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Computed Chaos or Numerical Errors | | Lun-Shin Yao
; | | Date: |
22 Jun 2005 | | Subject: | Chaotic Dynamics; Classical Analysis and ODEs; Dynamical Systems; Numerical Analysis; Computational Physics; Fluid Dynamics | nlin.CD math.CA math.DS math.NA physics.comp-ph physics.flu-dyn | | Abstract: | Discrete numerical methods with finite time steps represent a practical technique to solve non-linear differential equations. This is particularly true for chaos since no analytical chaotic solution is known today. Using the Lorenz equations as an example it is demonstrated that computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, it is well known that chaotic differential equations are unstable, and that any small error can be amplified exponentially near an unstable manifold. The more serious and less-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 ensure that, even if the computed results are bounded; their independence of time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. Due to the explosive amplification of numerical errors, no computed chaotic solution of differential equations that is independent of integration-time step has been found. | | Source: | arXiv, nlin.CD/0506045 | | Services: | Forum | Review | PDF | Favorites | |
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