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.
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 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.