4.3 Systems of ODEs and Phase Plane Analysis

Ordinary differential equations (ODEs) are crucial in mathematical physics. They describe how systems change over time. This section covers converting higher-order ODEs into systems of first-order equations and solving linear ODE systems using eigenvalues and eigenvectors.

Phase plane analysis is a powerful tool for understanding ODE solutions visually. It involves studying equilibrium points, their stability, and interpreting phase portraits. This approach helps predict long-term behavior of systems without solving equations explicitly.

Systems of ODEs

Conversion of higher-order ODEs

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• Converts an $n$-th order ODE into a system of $n$ first-order ODEs by introducing new variables for each derivative up to order $n-1$
  • Rewrites the original ODE as a system of equations representing relationships between consecutive derivatives
  • Resulting system is equivalent to the original higher-order ODE (mass-spring-damper system, beam deflection)

Solution of linear ODE systems

• Solves systems of linear ODEs $\mathbf{x}'(t) = A\mathbf{x}(t)$ using eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$ of constant matrix $A$
  • Eigenvalues found by solving characteristic equation $\det(A - \lambda I) = 0$
  • Eigenvectors found by solving $(A - \lambda_i I)\mathbf{v}_i = \mathbf{0}$ for each eigenvalue
  • General solution is linear combination of eigensolutions: $\mathbf{x}(t) = \sum_{i} c_i e^{\lambda_i t} \mathbf{v}_i$ with constants $c_i$ determined by initial conditions
• Handles cases of repeated eigenvalues or non-real eigenvalues using generalized eigenvectors or complex exponentials (coupled oscillators, population dynamics)

Phase Plane Analysis

Stability of equilibrium points

• Analyzes stability of equilibrium points (constant solutions or fixed points where $\mathbf{x}'(t) = \mathbf{0}$) in phase plane
• Classifies equilibrium points based on eigenvalues of Jacobian matrix $J$ evaluated at the point:
  1. Stable node: negative real eigenvalues
  2. Unstable node: positive real eigenvalues
  3. Saddle point: eigenvalues with both positive and negative real parts
  4. Center: purely imaginary eigenvalues
  5. Spiral: complex eigenvalues with nonzero real parts
• Assesses Lyapunov stability: equilibrium point is stable if nearby solutions remain close for all future time
  • Asymptotic stability: nearby solutions converge to equilibrium as $t \to \infty$ (pendulum, chemical reactions)

Interpretation of phase portraits

• Interprets qualitative behavior of solutions from phase portrait (graphical representation in phase plane)
  • Each point represents a system state; trajectories show evolution over time
• Sketches phase portrait by:
  1. Finding equilibrium points setting $\mathbf{x}'(t) = \mathbf{0}$
  2. Determining stability of each equilibrium point using Jacobian matrix
  3. Sketching representative trajectories guided by nullclines (curves where one component of $\mathbf{x}'(t)$ is zero)
• Identifies key features:
  • Basins of attraction
  • Separatrices
  • Limit cycles
• Determines long-term behavior of solutions from different initial conditions (predator-prey models, nonlinear oscillators)