## References for Chaos Part 0

The literature references cited in Chaos Part 0 are listed in this post. Maybe this will become a Pages.

**1. Edward N. Lorenz**, “Deterministic Nonperiodic Flow”, Journal of the Atmospheric Sciences, Vol. 20, pp. 130-141, 1963.

Abstract

Finite systems of deterministic ordinary differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to process bounded numerical solutions.

A simple system representing cellular convection is solved numerically. All of the numerical solutions are found to be stable, and almost all of them are nonperiodic.

The feasibility of very-long-range weather prediction is examined in the light of these results.

**2. Edward N. Lorenz**, “Climate Change as a Mathematical Problem”, Journal of Applied Meteorology”, Vol. 9. No. 3, pp. 325-329, 1970.

Abstract

Formulating reasonable hypotheses regarding climate change requires physical insight and ingenuity, but subsequently testing these hypotheses demands quantitative computation. Many features of today’s climate have been reproduced by mathematical models (equations arranged for numerical solution by digital computers), similar to those used in weather prediction. Models currently in use generally predict only the atmosphere, and pre-specify the state of the environment (oceans, land surfaces, sun, etc.). Newer models, where certain environmental conditions enter as additional dependent variables, should be suitable for testing climatic-change hypotheses. Aspects of the atmosphere which play no role in these hypotheses may be highly simplified. A super-model where virtually all not-strictly-constant features of the atmosphere and its environment enter as variables may ultimately lead to an acceptable theory of climate change.

**3. Edward N. Lorenz**, “Computational Chaos – A Prelude to Computational Instability”, Physica D, Vol. 35, pp. 299-317, 1989.

Abstract

Chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment r. In two simple examples we find that, as r increases, chaos first sets in when an attractor A acquires two distinct points that map to the same point. This happens when A acquires slopes of the same sign, in a rectifying coordinate system, at two consecutive intersections with the critical curve. Chaotic and quasi-periodic behavior may then alternate within a range of rc before computational instability finally prevails. Bifurcations to and from chaos and transitions to computational instability are highly scheme-dependent, even among differencing schemes of the same order. Systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings.

**4. E. N. Lorenz**, “Can Chaos and Intransitivity Lead to Interannual Variability?”, Tellus, Vol. 42A, pp. 378-389, 1990.

Abstract

We suggest that the atmosphere-ocean-earth system is unlikely to be intransitive, i.e., to admit two or more possible climates, any one of which, once established, will persist forever. Our reasoning is that even if the system would be intransitive if the external heating could be held fixed, say as in summer, the new heating patterns that actually accompany the advance of the seasons will break up any established summer circulation, and an alternative circulation may develop during the following summer, particularly if chaos has prevailed during the intervening winter. We introduce a very-low-order geostrophic baroclinic “general circulation” model, which may be run with or without seasonal variations of heating. Under perpetual summer conditions the model is intransitive, admitting either weakly oscillating or strongly oscillating westerly flow, while under perpetual winter conditions it is chaotic. When seasonal variations of heating are introduced, weak oscillations prevail through some summers and strong oscillations prevail through others, thus lending support to our original suggestion. We develop some additional properties of the model as a dynamical system, and we speculate as to whether its behavior has a counterpart in the real world.

**5. E. N. Lorenz**, “Irregularity: A Fundamental Property of the Atmosphere”, Tellus, Vol. 36A, pp. 98-110, 1984.

**6. E. N. Lorenz**, “Chaos, Spontaneous Climatic Variations and Detection of the Greenhouse Effect”, Published in Greenhouse-Gas-Induced Climatic Change: A Critical Appraisal of Simulations and Observations, M. E. Schlesinger, Editor, Elsevier, pp. 445-453, 1991.

**7. Roger A. Pielke and Xubin Zeng**, “Long-Term Variability of Climate”, Journal of the Atmospheric Sciences, Vol. 51, No. 1, pp. 155-159, 1994.

Abstract

In this research note, we address the following general question: In a nonlinear dynamical system (such as the climate system), can a known short-periodic variation lead to significant long-term variability? It is known from chaos studies (e.g., Lorenz 1991) that any perturbations in chaotic dynamic systems can lead to a red-noise spectrum; however, whether a significant long-term variability can be induced is unknown. To perform this study, an idealized nonlinear model developed by Lorenz (1984,1990) is used. The model and the results are presented in sections 2 and 3, respectively. Finally, the implications of our research to the understanding of the natural variability of the climate system due to internal dynamics will be discussed in section 4.

