Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-06T14:11:22.191Z Has data issue: false hasContentIssue false

5 - Non-analytic dynamics

Published online by Cambridge University Press:  05 July 2014

Bruce J. West  and
Paolo Grigolini
Bruce J. West
Affiliation:
Army Research Office
Paolo Grigolini
Affiliation:
University of North Texas
Get access

Summary

In this chapter we investigate one procedure for describing the dynamics of complex webs when the differential equations of ordinary dynamics are no longer adequate, that is, the webs are fractal. We described some of the essential features of fractal functions earlier, starting from the simple dynamical processes described by functions that are fractal, such as the Weierstrass function, which are continuous everywhere but are nowhere differentiable. This idea of non-differentiability suggests introducing elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. The relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, changes in time of phenomena that are described by fractal functions are probably best described by fractional equations of motion. In any event, this perspective is the one we developed elsewhere [31] and we find it useful here for discussing some properties of complex webs.

The separation of time scales in complex physical phenomena allows smoothing over the microscopic fluctuations and the construction of differentiable representations of the dynamics on large space scales and long time scales. However, such smoothing is not always possible.

Type
Chapter
Information
Complex Webs
Anticipating the Improbable
 , pp. 224 - 261
Publisher: Cambridge University Press
Print publication year: 2010

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

[1] E., Bacry, J., Delour and J. F., Muzy, “Multifractal random walk,” Phys.Rev. E 64, 026103–1 (2001).Google Scholar
[2] J. B., Bassingthwaighte, L. S., Liebovitch and B. J., West, Fractal Physiology, Oxford: Oxford University Press (1994).Google Scholar
[3] K. V., Bury, Statistical Models in Applied Science, New York: John Wiley (1975).Google Scholar
[4] K., Falconer, Fractal Geometry, New York: John Wiley (1990).Google Scholar
[5] J., Feder, Fractals, New York: Plenum Press (1988).Google Scholar
[6] P., Grigolini, A., Rocco and B.J., West, “Fractional calculus as a macroscopic manifestation of randomness,” Phys. Rev. E 59, 2603 (1999).Google Scholar
[7] L., Griffin, D. J., West and B. J., West, “Random stride intervals with memory,” J. Biol. Phys. 26, 185–202 (2000).Google Scholar
[8] J. M., Hausdorff, C.-K., Peng, Z., Ladinet al. “Is walking a random walk? Evidence for long-range correlations in stride interval of human gait,” J. Appl. Physiol. 78 (1), 349-358 (1995).Google Scholar
[9] R., Hilfer, ed., Applications of Fractional Calculus in Physics, Singapore: World Scientific (2000).Google Scholar
[10] V. M., Kenkre, E.W., Montroll and M. F., Shlesinger, “Generalized master equations for continuous-time random walks,” J. Statist. Phys. 9, 45 (1973).Google Scholar
[11] P., Lévy, Théorie de l'addition des variables aléatoires, Paris: Gauthier-Villars (1937)Google Scholar
[12] S.C., Lim and L. P., Teo, “Generalized Whittle-Matérn random field with application to modeling,” J. Phys.: Math. Theor. 42, 105202 (2009).Google Scholar
[13] E., Lutz, “Fractional Langevin equation,” Phys. Rev. E 64, 051106 (2001).Google Scholar
[14] R. L., Magin, Fractional Calculus in Bioengineering, Connecticut: Begell House (2006).Google Scholar
[15] F., Mainardi, R., Gorenflo and A., Vivoli, “Renewal processes of Mittag-Leffler and Wright type,” Fractional Calculus and Applied Analysis 8, 7–38 (2005).Google Scholar
[16] B.B., Mandelbrot, The Fractal Geometry of Nature, San Francisco, CA: W.J. Freeman (1983).Google Scholar
[17] P., Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge: Cambridge University Press (1998).Google Scholar
[18] K. S., Miller and B., Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: John Wiley & Sons (1993).Google Scholar
[19] E. W., Montroll and G. H., Weiss, “Random walks on lattice. II,” J. Math. Phys. 6, 167–181 (1965).Google Scholar
[20] C. K., Peng, J., Mietus, J. M., Hausdorffet al., “Long-range autocorrelations and non-Gaussian behavior in the heartbeat,” Phys. Rev. Lett. 70, 1343 (1993).Google Scholar
[21] Yu. N., Rabotnov, Elements of Hereditary Solid Mechanics, Moscow: Mir (1980).Google Scholar
[22] A., Rocco and B. J., West, “Fractional calculus and the evolution of fractal phenomena,” PhysicaA 265, 535 (1999).Google Scholar
[23] D., Ruelle, Chaotic Evolution and Strange Attractors, Cambridge: Cambridge University Press (1989).Google Scholar
[24] S.G., Samko, A.A., Kilbas and O.I., Marichev, Fractional Integrals and Derivatives, Amsterdam: Gordon and Breach Science Publishers (1993).Google Scholar
[25] N., Scafetta, L., Griffin and B.J., West, “Holder exponent spectra for human gait,” Physica A 328, 561–583 (2003).Google Scholar
[26] I., Sokolov, J., Klafter and A., Blumen, “Fractional kinetics,” Physics Today, Nov., 48-54 (2002).Google Scholar
[27] H. E., Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford: Oxford University Press (1979).Google Scholar
[28] W., Weibull, “A statistical theory of strength of materials,” Ing. Vetenskaps Akad. Handl. No. 151 (1939).Google Scholar
[29] G., Weiss, Aspects and Applications of the Random Walk, Amsterdam: North-Holland (1994).Google Scholar
[30] B. J., WestandT., Nonnenmacher, “An ant in a gurge,” Phys. Lett. A 278, 255–259 (2001).Google Scholar
[31] B.J., West, M., Bologna and P., Grigolini, Physics of Fractal Operators, New York: Springer (2003).Google Scholar
[32] B.J., West, M., Latka, M., Glaubic-Latka and D., Latka, “Multifractality of cerebral blood flow,” Physica A 318, 453–460 (2003).Google Scholar
[33] G. M., Zalslavsky, R. Z., Sagdeev, D. A., Usikov and A. A., Chernikov, Weak Chaos and Quasi-Regular Patterns, Cambridge: Cambridge University Press (1991).Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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.

Available formats
×

Save book to Dropbox

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

Available formats
×

Save book to Google Drive

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

Available formats
×