If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a step length which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval. (Lambert, J. D. (1992), Numerical Methods for Ordinary Differential Systems, New York: Wiley.)

 

An IVP [initial value problem] is stiff in some interval $[0,b]$ if the step size needed to maintain stability of the forward Euler method is much smaller than the step size required to represent the solution accurately. (Ascher, U. M. and Petzold, L. P. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia:SIAM.)

 

Stiff equations are equations where certain implicit methods, in particular BDF, perform better, usually tremendously better, than explicit ones. (C. F. Curtiss & J. O. Hirschfelder (1952): Integration of stiff equations. PNAS, vol. 38, p. 235-243)

1. A linear constant coefficient system is stiff if all of its eigenvalues have negative real part and the stiffness ratio is large.

2. Stiffness occurs when stability requirements, rather than those of accuracy, constrain the step length. [Note that this "observation" is essentially the definition from Ascher and Petzold.]

3. Stiffness occurs when some components of the solution decay much more rapidly than others.

If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a step length which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval. (Lambert, J. D. (1992), Numerical Methods for Ordinary Differential Systems, New York: Wiley.)

An IVP [initial value problem] is stiff in some interval $[0,b]$ if the step size needed to maintain stability of the forward Euler method is much smaller than the step size required to represent the solution accurately. (Ascher, U. M. and Petzold, L. P. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia:SIAM.)

Stiff equations are equations where certain implicit methods, in particular BDF, perform better, usually tremendously better, than explicit ones. (C. F. Curtiss & J. O. Hirschfelder (1952): Integration of stiff equations. PNAS, vol. 38, p. 235-243)

1. A linear constant coefficient system is stiff if all of its eigenvalues have negative real part and the stiffness ratio is large.

2. Stiffness occurs when stability requirements, rather than those of accuracy, constrain the step length. [Note that this "observation" is essentially the definition from Ascher and Petzold.]

3. Stiffness occurs when some components of the solution decay much more rapidly than others.

I can give you a good example of a small stiff system without very small eigenvalues. Since it's one-dimensional, its Jacobian matrix can only have one eigenvalue.

In Hairer and Wanner's book, they cite Curtiss and Hirschfelder's paper when giving the example ODE

$$\dot{y}(x) = -50(y(x) - \cos(x)).$$

(Wanner, G., Hairer, E., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (2002), Springer.) This book also gives several other examples in its first section (Part IV, section 1) that illustrate many other examples of stiff equations.

Probably the best example I could come up with would be integrating any sort of large combustion reaction system in chemical kinetics under conditions that result in ignition. The system of equations will be stiff until ignition, and then it will no longer be stiff because the system has passed an initial transient. The ratio of largest to smallest eigenvalue should not be large except around the ignition event, though such systems tend to confound stiff integrators unless you set exceedingly strict integration tolerances.

The book by Hairer and Wanner also gives several other examples in its first section (Part IV, section 1) that illustrate many other examples of stiff equations. (Wanner, G., Hairer, E., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (2002), Springer.)