**8. J. C. Sprott**, “Chaos from Euler Solution of ODEs”, September 19, 1997.

**9. Roger A. Pielke**, “Climate Prediction as an Initial Value Problem”, Bulletin of the American Meteorology Society, Vol. 79, pp. 2743-2746, 1998.

**10. E. A. Burroughs**, E. A. Coutsias and L. A. Romero, “A Reduced-order Partial Differential Equation Model for the Flow in a Thermosyphon”, Journal of Fluid Mechanics, Vol. 543, pp. 203-237, 2005.

Abstract

Flow in a closed loop thermosyphon heated from below exhibits a sequence of bifurcations with increasing Grashof number. Using the Navier-Stokes equations in the Boussinesq approximation we have derived a model where, in the case of a slender circular loop, the first Fourier modes exactly decouple from all other Fourier modes, leaving a system of three coupled nonlinear partial differential equations that completely describes the flow in the thermosyphon. We have characterized the flow through two bifurcations, identifying stable periodic solutions for flows of Prandtl number greater than 18.5, a much lower value than predicted previously. Because of the quadratic nonlinearity in this system of equations, it is possible to find the global stability limit, and we have proved it is identical to the first bifurcation point.

The numerical study of the model equations is based on a highly accurate Fourier-

Chebyshev spectral method, combined with asymptotic analysis at the various bifurcation points. Three-dimensional computations with a finite element method computational fluid dynamics code (MPSalsa), are also pursued. All three approaches are in close agreement.

**11. Elizabeth Burroughs**, “Convection in a Thermosyphon: Bifurcation and Stability Analysis”, PhD Dissertation, The University of New Mexico, 2003.

Abstract

When a closed vertical loop of fluid is heated from below, the more buoyant hot fluid at the bottom of the loop creates an unstable configuration. In flows such as this one, it is the interplay between the buoyancy, causing the fluid to tend to rise, and the viscosity, causing the fluid to resist flow, that produces the motion of the fluid.

Using the Navier-Stokes equations to model this flow, one arrives at a sequence of bifurcation problems. Historically, researchers have been interested in flow in a thermosyphon both as a bifurcation and an engineering problem. I have derived a model where, in the case of a circular loop, the first Fourier modes exactly decouple from all other Fourier modes, leaving a system of three coupled nonlinear PDEs that completely describe the flow in the thermosyphon. I have characterized the flow through two bifurcations and found a global stability limit, thereby narrowing the location of a third bifurcation. This model has identified periodic solutions for flows of Prandtl number greater than 19, a phenomenon that other, more simplified models do not.

Finding the trivial solution (pure conduction) and linearizing about this solution, one arrives at an eigenvalue problem from which the onset of convection can be found. Using continuation in Grashof number, one can follow this solution branch to the Hopf bifurcation, which changes from sub- to supercritical at Prandtl number ~ 19.

I numerically analyze the equations using a spectral method with Chebyshev basis functions for the space dimension and use an implicit-explicit scheme to discretize the time dimension. The results obtained agree with those found analytically, where possible, and those obtained from the full three-dimensional equations with a FEM CFD code (MPSalsa), where available.

Because of the quadratic nonlinearity in this system of equations, it is possible to find the global stability limit, and I have proven it is identical to the first bifurcation point.

**12. Elizabeth A. Burroughs**, Richard B. Lehoucq, Louis A. Romero, and Andrew G. Salinger, “Linear Stability of Flow in a Differentially Heated Cavity via Large-Scale Eigenvalue Calculations”, Preprint submitted to International Journal of Numerical Methods for Heat and Fluid Flow, July 2002.

