  "Stiff" differential equations are commonplace in engineering and dynamical
systems. To solve them we need flexible integrators that can deal with
rapidly-changing righthand sides. This tutorial describes the application of
"adaptive" [ variable timestep ] integrators to "stiff" mechanical problems
encountered in modern applications of Gibbs' 1902 statistical mechanics. Linear
harmonic oscillators subject to nonlinear thermal constraints can exhibit
either stiff or smooth dynamics. Two closely-related examples, Nos\'e's 1984
dynamics and Nos\'e-Hoover 1985 dynamics, are both based on Hamiltonian
mechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Both
these dynamics are consistent with Gibbs' canonical ensemble. Nos\'e's dynamics
is "stiff" and can present severe numerical difficulties. Nos\'e-Hoover
dynamics, though it follows exactly the same trajectory, is "smooth" and
relatively trouble-free. Our tutorial emphasises the power of adaptive
integrators to resolve stiff problems like the Nos\'e oscillator. The solutions
obtained illustrate the power of computer graphics to enrich numerical
solutions. Adaptive integration with computer graphics are basic to an
understanding of dynamical systems and statistical mechanics. These tools lead
naturally into the visualization of intricate fractal structures formed by
chaos as well as elaborate knots tied by regular nonchaotic dynamics. This work
was invited by the American Journal of Physics.
