  Discrete numerical methods with finite time-steps represent a practical
technique to solve initial-value problems involving nonlinear differential
equations. These methods seem particularly useful to the study of chaos since
no analytical chaotic solution is currently available. Using the well-known
Lorenz equations as an example, it is demonstrated that numerically computed
results and their associated statistical properties are time-step dependent.
There are two reasons for this behavior. First, chaotic differential equations
are unstable so that any small error is amplified exponentially near an
unstable manifold. The more serious and lesser-known reason is that stable and
unstable manifolds of singular points associated with differential equations
can form virtual separatrices. The existence of a virtual separatrix presents
the possibility of a computed trajectory actually jumping through it due to the
finite time-steps of discrete numerical methods. Such behavior violates the
uniqueness theory of differential equations and amplifies the numerical errors
explosively. These reasons imply that, even if computed results are bounded,
their independence on time-step should be established before accepting them as
useful numerical approximations to the true solution of the differential
equations. However, due to these exponential and explosive amplifications of
numerical errors, no computed chaotic solutions of differential equations
independent of integration-time step have been found. Thus, reports of computed
non-periodic solutions of chaotic differential equations are simply
consequences of unstably amplified truncation errors, and are not approximate
solutions of the associated differential equations.
