808 research outputs found
Nonexistence of Chaotic Solutions of Nonlinear Differential Equations
We discuss some important issues arising from computational efforts in
dynamical systems and fluid dynamics. Various individuals have misunderstood
these issues since the onset of these problem areas; indeed, they have been
routinely misinterpreted, and even viewed as "laws" by some. This paper hopes
to stimulate appropriate corrections and to realign thinking, with the overall
goal being sound future progress in dynamical systems and fluid dynamics.Comment: 12 page
Computed Chaos or Numerical Errors
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.Comment: pages 24, Figures
Comment on "Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design" Teixeira et al. (2007)
Comment on "Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical
Convergence, Truncation Error Growth, and Ensemble Design" Teixeira et al.Comment: 3 page
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