TY - JOUR

T1 - Integrating the nonintegrable

T2 - Analytic structure of the Lorenz system revisited

AU - Levine, G.

AU - Tabor, M.

N1 - Funding Information:
This work is supported by Department of Energy grant DE-FGO2-84ER-13190. M.T. is an Alfred P. Sloan Research Fellow.

PY - 1988

Y1 - 1988

N2 - A study of the complex time analytic structure of the Lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one (time-dependent) integral of motion exists. Furthermore, the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral. Recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument. The analytic techniques used in these studies, a systematic resummation of a logarithmic psi-series, appears to be quite general and can provide explicit representations of a solution-even in the chaotic regimes-in the neighborhood of a given movable singularity. Furthermore, we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems.

AB - A study of the complex time analytic structure of the Lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one (time-dependent) integral of motion exists. Furthermore, the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral. Recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument. The analytic techniques used in these studies, a systematic resummation of a logarithmic psi-series, appears to be quite general and can provide explicit representations of a solution-even in the chaotic regimes-in the neighborhood of a given movable singularity. Furthermore, we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems.

UR - http://www.scopus.com/inward/record.url?scp=45449123299&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45449123299&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(98)90018-5

DO - 10.1016/S0167-2789(98)90018-5

M3 - Article

AN - SCOPUS:45449123299

VL - 33

SP - 189

EP - 210

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-3

ER -