Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-xt4p2 Total loading time: 0.336 Render date: 2022-05-23T11:18:20.846Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Discrete-time and continuous-time modelling: some bridges and gaps

Published online by Cambridge University Press:  01 April 2007

HUBERT KRIVINE
Affiliation:
LPTMS, Bâtiment 100, Université Paris-Sud, F-91405 Orsay Email: hubert.krivine@u-psud.fr
ANNICK LESNE
Affiliation:
LPTMC UMR7600, Université Pierre et Marie Curie-Paris 6, 4 place Jussieu, F-75252 Paris Email: lesne@ihes.fr Institut des Hautes Études Scientifiques, 35 route de Chartres, F-91440 Bures-sur-Yvette, France Email: treiner@ccr.jussieu.fr
JACQUES TREINER
Affiliation:
LPTMS, Bâtiment 100, Université Paris-Sud, F-91405 Orsay Email: hubert.krivine@u-psud.fr

Extract

The relationship between continuous-time dynamics and the corresponding discrete schemes, and its generally limited validity, is an important and widely acknowledged field within numerical analysis. In this paper, we propose another, more physical, viewpoint on this topic in order to understand the possible failure of discretisation procedures and the way to fix it. Three basic examples, the logistic equation, the Lotka–Volterra predator–prey model and Newton's law for planetary motion, are worked out. They illustrate the deep difference between continuous-time evolutions and discrete-time mappings, hence shedding some light on the more general duality between continuous descriptions of natural phenomena and discrete numerical computations.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Borrelli, R. L. and Coleman, C. S. (1998) Differential equations: A modeling perspective, Wiley.Google Scholar
Coullet, P., Monticelli, M. and Treiner, J. (2004) L'algorithme de Newton-Hooke. Bulletin de l'Union des Physiciens 861 193206.Google Scholar
Coullet, P. and Tresser, C. (1978) Itération d'endomorphismes et groupe de renormalisation. CR Acad. Sc. Paris 287A 577580.Google Scholar
Devaney, R. L. (1989) An introduction to chaotic dynamical systems, Addison-Wesley.Google Scholar
Evans, D. J. and Morriss, G. P. (1990) Statistical mechanics of nonequilibrium liquids, Chapter 10, Academic Press.Google Scholar
Feigenbaum, M. (1978) Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19 2552.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2002) Geometric numerical integration, Springer.CrossRefGoogle Scholar
Hubbard, J. and West, B. (1991) Differential equations, a dynamical systems approach, Springer.Google Scholar
Korsch, H. J. and Jodl, H. J. (1998) Chaos: a program collection for the PC, Springer.Google Scholar
Iooss, G. and Joseph, D. D. (1981) Elementary stability and bifurcation theory, Springer.Google Scholar
Kolmogorov, A. N. (1936) Sulla teoria di Volterra della lotta per l'esistenza. Giornale Istituto Ital. Attuari 7 7480.Google Scholar
Lotka, A. J. (1920) Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA 6 410415.CrossRefGoogle ScholarPubMed
May, R. M. (1973) Stability and complexity in model ecosystems, Princeton University Press.Google ScholarPubMed
May, R. M. (1976) Simple mathematical models with very complicated dynamics. Nature 261 459467.CrossRefGoogle ScholarPubMed
Mendes, E. and Letellier, C. (2004) Displacement in the parameter space versus spurious solution of discretization with large time step. J. Phys. A 37 12031218.CrossRefGoogle Scholar
Mickens, R. E. (2002) Nonstandard finite difference schemes of differential equations. Journal of Difference Equations and Applications 2 823847.CrossRefGoogle Scholar
Murray, J. D. (2002) Mathematical biology, 3rd edition, Springer.Google Scholar
Nyquist, H. (1928) Certain topics in telegraph transmission theory. AIEE Trans. 47 617644.Google Scholar
Peitgen, H. O., Jürgens, H. and Saupe, D. (1992) Chaos and fractals, Springer.CrossRefGoogle Scholar
Sanz-Serna, J. M. (1992) Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica 1 243286.CrossRefGoogle Scholar
Schuster, H. G. (1984) Deterministic chaos, Physik-Verlag, Wienheim.Google Scholar
Shannon, C. (1949) Communication in the presence of noise. Proceedings of the IRE 37 10–21. (Reprinted in Proceedings of the IEEE 86 447–457 (1998).)Google Scholar
Tabor, M. (1989) Chaos and integrability in non linear dynamics. An introduction, Wiley.Google Scholar
Verhulst, P. F. (1838) Notice sur la loi que la population suit dans son accroissement. Corresp. Math. et Phys. 10 1321.Google Scholar
Volterra, V. (1931) Leçons sur la théorie mathématique de la lutte pour la vie, Gauthiers-Villars, Paris.Google Scholar
Yamaguti, M. and Matano, H. (1979) Euler's finite difference scheme and chaos. Proc. Japan Acad. Series A 55 7880.CrossRefGoogle Scholar
5
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Discrete-time and continuous-time modelling: some bridges and gaps
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Discrete-time and continuous-time modelling: some bridges and gaps
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Discrete-time and continuous-time modelling: some bridges and gaps
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *