TY - JOUR

T1 - On the singularity analysis of intersecting separatrices in near-integrable dynamical systems

AU - Bountis, Tassos

AU - Papageorgiou, V.

AU - Bier, M.

N1 - Funding Information:
This work was supported in part by the NSF grant no. DMS-8314172 and the D.O.E. contract DE-FG02-84ER40144. Two of the authors (T.B. and V.P.) also wish to acknowledge discussions with Prof. Drs. H. Capel, R. Helleman and J. Hijmans and the hospitality of the Institute of
Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1987

Y1 - 1987

N2 - It has been recently proved by S. L. Ziglin that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglin's theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows x ̇=f(x) + εg(x,t), with x≡(x,y) and g(x,t)=g(x,t+2π), which for ε = 0 possess the Painlevé property. On the other hand, the theoretically expected logarithmic singularities for ε ≠ 0 are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painlevé property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting "chimney" patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes "denser" and each chimney is seen to gradually "collapse" towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.

AB - It has been recently proved by S. L. Ziglin that transversal intersections of separatrices (invariant manifolds) in near-integrable Hamiltonian systems of two degrees of freedom imply the existence of multi-valued solutions with infinitely many Riemann sheets. Ziglin's theorem is illustrated here on a simple example and then extended and applied to non-Hamiltonian, analytic flows x ̇=f(x) + εg(x,t), with x≡(x,y) and g(x,t)=g(x,t+2π), which for ε = 0 possess the Painlevé property. On the other hand, the theoretically expected logarithmic singularities for ε ≠ 0 are obtained explicitly in solutions near the intersecting separatrices. Thus, we conjecture that dynamical systems with the Painlevé property, can have no separatrix intersections and hence no strange attractors, etc. These singularities are then numerically located and found to form certain very interesting "chimney" patterns in the complex t-plane, on which they accumulate densely. The upper part of the chimneys (away from the Re t axis) is asymptotically quite insensitive to changes in parameter values or initial conditions. The singularity pattern itself, however, becomes "denser" and each chimney is seen to gradually "collapse" towards the Re t axis, as the amplitude of the driving term increases and the motion becomes more chaotic.

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U2 - 10.1016/0167-2789(87)90081-9

DO - 10.1016/0167-2789(87)90081-9

M3 - Article

AN - SCOPUS:0040383970

VL - 24

SP - 292

EP - 304

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-3

ER -