Abstract

We locate the onset of oscillatory instability in the flow in a differentially heated cavity of aspect ratio 2 by computing a steady state and analyzing the stability of the system via eigenvalue approximation. We discuss the choosing of parameters for the Cayley transformation so that the calculation of selected eigenvalues of the transformed system will most reliably answer the question of stability. We also present an argument that due to the symmetry of the problem, the first two unstable modes will have eigenvalues that are nearly identical, and our numerical experiments confirm this. We also locate a codimension 2 bifurcation signifying where there is a switch in the mode of initial instability. The results were obtained using a parallel finite element CFD code (MPSalsa) along with an Arnoldi-based eigensolver (ARPACK), a preconditioned Krylov method code for the necessary linear solves (Aztec), and a stability analysis library (LOCA).

**13. E. A. Burroughs**, L. A. Romero, R. B. Lehoucq, and A. G. Salinger , “Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows”.

Abstract

We present results for large scale linear stability analysis of buoyancy driven fluid flows using a parallel finite element CFD code (MPSalsa) along with a general purpose eigensolver (ARPACK). The goal of this paper is to examine both the capabilities and limitations of such an approach, with particular focus on solving large problems on massively parallel computers using iterative methods. We accomplish our goal by solving a large variety of two and three dimensional problems of varying difficulty, comparing our results (whenever possible) to semi-analytical results. We also carefully explain how we successfully combined Cayley transformations with an Arnoldi based eigensolver and preconditioned Krylov methods for the necessary linear solves.

For problems where the advective terms are not significant, we achieve excellent convergence of the computed eigenvalues as we refine the finite element mesh. We also successfully solve advectively dominated problems, but the convergence is slower. We believe that the main difficulties arise not from problems with the eigensolver, but from the accuracy of the finite element discretization. Therefore, we believe that our results are as reliable as using transient integration but are more efficiently computed. The largest eigenvalue problem we solve has over 16 million unknowns on 2048 processors.

**14. Luca Dieci**, Robert D. Russell, and Erik S. van Vleck, “On the Computation of Lyapunov Exponents for Continuous Dynamical Systems”, Society for Industrial and Applied Mathematics Journal of Numerical Analysis Vol. 34, No. 1, pp. 402-423, February 1997.

Abstract

In this paper, we consider discrete and continuous QR algorithms for computing all of the Lyapunov exponents of a regular dynamical system. We begin by reviewing theoretical results for regular systems and present general perturbation results for Lyapunov exponents. We then present the algorithms, give an error analysis of them, and describe their implementation. Finally, we give several numerical examples and some conclusions.

**15. Julyan** H. E. Cartwright and Oreste Piro, “The Dynamics of Runge-Kutta Methods”, International Journal Bifurcation and Chaos, Vol. 2, pp. 427-449, 1992.

Abstract

The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge-Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

**16. Divakar Viswanath** , Lyapunov Exponents from Random Fibonacci

Sequences to the Lorenz Equations, Ph D Dissertation Cornell University, August 1998.

Abstract

Lyapunov exponents give a way to capture the central features of chaos and of stability in both deterministic and stochastic systems using just a few real numbers. However, exact analytic determination of Lyapunov exponents is rarely possible, and as we will show, even an accurate numerical computation is not a trivial task.

One of the principal results of this thesis is about random Fibonacci sequences. Random Fibonacci sequences are defined by t1 = t2 = 1 and tn = Â±tn-1 Â± tn-2 for n > 2, where each Â± sign is independent and either + or – with probability 1/2.

Using Stern-Brocot sequences, we prove that

with probability 1.

Other contributions of this thesis include formulas for condition numbers of random triangular matrices and an accurate computation of the Lyapunov exponents of the Lorenz equations.

**17. Gabriel James Lord**, “Analysis of Numerical Methods Suitable for Computing Lyapunov Exponents”, School of Mathematical Sciences, University of Bath, 1995. ftp://ftp.maths.bath.ac.uk/pub/preprints/maths9502.ps.gz.

Abstract

Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth-order numerical schemes to preserve orthonormality. Convergence is shown for the particular case when the explicit Euler’s method is taken as the arbitrary method. This motivates looking at more general systems of ordinary differential equations which preserve orthonormality. An arbitrary convergent numerical method is modified to preserve orthonormality and convergence discussed in this general case. In each case numerical results are presented for the estimation of Lyapunov exponents for the Lorenz equations and results compared with those of other authors.

**12. Alicia Serfaty de Markus** and Ronald E. Mickens, “Suppression of Numerically Induced Chaos with Nonstandard Finite Difference Schemes”, Journal of Computational and Applied Mathematics, Vol. 106, pp. 317-324, 1999.

