
  • Stable eigenvectors correspond to the stable manifold
  • Unstable eigenvectors correspond to the unstable manifold

Lyapunov Stability

Stability Concepts

• Lyapunov stability refers to the behavior of a system near an equilibrium point
• An equilibrium is stable if nearby solutions remain close to it for all future time
• Asymptotic stability occurs when nearby solutions not only remain close but also converge to the equilibrium as time approaches infinity
• Unstable equilibrium implies that nearby solutions diverge from the equilibrium over time

Lyapunov Functions

• Lyapunov functions are scalar-valued functions used to determine stability without explicitly solving the system
• A Lyapunov function $V(x)$ must satisfy certain conditions:
  • $V(x)$ is positive definite (strictly positive except at the equilibrium where it is zero)
  • The time derivative of $V(x)$ along the system's trajectories is negative semi-definite (non-positive)
• If a Lyapunov function exists satisfying these conditions, the equilibrium is stable
• If the time derivative is strictly negative, the equilibrium is asymptotically stable

Stable and Unstable Manifolds

Invariant Manifolds

• Stable and unstable manifolds are invariant sets associated with an equilibrium point
• The stable manifold consists of all initial conditions that converge to the equilibrium as time approaches infinity
  • Solutions starting on the stable manifold remain on it and approach the equilibrium
  • The stable manifold is tangent to the stable eigenvectors at the equilibrium
• The unstable manifold consists of all initial conditions that diverge from the equilibrium as time approaches negative infinity
  • Solutions starting on the unstable manifold remain on it and move away from the equilibrium
  • The unstable manifold is tangent to the unstable eigenvectors at the equilibrium

Role in Dynamics

• Stable and unstable manifolds play a crucial role in understanding the global behavior of a dynamical system
• They determine the basins of attraction and the separatrices between different regions of the phase space
• The intersection of stable and unstable manifolds can lead to complex dynamics, such as homoclinic or heteroclinic orbits
• Understanding the geometry of these manifolds helps in analyzing the qualitative behavior of the system, including stability, bifurcations, and chaos