A differential equation involves velocities or rates of change. More precisely, it is an equation for a missing mathematical function (or functions) in terms of the derivatives of that function. Starting with simple examples presented graphically, see why differential equations are one of the most powerful tools in mathematics.
Using a limited-growth population model (also known as a logistic growth model), investigate several ways to visualize solutions to autonomous first-order differential equations—those that involve only the first derivative and that do not depend on time. Plot slope-field and solution graphs, and learn about a pictorial tool called a phase line.
Explore the concepts of source, sink, and node. These are the three types of equilibrium solutions to differential equations, which govern the behavior of nearby solutions on a graph. Then turn to the existence and uniqueness theorem, perhaps the most important theorem regarding first-order differential equations.
Sometimes tiny differences in the value of a parameter in a differential equation can lead to drastic changes in the behavior of solutions—a phenomenon called bifurcation. Probe an example involving the harvesting rate of fish, finding the bifurcation point at which fish stocks suddenly collapse.
Turning from the qualitative computer-based approach, try your hand at the standard methods of solving differential equations, specifically those for linear and separable first-order equations. Professor Devaney first reviews integration—the technique from calculus used to solve the examples, including one problem illustrating Newton's law of cooling.
Computers have revolutionized the solution of differential equations. But how do they do it? Learn one simple approach, Euler's method, which allows a very straightforward approximation of solutions. Test it using one of the most surprisingly powerful tools for analyzing differential equations: spreadsheets.
Embark on the second part of the course: systems of differential equations. These are collections of two or more differential equations for missing functions. An intriguing example is the fluctuating population of foxes and rabbits in a predator-prey relationship, each represented by a differential equation.
Advancing to second-order differential equations (those involving both the first and second derivatives), examine a mass-spring system, also known as a harmonic oscillator. Taking three different views of the system, watch its actual motion, its solutions in the phase plane, and the graph of its changing position and velocity.
Consider cases of a spring with no or very little friction. In solving these differential equations, encounter one of the most beautiful and important formulas in all of mathematics, Euler's formula, which shows the deep link between complex exponential functions and trigonometric functions.
Analyze what happens when force is applied to a spring in a periodic fashion. The resulting motions are very different depending on the relationship of the natural frequency and the forcing frequency. When these are the same, disaster strikes—a phenomenon that may have contributed to the famous collapse of the Tacoma Narrows Bridge.
Begin a series of lectures on linear systems of differential equations by delving into linear algebra, which provides tools for solving these problems. Review vector notation, matrix arithmetic, the concept of the determinant, and the conditions under which equilibrium solutions arise.
Explore more ideas from linear algebra, learning about eigenvalues and eigenvectors, which are the key to finding straight-line solutions for linear systems of differential equations. From these special solutions, all possible solutions can be generated for any given initial conditions.
Professor Devaney summarizes the steps for solving linear systems of differential equations, pointing out that complex eigenvalues are one possibility. Discover that in this case Euler's formula is used, which yields solutions that depend on both exponential and trigonometric functions.
Turn to the special cases of repeated eigenvalues and zero eigenvalues. Then end this part of the course with a computer visualization of all possible types of phase planes for linear systems, seeing their connection to the bifurcation diagrams from Lecture 4.
Most applications of differential equations arise in nonlinear systems. Begin your study of these challenging problems with a nonlinear model of a predator-prey relationship. Learn to use an analytical tool called the nullcline to get a global picture of the behavior of solutions in such systems.
Experiment with another tool for coping with nonlinear systems: linearization. Given an equilibrium point for a nonlinear system, it's possible to approximate the behavior of nearby solutions by dropping the nonlinear terms and considering the corresponding linearized system, which involves an expression called the Jacobian matrix.
Combine linearization and nullclines to analyze what happens when two species compete. The resulting system of differential equations depends on several different parameters, yielding many possible outcomes—from rapid extinction of one species to a coexistence equilibrium for both. As the parameters change, bifurcations arise.
Use nullclines and linearization to investigate a startling phenomenon in chemistry. Before the 1950s, it was thought that all chemical reactions tended to equilibrium. But the Russian chemist Boris Belousov discovered a reaction that oscillated for hours. Your analysis shows how differential equations can model this process.
Focusing on the nonlinear behavior of a pendulum, learn new ways to deal with nonlinear systems of differential equations. These include Hamiltonian and Lyapunov functions. A Hamiltonian function remains constant along all solutions of special differential equations, while a Lyapunov function decreases along all solutions.
Study the behavior of a periodically forced nonlinear pendulum to see how tiny changes in the initial position lead to radically different outcomes. To understand this chaotic behavior, turn to the Lorenz equation from meteorology, which was the first system of differential equations to exhibit chaos.
Mathematicians understand chaotic behavior in certain differential equations by reducing them to an iterated function (also known as a difference equation). Try several examples using a spreadsheet. Then delve deeper into the subject by applying difference equations to the discrete logistic population model.
Continuing with the discrete logistic population model, notice that fixed and periodic points play the role in difference equations that equilibrium points play in differential equations. Also investigate Sharkovsky's theorem from 1964, a result that heralded the first use of the word "chaos" in the science literature.
How do mathematicians understand chaotic behavior? Starting with a simple function that is behaving chaotically, move off the real line and onto what at first appears to be a much more complicated space, but one that is an ideal setting for analyzing chaos.
What is the big picture of chaos that is now emerging? Center your investigation on the complex plane, where iterated functions produce shapes called fractals, including the Mandelbrot and Julia sets. Close by considering how far you've come—from the dawn of differential equations in the 17th century to fractals and beyond.