Definition and Basic Concepts
Linear Systems Overview
Linear systems: sets of ordinary differential equations (ODEs) where dependent variables and derivatives appear linearly. Form: dx/dt = A(t)x + f(t), x vector, A(t) matrix, f(t) vector function. Solutions: functions satisfying system equations.
Order and Dimension
Order: highest derivative order in system. Dimension: number of coupled equations (size of vector x). Typical focus: first-order systems.
Homogeneous vs Nonhomogeneous
Homogeneous system: f(t) = 0 for all t. Nonhomogeneous: f(t) ≠ 0. Homogeneous systems form basis of solution theory.
Matrix Formulation
Vector-Matrix Notation
System representation: dx/dt = A(t)x + f(t). Vector x ∈ ℝⁿ, matrix A(t) ∈ ℝⁿˣⁿ, forcing vector f(t) ∈ ℝⁿ. Compact, enables linear algebra tools.
Time-Invariant vs Time-Variant
Time-invariant: A(t) = constant matrix A. Time-variant: A(t) varies with t. Solution techniques differ accordingly.
Initial Value Problem (IVP)
Given x(t₀) = x₀, seek x(t) satisfying system and initial condition. Existence and uniqueness guaranteed under continuity and boundedness conditions.
Homogeneous Linear Systems
General Form
dx/dt = A(t)x. Solutions form vector space. Superposition principle applies: linear combinations of solutions are solutions.
Solution Structure
General solution: linear combination of n linearly independent solutions. Basis of solution space of dimension n.
Fundamental Set of Solutions
Set {x₁(t), x₂(t), ..., xₙ(t)} linearly independent solutions. Any solution: x(t) = c₁x₁(t) + ... + cₙxₙ(t).
Nonhomogeneous Linear Systems
General Form
dx/dt = A(t)x + f(t), f(t) ≠ 0. Solutions: sum of homogeneous solution and particular solution.
Particular Solution
Any solution satisfying nonhomogeneous equation, not necessarily satisfying initial conditions.
Superposition Principle
General solution = homogeneous general solution + particular solution. Enables construction of full solution from parts.
Eigenvalues and Eigenvectors
Definition
Eigenvalue λ and eigenvector v satisfy: Av = λv. Key for diagonalization and system decoupling.
Role in Solutions
Eigenvalues determine solution behavior: growth, decay, oscillations. Eigenvectors provide directions in solution space.
Computation
Characteristic polynomial: det(A - λI) = 0. Solutions λ found via polynomial roots. Eigenvectors from null space of (A - λI).
Fundamental Matrix Solution
Definition
Fundamental matrix Φ(t): square matrix whose columns form fundamental set of solutions. Satisfies dΦ/dt = AΦ, Φ(t₀)=I.
Properties
Invertible for all t. Enables expression of general solution as x(t) = Φ(t)c for constant vector c.
Variation of Parameters
Method to find particular solutions using Φ(t). Formula: x_p(t) = Φ(t) ∫ Φ⁻¹(s)f(s) ds.
dΦ/dt = A(t)Φ(t), Φ(t₀) = Ix(t) = Φ(t)c + Φ(t) ∫ₜ₀ᵗ Φ⁻¹(s) f(s) dsPhase Plane Analysis
Concept
Graphical representation of solutions in state space (x₁ vs x₂). Visualizes trajectories, equilibria, and stability.
Critical Points
Points where dx/dt = 0. Typically equilibria. Classification via eigenvalues of Jacobian at point.
Types of Equilibria
Node, saddle, focus, center. Determined by eigenvalue sign and real/complex nature.
| Equilibrium Type | Eigenvalues | Behavior |
|---|---|---|
| Stable Node | Real, negative | Converges to equilibrium |
| Saddle Point | Real, opposite signs | Unstable, trajectories diverge |
| Focus/Spiral | Complex conjugates | Spiral in/out behavior |
| Center | Purely imaginary | Closed orbits, neutral stability |
Stability Theory
Lyapunov Stability
Equilibrium stable if solutions remain close for small perturbations. Asymptotic stability requires solutions tend to equilibrium.
Eigenvalue Criterion
Stable if all eigenvalues of A have negative real parts. Unstable if any eigenvalue has positive real part.
Stability for Time-Varying Systems
More complex; requires Lyapunov functions or other advanced techniques. Uniform stability concept used.
Solution Methods
Diagonalization
If A diagonalizable, transform system into uncoupled scalar ODEs. Solutions easily found by exponentials.
Jordan Canonical Form
Used when A not diagonalizable. Jordan blocks yield generalized eigenvectors and polynomial-exponential solutions.
Variation of Parameters
Method for nonhomogeneous systems using fundamental matrix. Integral formula for particular solution.
Laplace Transform
Transforms system into algebraic equations in complex domain. Useful for constant coefficient systems with initial conditions.
Applications
Mechanical Systems
Model coupled oscillators, damped vibrations, multi-degree-of-freedom systems via linear ODE systems.
Electrical Circuits
RLC circuit analysis with multiple loops/nodes leads to linear systems of ODEs for voltages/currents.
Population Dynamics
Interacting species modeled by Lotka-Volterra linearized systems near equilibria for stability and behavior prediction.
Numerical Approaches
Euler and Runge-Kutta Methods
Explicit time-stepping schemes for approximate solutions. Stability and step size considerations critical.
Matrix Exponential Computation
Numerical methods to compute e^{At}: scaling and squaring, Padé approximants, Krylov subspace methods.
Software Tools
MATLAB, Mathematica, Python (SciPy), specialized ODE solvers for linear systems with built-in functions.
Worked Examples
Example 1: 2x2 Constant Coefficient Homogeneous System
System: dx/dt = Ax, A = [[3,1],[0,2]]. Find general solution.
A = [3 1; 0 2]Eigenvalues: λ₁=3, λ₂=2Eigenvectors: v₁=[1;0], v₂=[1;0]General solution:x(t) = c₁ e^{3t} [1;0] + c₂ e^{2t} [1;0]Example 2: Nonhomogeneous System with Variation of Parameters
Given dx/dt = Ax + f(t), with A = [[0,1],[-1,0]], f(t) = [0; cos t]. Find particular solution.
Step 1: Compute fundamental matrix Φ(t)Step 2: Evaluate integral x_p(t) = Φ(t) ∫ Φ⁻¹(s) f(s) dsStep 3: Add homogeneous solution for general solutionReferences
- Coddington, E.A., Levinson, N., The Theory of Ordinary Differential Equations, McGraw-Hill, 1955, pp. 1-400.