Abstract

It has been previously shown that despite its simplicity, appropriate nonstandard schemes greatly improve or eliminate numerical instabilities. In this work we construct several standard and nonstandard finite-difference schemes to solve a system of three ordinary nonlinear differential equations that models photoconductivity in semiconductors and for which it has been shown that integration with a conventional fourth-order Runge-Kutta algorithm produces numerical-induced chaos. It was found that a simple nonstandard forward Euler scheme successfully eliminates these numerical instabilities. In order to help determine the best finite-difference scheme, it was found useful to test the local stability of the scheme by direct inspection of the eigenvalues dependent on the step size.

**13. Alicia Serfaty de Markus**, “Detection of the Onset of Numerical Chaotic Instabilities by Lyapunov Exponents”, Discrete Dynamics in Nature and Society, Vol. 6, pp. 121-128, 2001.

Abstract

It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.

**14. Eduardo M. A. M. Mende**s and S. A. Billings, “A Note on Discretization of Nonlinear Differential Equations”, Chaos, Vol. 12, No. 1, pp. 66-71, 2002. DOI: 10.1063/1.1445783.

Abstract

An important issue when integrating nonlinear differential equations on a digital computer is the choice of the time increment or step size. For example, it is known that if this quantity is not sufficiently short, spurious chaotic motions may be induced when integrating a system using several of the well-known methods available in the literature. In this paper, a new approach to discretize differential equations is analyzed in light of computational chaos. It will be shown that the fixed points of the continuous system are preserved under the new discretization approach and that the spurious fixed points generated by higher order approximations depend upon the increment

parameter.

**15. Christophe Letellier**, Saber Elaydi, Luis A. Aguirre, and Aziz Alaoui, “Difference Equations versus Differential Equations, a Possible Equivalence for the Rossler System?” , Physica D, Vol. 195, pp. 29-49, 2004.

Abstract

When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete. In numerical investigations a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rossler system as long as the time step is less than the threshold value associated with the Nyquist criterion.

**16. Wayne H. Enright**, Desmond J. Higham, Brynjulf Owren, and Philip W. Sharp, “A Survey of the Explicit Runge-Kutta Method”.

Abstract

Research in explicit Runge-Kutta methods is producing continual improvements to the original algorithms, and the aim of this survey is to relate the current state-of-the-art. In drawing attention to recent advances, we hope to provide useful information for those who apply numerical methods.

We describe recent work for high order and Hamiltonian problems, and “continuous” formulas for dense output. We also give a thorough review of implementation details. Modern techniques are described for the tasks of controlling the global error, detecting stiffness, and detecting and handling discontinuities and singularities. We also discuss some important software issues.

**17. E. Adams**, W. F. Ames, W. Kahn, W. Rufeger and H. Spreuer, “Computational Chaos May Be Due to a Single Local Error”, Journal of Computational Physics, Vol. 104, No. 1, pp. 214-250, 1993.

Abstract

Nonlinear ordinary differential equations and arbitrary difference methods are considered which satisfy conditions for the convergence of a sequence of true difference solutions. This convergence does not prevent “diversions” of computed difference approximations, a property which is defined here. The occurrence of diversions is demonstrated in examples, namely the restricted three body problem and the Lorenz equations. This occurrence is practically unpredictable. In the applied literature, this property has been used to define “(dynamical) chaos.” Therefore, observed chaos for solutions of ODEs is not necessarily a consequence of a sensitive dependency on the initial vector but, rather, may be due to a corresponding dependency on computational errors.

**18. Nedialko S. Nedialkov**, “Interval Tools for ODEs and DAEs”.

Abstract

We overview the current state of interval methods and software for computing bounds on solutions in initial value problems (IVPs) for ordinary differential equations (ODEs). We introduce the VNODE-LP solver for IVP ODEs, a successor of the authorâ€™s VNODE package. VNODE-LP is implemented entirely using literate programming. A major goal of the VNODE-LP work is to produce an interval solver such that its correctness can be verified by a human expert, similar to how mathematical results are certified for correctness. We also discuss the state in computing bounds on solutions in differential algebraic equations.

