The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. The program "predprey" studies this model.

Predator-Prey Equations

The Lorenz chaotic attractor was discovered by Edward Lorenz in 1963 when he was investigating a simplified model of atmospheric convection. It is a nonlinear system of three differential equations. The program "lorenzgui" studies this model.

Lorenz Attractor and Chaos

Throw a rectangular box with sides of three different lengths into the air. You can get the box to tumble stably about its longest axis or its shortest axis. But if you try to make it tumble about it middle axis, you will find the motion is unstable.

Tumbling Box

An ODE involving higher order derivatives is rewritten as a vector system involving only first order derivatives. The classic Van der Pol nonlinear oscillator is provided as an example. The VdP equation becomes stiff as the parameter is increased.

Systems of Equations

The MATLAB documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite.

The MATLAB ODE Suite

A problem is said to be stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results. The flame model demonstrates stiffness.

Stiffness, ODE23s, ODE15s

ODE45 is usually the function of choice among the ODE solvers. It compares 4th and 5th order methods to estimate error and determine step size.

ODE45

The digits in the name of a MATLAB ODE solver reflect its order and resulting accuracy. A method is said to have order p if cutting the step size in half reduces the error in one step by a factor of two to the power p+1.

Order, Naming Conventions

ODE23 compares 2nd and 3rd order methods to automatically choose the step size and maintain accuracy. It is the simplest MATLAB solver that has automatic error estimate and continuous interpolant. ODE23 is suitable for coarse accuracy requirements.

Estimating Error, ODE23

ODE2 implements a midpoint method with two function evaluations per step. This method is twice as accurate as Euler's method. A nonlinear equation defining the sine function provides an example. An exercise involves implementing a trapezoid method.

Midpoint Method, ODE2

Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler

A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λ is the eigenvalue matrix.

Diagonalizing a Matrix

The incidence matrix has a row for every edge, containing -1 and +1 to show which two nodes are connected by that edge.

Incidence Matrices of Graphs

The direction field has an arrow with slope at each point coming from the differential equation. Arrows with the same slope lie along an "isocline".

Pictures of Solutions

Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution.

Laplace Transform: First Order Equation

With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation.

Unforced Damped Motion

Two equations with a constant matrix are stable (solutions approach zero) when the trace is negative and the determinant is positive.

Eigenvalues and Stability: 2 by 2 Matrix, A

A critical point is a constant solution to the differential equation. The slope of the right hand side decides stability or instability.

Linearization at Critical Points

With two equations,  the two linearized equations use the 2 by 2 matrix of partial derivatives of the right hand sides.

Linearization of Two Nonlinear Equations

Combine null solutions to find a particular solution for any right hand side. But it may involve a difficult integral.

Variation of Parameters

Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exponent or eigenvalue.

Phase Plane Pictures: Source, Sink, Saddle

When the input force is an impulse, the output is the impulse response. For all inputs the response is a "convolution" with the impulse response.

Laplace Transforms and Convolution

An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combinations of those solutions lead to all solutions.

Solving Linear Systems

A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.

The Big Picture of Linear Algebra

A graph has nodes connected by edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.

Graphs

A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular).

Two First Order Equations: Stability

Diagonalizing a matrix also diagonalizes all its powers.

Powers of Matrices and Markov Matrices

Vectors are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension = number of basis vectors.

Independence, Basis, and Dimension

Imaginary exponents with pure oscillation provide a "center" in the phase plane. The point (position, velocity) travels forever around an ellipse.

Phase Plane Pictures: Spirals and Centers

The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.

Impulse Response and Step Response

A box in the air can rotate stably around its shortest and longest axes. Around the middle axis it tumbles wildly.

The Tumbling Box in 3-D

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