The fractal dimension of the Lorenz attractor Export

@article{citeulike:5778476, abstract = {It is noted that the box-counting algorithm suggested by Takens has been successfully applied to two-dimensional maps, a set of three ordinary differential equations, and a delay differential equation. The algorithm, however, fails to converge for the Lorenz (1963) equations. This convergence problem has been encountered by others for the Lorenz equations and for the Curry model. As has been suggested, a major reason for the problem is that some parts of the attractor are visited with a probability that tends to zero (Doerfle and Graham, 1983). The rate at which this probability tends to zero is algebraic in the region of the attractor that yields a one-dimensional cusp-shaped return map. If the algebraic decay rate holds on the whole attractor, it will imply an algebraically slow rate of convergence in time of the number of boxes visited by a solution trajectory (over some intermediate time range). This indicates an extrapolation in time to find the total number of boxes convering the attractor. Taken's box-counting algorithm for computing the fractal dimension of a strange attractor is applied here to the Lorenz equations. After discussing a convergence problem, an approximate dimension is computed.}, author = {{McGuinness}, M. J.}, citeulike-article-id = {5778476}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/0375-9601(83)90052-X}, citeulike-linkout-1 = {http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983PhLA...99....5M}, doi = {10.1016/0375-9601(83)90052-X}, journal = {Physics Letters A}, keywords = {algorithms, attractors, box-counting, dimensions, dynamical-systems, fractals, lorenz-attractor, math}, month = {November}, pages = {5--9}, posted-at = {2009-09-12 20:34:55}, priority = {2}, title = {The fractal dimension of the Lorenz attractor}, url = {http://dx.doi.org/10.1016/0375-9601(83)90052-X}, volume = {99}, year = {1983} }

TY - JOUR ID - citeulike:5778476 L3 - citeulike-article-id:5778476 N2 - It is noted that the box-counting algorithm suggested by Takens has been successfully applied to two-dimensional maps, a set of three ordinary differential equations, and a delay differential equation. The algorithm, however, fails to converge for the Lorenz (1963) equations. This convergence problem has been encountered by others for the Lorenz equations and for the Curry model. As has been suggested, a major reason for the problem is that some parts of the attractor are visited with a probability that tends to zero (Doerfle and Graham, 1983). The rate at which this probability tends to zero is algebraic in the region of the attractor that yields a one-dimensional cusp-shaped return map. If the algebraic decay rate holds on the whole attractor, it will imply an algebraically slow rate of convergence in time of the number of boxes visited by a solution trajectory (over some intermediate time range). This indicates an extrapolation in time to find the total number of boxes convering the attractor. Taken's box-counting algorithm for computing the fractal dimension of a strange attractor is applied here to the Lorenz equations. After discussing a convergence problem, an approximate dimension is computed. JF - Physics Letters A EP - 9 TI - The fractal dimension of the Lorenz attractor VL - 99 SP - 5 KW - algorithms KW - attractors KW - box-counting KW - dimensions KW - dynamical-systems KW - fractals KW - lorenz-attractor KW - math AU - {McGuinness}, MJ PY - 1983/November// UR - http://dx.doi.org/10.1016/0375-9601(83)90052-X ER -