**19. **COSY Infinity Site and publications list.

What Is COSY?

COSY is a system for the use of various advanced concepts of modern scientific computing. COSY currently has more than 1000 registered users and has been extensively cross-checked and verified. The COSY system consists of the following parts.

1) A collection of advanced highly optimized Data Types. In particular:

– The Differential Algebraic types for high-order multivariate study of ODEs, Flows, and PDEs. Also allow high-order multivariate automatic differentiation.

– The Taylor Model type for rigorous high-order computing with often far-reaching suppression of dependency. Tools for range bounding, derivative-based box rejection, constraint satisfaction, ODEs and PDEs.

2) The COSYScript environment for the use of these types. It is object oriented and supports polymorphism, has a compact and simple syntax, and is compiled and executed on the fly. It has built-in optimization constructs, and is used for high turn around simulation.

3) Interfaces for C++ and F90 to seamlessly use the types in external programs in these object oriented languages.

What Else Is COSY INFINITY?

COSY is also a collection of application packages. The following are the major packages currently available:

1) The Beam Physics package cosy.fox (see below)

2) The Rigorous Computing package TM.fox

3) The Rigorous Global Optimizer COSY-GO

4) The Rigorous Verified Integrator COSY-VI

**20. Ole Stauning**, “Automatic Validation of Numerical Solutions” Ph. D. Dissertation, Technical University of Denmark, 1997.

Abstract

This thesis is concerned with “Automatic Validation of Numerical Solutions”. The basic theory of interval analysis and self-validating methods is introduced. The mean value enclosure is applied to discrete mappings for obtaining narrow enclosures of the iterates when applying these mappings with intervals as initial values. A modification of the mean value enclosure of discrete mappings is considered, namely the extended mean value enclosure which in most cases leads to even better enclosures. These methods have previously been described in connection with discretizing solutions of ordinary differential equations, but in this thesis, we describe how to use the methods for enclosing iterates of discrete mappings, and then later use them for discretizing solutions of ordinary differential equations.

The theory of automatic differentiation is introduced, and three methods for obtaining derivatives are described: The forward, the backward, and the Taylor expansion methods. The three methods have been implemented in the C++ program packages FADBAD/TADIFF. Some examples showing how to use the three methods are presented. A feature of FADBAD/TADIFF not present in other automatic differentiation packages is the possibility to combine the three methods in an extremely flexible way. We examine some applications where this flexibility is very useful.

A method for Taylor expanding solutions of ordinary differential equations is presented, and a method for obtaining interval enclosures of the truncation errors incurred, when truncating these Taylor series expansions is described. By combining the forward method and the Taylor expansion method, it is possible to implement the (extended) mean value enclosure of a truncated Taylor series expansion with enclosures of the truncation errors. A C++ program package ADIODES, using this method has been developed. (ADIODES is an abbreviation of “Automatic Differentiation Interval Ordinary Differential Equation Solver”).

ADIODES is used to prove existence and uniqueness of periodic solutions to specific ordinary differential equations occurring in dynamical systems theory. These proofs of existence and uniqueness are difficult or impossible to obtain using other known methods. Also, a method for solving boundary value problems is described.

Finally a method for enclosing solutions to a class of integral equations is described. This method is based on the mean value enclosure of an integral operator and uses interval Bernstein polynomials for enclosing the solution. Two numerical examples are given, using two orders of approximation and using different numbers of discretization points.

**21. Joao Teixeira**, Carolyn A. Reynolds, and Kevin Judd “Time Step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth, and Ensemble Design”, Journal of the Atmospheric Sciences, Vol. 64, pp. 175-189, 2007.

