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"Stiff" differential equations are commonplace in engineering and dynamicalsystems. To solve them we need flexible integrators that can deal withrapidly-changing righthand sides. This tutorial describes the application of"adaptive" [ variable timestep ] integrators to "stiff" mechanical problemsencountered in modern applications of Gibbs' 1902 statistical mechanics. Linearharmonic oscillators subject to nonlinear thermal constraints can exhibiteither stiff or smooth dynamics. Two closely-related examples, Nos\'e's 1984dynamics and Nos\'e-Hoover 1985 dynamics, are both based on Hamiltonianmechanics, as was ultimately clarified by Dettmann and Morriss in 1996. Boththese dynamics are consistent with Gibbs' canonical ensemble. Nos\'e's dynamicsis "stiff" and can present severe numerical difficulties. Nos\'e-Hooverdynamics, though it follows exactly the same trajectory, is "smooth" andrelatively trouble-free. Our tutorial emphasises the power of adaptiveintegrators to resolve stiff problems like the Nos\'e oscillator. The solutionsobtained illustrate the power of computer graphics to enrich numericalsolutions. Adaptive integration with computer graphics are basic to anunderstanding of dynamical systems and statistical mechanics. These tools leadnaturally into the visualization of intricate fractal structures formed bychaos as well as elaborate knots tied by regular nonchaotic dynamics. This workwas invited by the American Journal of Physics.

A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems : Comparing the Nos\'e and Nos\'e-Hoover Oscillator Dynamics