1,058 research outputs found
Numerical Shadowing Near Hyperbolic Trajectories
This is the published version, also available here: http://dx.doi.org/10.1137/0916068.Shadowing is a means of characterizing global errors in the numerical solution of initial value ordinary differential equations by allowing for a small perturbation in the initial condition. The method presented in this paper allows for a perturbation in the initial condition and a reparameterization of time in order to compute the shadowing distance in the neighborhood of a periodic orbit or more generally in the neighborhood of an attractor. The method is formulated for one-step methods and both a serial and parallel implementation are applied to the forced van der Pol equation, the Lorenz equation and to the approximation of a periodic orbit
Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field
Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local 'hot spot' on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity held further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Peclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Peclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field
Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors
We consider a chain of oscillators with hyperbolic chaos coupled via
diffusion. When the coupling is strong the chain is synchronized and
demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent.
With the decay of the coupling the second and the third Lyapunov exponents
approach zero simultaneously. The second one becomes positive, while the third
one remains close to zero. Its finite-time numerical approximation fluctuates
changing the sign within a wide range of the coupling parameter. These
fluctuations arise due to the unstable dimension variability which is known to
be the source for non-hyperbolicity. We provide a detailed study of this
transition using the methods of Lyapunov analysis.Comment: 24 pages, 13 figure
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
Validity of numerical trajectories in the synchronization transition of complex systems
We investigate the relationship between the loss of synchronization and the
onset of shadowing breakdown {\it via} unstable dimension variability in
complex systems. In the neighborhood of the critical transition to strongly
non-hyperbolic behavior, the system undergoes on-off intermittency with respect
to the synchronization state. There are potentially severe consequences of
these facts on the validity of the computer-generated trajectories obtained
from dynamical systems whose synchronization manifolds share the same
non-hyperbolic properties.Comment: 4 pages, 4 figure
Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions
Due to the Heisenberg uncertainty principle, various classical systems
differing only on the scale smaller than Planck's cell correspond to the same
quantum system. This fact is used to find a unique semiclassical representation
without the Van Vleck determinant, applicable to a large class of correlation
functions expressible as quantum fidelity. As in the Feynman path integral
formulation of quantum mechanics, all contributing trajectories have the same
amplitude: that is why it is denoted the ``dephasing representation.'' By
relating the present approach to the problem of existence of true trajectories
near numerically-computed chaotic trajectories, the approximation is made
rigorous for any system in which the shadowing theorem holds. Numerical
implementation only requires computing actions along the unperturbed
trajectories and not finding the shadowing trajectories. While semiclassical
linear-response theory was used before in quasi-integrable and chaotic systems,
here its validity is justified in the most generic, mixed systems. Dephasing
representation appears to be a rare practical method to calculate quantum
correlation functions in nonuniversal regimes in many-dimensional systems where
exact quantum calculations are impossible.Comment: 5 pages, 1 figure, to appear in Phys. Rev. E (R
Aggressive shadowing of a low-dimensional model of atmospheric dynamics
Predictions of the future state of the Earth's atmosphere suffer from the
consequences of chaos: numerical weather forecast models quickly diverge from
observations as uncertainty in the initial state is amplified by nonlinearity.
One measure of the utility of a forecast is its shadowing time, informally
given by the period of time for which the forecast is a reasonable description
of reality. The present work uses the Lorenz 096 coupled system, a simplified
nonlinear model of atmospheric dynamics, to extend a recently developed
technique for lengthening the shadowing time of a dynamical system. Ensemble
forecasting is used to make forecasts with and without inflation, a method
whereby the ensemble is regularly expanded artificially along dimensions whose
uncertainty is contracting. The first goal of this work is to compare model
forecasts, with and without inflation, to a true trajectory created by
integrating a modified version of the same model. The second goal is to
establish whether inflation can increase the maximum shadowing time for a
single optimal member of the ensemble. In the second experiment the true
trajectory is known a priori, and only the closest ensemble members are
retained at each time step, a technique known as stalking. Finally, a targeted
inflation is introduced to both techniques to reduce the number of instances in
which inflation occurs in directions likely to be incommensurate with the true
trajectory. Results varied for inflation, with success dependent upon the
experimental design parameters (e.g. size of state space, inflation amount).
However, a more targeted inflation successfully reduced the number of forecast
degradations without significantly reducing the number of forecast
improvements. Utilized appropriately, inflation has the potential to improve
predictions of the future state of atmospheric phenomena, as well as other
physical systems.Comment: 14 pages, 16 figure
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