Abstract

Computational models based on discrete dynamical equations are a successful way of approaching the problem of predicting or forecasting the future evolution of dynamical systems. For linear and mildly nonlinear models, the solutions of the numerical algorithms on which they are based converge to the analytic solutions of the underlying differential equations for small time steps and grid sizes. In this paper, the authors investigate the time step sensitivity of three nonlinear atmospheric models of different levels of complexity: the Lorenz equations, a quasigeostrophic (QG) model, and a global weather prediction system (NOGAPS). It is illustrated here how, for chaotic systems, numerical convergence cannot be guaranteed forever. The time of decoupling of solutions for different time steps follows a logarithmic rule (as a function of time step) similar for the three models. In regimes that are not fully chaotic, the Lorenz equations are used to illustrate how different time steps may lead to different model climates and even different regimes. A simple model of truncation error growth in chaotic systems is proposed. This model decomposes the error onto its stable and unstable components and reproduces well the short- and medium-term behavior of the QG model truncation error growth, with an initial period of slow growth (a plateau) before the exponential growth phase. Experiments with NOGAPS suggest that truncation error can be a substantial component of total forecast error of the model. Ensemble simulations with NOGAPS show that using different time steps may be a simple and natural way of introducing an important component of model error in ensemble design.

**22. Lun-Shin Yao**, “What’s new on Computed Lorenz Strange Attractors: Chaos or Numerical Errors”, 2003.

Abstract

Discrete numerical methods with finite time steps are probably the most practical technique, if not the only one, to solve non-linear differential equations. This is particularly true for chaos. Using the Lorenz equations as an example, we clearly demonstrate that a trajectory mistakenly penetrates the virtual separatrix due to finite time steps and truncation errors. This leads to the explosive growth of numerical errors; hence, these computed “solutions” are unshadowable and are not valid solutions. This is generic for chaotic systems and occurs repeatedly since the l-lemma guarantees a trajectory travels arbitrarily close to the inset of a saddle point over and over again. Hence, numerical solutions of differential equations sensitive to the integration-time step with finite statistical properties are divergent, bounded numerical errors. At present, all numerical methods introduce truncation errors; consequently, they are incapable of solving chaotic differential equations. The convergence of these numerical chaotic solutions has never been proven. Realistically, it cannot be proven; nevertheless, it is common practice, perhaps out of convenience and for lack of an alternative, to blindly accept them as adequate solutions to these problems!

**23. Lun-Shin Yao**, “Computed Chaos or Numerical Errors”, 2006.

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.

**24. Lun-Shin Yao**, “Is a Direct Numerical Simulation of Chaos or Turbulence Possible: A Study of a Model Non-Linearity”, 2002.

Abstract

There are many subtle issues associated with solving the Navier-Stokes equations. In this paper, several of these issues, which have been observed previously in research involving the Navier-Stokes equations, are studied within the framework of the one-dimensional Kuramoto-Sivashinsky (KS) equation, a model nonlinear partial-differential equation. This alternative approach is expected to more easily expose major points and hopefully identify open questions that are related to the Navier-Stokes equations. In particular, four interesting issues are discussed. The first is related to the difficulty in defining regions of linear stability and instability for a time-dependent governing parameter; this is equivalent to a time-dependent base flow for the Navier-Stokes equations. The next two issues are consequences of nonlinear interactions. These include the evolution of the solution by exciting its harmonics or sub-harmonics (Fourier components) simultaneously in the presence of a constant or a time-dependent governing parameter; and the sensitivity of the long-time solution to initial conditions. The final issue is concerned with the lack of convergent numerical chaotic solutions, an issue that has not been previously studied for the Navier-Stokes equations. The last two issues, consequences of nonlinear interactions, are not commonly known. Conclusions and questions uncovered by the numerical results are discussed. The reasons behind each issue are provided with the expectation that they will stimulate interest in further study.

**25. Lun-Shin Yao**, “Trouble of Non-Linearity”, 2004.

Abstract

All complex fluid motions, such as transition and turbulence, obeying the Navier-Stokes equations are non-linear phenomena. Some aspects of the non-linear terms of these equations are not well understood and are, in fact, misunderstood. The one-dimensional Kuramoto-Sivashinsky (KS) equation is used as a simple model non-linear partial differential equation to show some essential functions of its non-linear term and its consequences, which, we believe, are shared with other non-linear partial differential equations. We show that solutions of nonlinear partial differential equations above their critical parameters may be linearly stable, but are nonlinearly unstable. No stable solution exists above the critical parameter, contrary to the prediction of the linear-stability analysis. This is because a linearly stable disturbance can transfer energy simultaneously, not necessarily in cascade from small wave numbers to large wave numbers. An initial disturbance can breed its entire harmonics simultaneously. Second, we show that a longtime numerical chaotic solution cannot be achieved by a discrete numerical method.

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