Models Methods Software

Dan Hughes

Chaos and ODEs Part 1d: Calculations and Results

The calculations preformed with the equation systems are summarized in the following discussions. The focus had been on testing for convergence of the numerical solution methods to solutions of the continuous equations. By convergence I mean that as the size of the discrete increment for the independent variable is reduced the calculated values for all dependent variables approach limiting constant values for all values of the independent variable.

None of the systems that are said to exhibit chaotic response have shown convergence. One of those, the Terman system, exhibits periodic response, not chaotic response. The Saltzman system was never intended to be an example for chaotic response.

Whenever a reference to a numerical solution method is stated in the following discussions it refers to the method outlined in Part 1c. The solution methods are listed here.

The explicit Euler method refers to Eq. (1.4).

The implicit Euler method refers to Eq. (1.5) and the implementation in the SEULEX solver routines.

The explicit midpoint method refers to Eq. (1.6).

The explicit Runge-Kutta method refers to Eq. (1.7).

The analytical Taylor series method refers to Eq. (1.8) and to Ascher and Petzold, pp. 73-74.

Off-the-shelve software refers to the DVERK solver routines.

Finite difference methods refers to equations like (1.11) through (1.13) while specific formulation will be stated in the following discussions.

The Adomian decomposition method (ADM) refers to the method discussed by Adomian and used in several of the papers listed in Part 1a.

All the calculations were done with Fortran programs. With the exceptions of the SEULEX and DVERK methods, all the software was developed by me. The source code for any calculations can be made available to anyone who wants it. To be usable you’ll need a Fortran compiler or you can convert the source to any of your favorite languages. The source code is research-grade software and is not intended to be production-grade software by any definition whatsoever. The programs are interactive only at the source-code-editor level. To change any of the runtime options you’ll have to make the changes in the source code and compile-link-execute.

A pdf version of the report(s) for each equation system will be uploaded at this blog and a link to the file(s) supplied in this post as they are completed, most likely in the following list.

Part 1d(1): The Saltzman System
Part 1d(2): The Lorenz 1963 System. A Comment and Reply.
Part 1d(3): The Lorenz 1984 System
Part 1d(4): The Chen System
Part 1d(5): The Rossler System
Part 1d(6): The Vadasz and Olek System
Part 1d(7): The Lotka-Volterra System
Part 1d(8): The Terman system
Part 1d(9): The Generalized Lorenz Systems

The reports are very likely to appear in random order and under no fixed time schedule.

Update February 4, 2008
I have uploaded a short writeup of some of the results for the Terman system. Let me know if you want to see other results, calculations with other solution methods, or results for other ranges of parameter values.

Update Sometime later in early February 2008
I uploaded a short writeup of some results for the Lorenz 1984 system.

Update March 18, 2008
I uploaded a Comment and Reply for a paper by Teixeira et al. published in the JAS.

Update April 2, 2008
The Tellus Comment is available if you want to pay for it.

Update April 17, 2008
Very sad news. Edward Lorenz has died. The New York Times has an obituary. His response to the above comment was likely among the last of his publications.


November 20, 2007 - Posted by | Chaos and Lorenz | , , ,


  1. That is definitely a huge program you have there Dan .
    Perhaps reducing the number of systems to study to the most “physical” would make the work easier .
    Do you intend to extend from ODE (that are often labelled as describing unphysical/ideal systems and therefore irrelevant) to PDE where you have all the physical world like Navier Stokes ?

    In any case I am following your posts with great interest .
    On a particular note , do you know the Ruelle Takens paper “On the nature of turbulence” showing that turbulence in N-S is chaotic ?

    Comment by Tom Vonk | December 3, 2007 | Reply

  2. Hello Tom,

    Yep, lots o’ material; several equation systems, several parameters in each system, several numerical methods for each system/parameter. I cannot not possibly run all combinations, and I dabble in other aspects off and on, too. Getting my results organized for documentation has proven to be a lot of work. I am trying to provide sufficient details that anyone can reproduce my results.

    A bunch of runs will not be a proof, of course, but I’m trying to head off questions like ‘did you try this or that’. But I have concluded, for myself at least, that there are no converged solutions for any system of ODEs said to exhibit chaotic response of complex dynamical systems. There are some ranges of the independent variables for which one might can say the numerical method has converged, but is it really convergence solely because the range is restricted?

    The Terman system is an interesting case. Smale’s horseshoe is said to exist for the system yet when I run the calculations out for extended time I get periodic solutions. There seems to be a single ‘chaotic’ response and then periodic response thereafter. This behavior is especially interesting because in some applications the ‘startup transient’ is ignored and stats on the calculated numbers are taken from that time onward. For the Terman case, this procedure would completely miss the only ‘chaotic’ response. And if one did not look at a plot of the results this fact would be missed.

    I would like to get to some physical equation system and will attempt to do that as I continue to work on the problem. I will study the Ruelle and Takens paper. The Saltzman system seems to be the closest to a physical system and it does not exhibit chaotic response. Lorenz had to fiddle around with the Saltzman system in order to get a system that is chaotic. I think it is a correct statement that results from none of the systems have been compared with any measured data of any kind.

    The issues of (1) shadowing and (2) using stats to make the calculated numbers useful remain open. On shadowing, however, if the calculated numbers do not satisfy the continuous equations, how can a true solution be shadowed? And the same concern holds for using stats; can averages of numbers that do not satisfy the continuous equations have any meaning?

    Lot and lots of questions and not many answers.

    Comment by Dan Hughes | December 3, 2007 | Reply

  3. Tellus A has accepted a Comment on this paper which arose from this paper. The Comment and a reply by Lorenz might make the March 2008 issue if we made all the deadlines on time.

    Comment by Dan Hughes | December 12, 2007 | Reply

  4. Damn it , I can’t access the links because :
    client IP is blocked because: Spider trap hit

    Whatever this might be .

    Comment by Tom Vonk | December 13, 2007 | Reply

  5. Tom Vonk,
    Regarding results of Ruelle and Takens, you may want to read their clarification,
    As V.I.Arnold said somewhere, the Ruelle-Takens construction occupies a “skinny” set of measure near zero in the space of dynamical systems spanned from N-S equations. Actually, the studies of turbulence in early 1960’s have been conducted at a group of mathematicians lead by Kolmogorov at Moscow University; they discovered a link of N-S equations with more generic “hyperbolical” systems. The problem appeared to be of tremendous complexity, no clean results were obtainable, so the honest scientist decided not to make a big deal of their “unsuccessful” efforts. Later some of their results were re-discovered by West, and widely publicized. As another physicist from Siberia, B.V. Chirikov used to mention in his lectures about dynamical systems, “strange attractors are strange only to strangers”.
    – Alexi

    Comment by Alexi Tekhasski | April 7, 2008 | Reply

  6. Alexi Thekasski

    I am aware of Kolmogorov work as well as of the Ruelle&Takkens “clarification” that you linked .
    I am also aware that the Ruelle&Takkens result doesn’t apply to a general case even if it is far from being “skinny” .
    As for “strange” attractors – it is only a word , they have indeed nothing strange like quark “colors” are not really colors .

    I do not know the russian papers from the 60ies as we are in the meantime 50 years later but if there are some new russian results that go beyond the 60ies “failures” and are available , links would be of course welcome .

    Comment by Tom Vonk | April 9, 2008 | Reply

  7. […] here, the numerical solution methods are summarized here, and some results are summarized here. For this post I have used the explicit Euler method although I’m certain that any other of […]

    Pingback by Averaging Planet Earths or Averaging Planet Xs « Models Methods Software | January 23, 2011 | Reply

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