The eigenvalue approach for homogeneous systems is a powerful tool for solving and analyzing systems of differential equations. It simplifies complex problems by transforming them into algebraic equations, making it easier to understand the long-term behavior of solutions.

This method uses eigenvalues and eigenvectors to break down the system into simpler components. By examining these components, we can predict stability, classify critical points, and determine the dominant behavior of solutions over time, which is crucial in many real-world applications.

Eigenvalues and eigenvectors for matrices

Calculating eigenvalues

• Eigenvalues (λ) satisfy the characteristic equation $det(A - λI) = 0$, where A represents the coefficient matrix and I denotes the identity matrix
• Obtain the characteristic polynomial by expanding $det(A - λI)$
  • Roots of this polynomial yield the eigenvalues
• Complex eigenvalues occur in conjugate pairs
• Algebraic multiplicity refers to an eigenvalue's multiplicity as a root of the characteristic polynomial
• Geometric multiplicity denotes the dimension of the corresponding eigenspace

Finding eigenvectors

• Eigenvectors (v) satisfy the equation $Av = λv$, where A represents the coefficient matrix and λ denotes an eigenvalue
• Calculate eigenvectors by solving the homogeneous system $(A - λI)v = 0$ for each eigenvalue
• Eigenspace for λ comprises the nullspace of $(A - λI)$, containing all associated eigenvectors
• Complex eigenvalues yield complex conjugate eigenvectors

Eigenvalue and eigenvector properties

• Non-zero vectors serve as eigenvectors
• Eigenvalues may be real or complex numbers
• For an n × n matrix, there exist n eigenvalues (counting multiplicity)
• Eigenvectors corresponding to distinct eigenvalues remain linearly independent
• Trace of a matrix equals the sum of its eigenvalues
• Determinant of a matrix equals the product of its eigenvalues

Solving homogeneous systems

General solution formulation

• Express general solution to $x' = Ax$ as a linear combination of eigenvector solutions: $x(t) = c₁e^{λ₁t}v₁ + c₂e^{λ₂t}v₂ + ... + c_ne^{λ_nt}v_n$
• Real, distinct eigenvalues produce exponential function terms multiplied by corresponding eigenvectors
• Complex conjugate eigenvalues yield solutions with sine and cosine functions multiplied by real and imaginary parts of eigenvectors
• Repeated eigenvalues may require generalized eigenvectors for a complete set of linearly independent solutions

Solution process

• Find eigenvalues of coefficient matrix A
• Calculate corresponding eigenvectors
• Form general solution using eigenvalue-eigenvector pairs
• Apply initial conditions to determine constants c₁, c₂, ..., c_n
• Eigenvalue method proves particularly effective for systems with constant coefficients
• Reduces differential equation system to a set of algebraic equations

Examples and applications

• Population dynamics (predator-prey models)
  • Two-species system yields 2 × 2 coefficient matrix
  • Eigenvalues and eigenvectors reveal population growth or decline patterns
• Mechanical systems (coupled oscillators)
  • Mass-spring systems produce second-order equations convertible to first-order systems
  • Eigenvalues determine natural frequencies and damping characteristics
• Electrical circuits (RLC circuits)
  • Convert circuit equations to state-space form
  • Eigenvalues indicate circuit behavior (overdamped, underdamped, critically damped)

Stability of critical points

Classification criteria

• Critical points (equilibrium solutions) occur where $Ax = 0$, typically at the origin for homogeneous systems
• Stability determined by real parts of eigenvalues of coefficient matrix A
• Asymptotically stable (sink) when all eigenvalues have negative real parts
• Unstable (source or saddle) if at least one eigenvalue has a positive real part
• Stability indeterminate for eigenvalues with zero real parts using linear analysis alone

Two-dimensional systems analysis

• Use trace and determinant of A to classify critical point types
• Node occurs with real, distinct eigenvalues of same sign
• Spiral arises from complex conjugate eigenvalues with non-zero real parts
• Center results from purely imaginary eigenvalues
• Saddle emerges from real eigenvalues of opposite signs

Phase portrait interpretation

• Phase portrait near critical point determined by eigenvalue nature (real, complex, repeated) and corresponding eigenvectors
• Stable node exhibits trajectories converging to critical point
• Unstable node shows trajectories diverging from critical point
• Stable spiral displays inward spiraling trajectories
• Unstable spiral reveals outward spiraling trajectories
• Saddle point demonstrates trajectories approaching along stable manifold and departing along unstable manifold

Long-term behavior of solutions

Dominant eigenvalue analysis

• Dominant eigenvalue (largest real part) determines asymptotic behavior as $t → ∞$
• Real, positive dominant eigenvalue leads to exponential growth along corresponding eigenvector direction
• Real, negative dominant eigenvalue results in exponential decay towards origin along eigenvector direction
• Complex conjugate dominant eigenvalues with positive real parts cause outward spiraling from origin
• Complex conjugate dominant eigenvalues with negative real parts produce inward spiraling towards origin

Special cases and considerations

• Conservative systems with purely imaginary eigenvalues exhibit periodic behavior without growth or decay
• Repeated eigenvalues may lead to polynomial growth in addition to exponential behavior
• Degenerate cases (e.g., zero eigenvalues) require careful analysis of higher-order terms

Modal analysis and applications

• Decompose solution into different modes associated with eigenvalue-eigenvector pairs
• Understand contribution of each mode to overall system behavior
• Applications in vibration analysis (structural engineering)
  • Identify natural frequencies and mode shapes of structures
• Signal processing and control systems
  • Design filters and controllers based on system modes
• Economic models
  • Analyze long-term trends and stability of economic systems