A simple model for chaotic scattering. I: Classical theory. (English) Zbl 0753.58037
Summary: We study a simple physical problem which displays classical chaotic (irregular) scattering. The dynamics is given by a mapping which has much in common with the “standard” map. Through the study of the dynamics we show that irregular scattering is a physical realization of transient chaos in a Hamiltonian system. We discuss the mechanism responsible for transient chaos in this system and derive analytical expressions for the various parameters which are needed to characterize it.
MSC:
| 58Z05 | Applications of global analysis to the sciences |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 34L25 | Scattering theory, inverse scattering involving ordinary differential operators |
Keywords:
scattering; chaotic systems; scattering theory; stochastic processes; classical mechanics; topological mapping; transients; equations of motion; dynamical systems; Hamiltonian function; time dependence; potential scattering; bifurcationCitations:
Zbl 0753.58038References:
| [1] | Silnikov, K., Dokl. Akad. Nauk. USSR, 133, No. 2, 303 (1960) |
| [2] | Alekseev, V. M., Math. USSR Sbornik, 7, 1 (1969) |
| [3] | Moser, J., Stable and Random Motions in Dynamical Systems (1973), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0271.70009 |
| [4] | Physica D (1988), to appear |
| [5] | Vázquez, E. C.; Jeffays, W. H.; Sivaramakrishnan, A., Physica D, 29, 84 (1987) |
| [6] | Grebogi, C.; Ott, E.; Yorke, J. A., Physica D, 7, 181 (1983) |
| [7] | Kantz, H.; Grassberger, P., Physica D, 17, 75 (1985) · Zbl 0597.58017 |
| [8] | Grebogi, C.; Ott, E.; Yorke, J. A., Phys. Rev. Lett., 57, 1284 (1986) |
| [9] | Grebogi, C.; McDonald, S. W.; Ott, E.; Yorke, J. A., Phys. Lett. A, 99, 415 (1983) |
| [10] | McDonald, S. W.; Grebogi, C.; Ott, E.; Yorke, J. A., Physica D, 17, 125 (1985) · Zbl 0588.58033 |
| [11] | Eckhardt, B., Physica D, 33, 89 (1988) · Zbl 0669.58050 |
| [12] | De Vogelaereand, R.; Bondart, M., Contribution to the theory of fast reaction rates, J. Chem. Phys., 23, 1236-1244 (1955) |
| [13] | Rankin, C. C.; Miller, W. H., Classical \(S\) matrix for linear reactive collisions of \(H+Cl_2\), J. Chem. Phys., 55, 3150-3156 (1971) |
| [14] | Gottdiener, I., The multiple-collision region in non-reactive atom-diatom collisions, Mol. Phys., 29, 1309-1316 (1973) |
| [15] | McGehee, R., Triple collision in the collinear three-body problem, Inventiones Math., 27, 191-227 (1974) · Zbl 0297.70011 |
| [16] | Agmon, N., Fine structure in the dependence of final conditions on initial conditions in classical collinear \(H_2+H\) dynamics, J. Chem. Phys., 76, 1300-1316 (1982) |
| [17] | Pollak, E.; Pechukas, P., Transition states, trapped trajectories and classical bond states embedded in the continuum, J. Chem. Phys., 69, 1218-1226 (1978) |
| [18] | Pollak, E.; Child, M. S., Classical mechanism of a collinear exchange reaction, J. Chem. Phys., 73, 4373-4380 (1980) |
| [19] | Pechukas, P., Transition state theory, Ann. Rev. Phys. Chem., 32, 1569-1577 (1981) |
| [20] | Pollak, E., Periodic Orbits and the Theory of Reactive Scattering, (Baer, M., Theory of Chemical Reaction Dynamics, vol. III (1987), CRC Publ: CRC Publ Boca Raton, FL) |
| [21] | Eckhardt, B.; Jung, C., Regular and irregular potential scattering, J. Phys. A:Math. Gen., 19, L829-L833 (1986) · Zbl 0618.34020 |
| [22] | Petit, J. M.; Henon, M., Satellite encounters, Icarus, 66, 536-555 (1986) |
| [23] | Spirig, F.; Waldvogel, J., The three-body problem with two small masses: A singular perturbation approach to the problem of Saturns coorbiting satellites, (Szebehely, B., Stability of the Solar System and its Minor Natural and Artificial Bodies (1985), Reidel: Reidel Dordrecht) |
| [24] | Aref, H., Integrable, chaotic and turbulent vortex motion in two dimensional flows, Ann. Rev. Fluid Mech., 15, 345-389 (1983) · Zbl 0562.76029 |
| [25] | Eckhardt, B.; Aref, H., Integrable and chaotic motion of four vortices II: Collision dynamics of vortex pairs, Phil. Trans. Roy. Soc. London (1988), to appear · Zbl 0661.76019 |
| [26] | Noid, D. W.; Gray, S. K.; Rice, S. A., Fractal behaviour in classical collisional energy transfer, J. Chem. Phys., 84, 2649-2652 (1986) |
| [27] | Eckhardt, B., Irregular scattering of vortex pairs, Europhys. Lett. (1988), to appear |
| [28] | Manakov, S. V.; Schur, L. N., Stochastic aspect of two-particle scattering, Sov. Phys. JETP, 37, 54-58 (1983) |
| [29] | (Model studies of one-dimensional double well and anharmonic potentials. Model studies of one-dimensional double well and anharmonic potentials, Theory, Chem. Phys. Lett., 84 (1981)), 144-150 |
| [30] | Jung, C., Poincaré map for scattering states, J. Phys. A:Math. Gen., 19, 1345-1353 (1986) · Zbl 0619.58025 |
| [31] | Eckhardt, B., Fractal properties of scattering singularities, J. Phys. A: Math. Gen., 20, 5971-5979 (1987) |
| [32] | Jung, C.; Scholz, H. J., Cantor set structures in the singularities of classical potential scattering, J. Phys. A: Math. Gen., 20, 3607-3617 (1987) · Zbl 0644.58036 |
| [33] | Jung, C., Can the integrability for Hamiltonian systems be decided by the knowledge of scattering data?, J. Phys. A: Math. Gen., 20, 1787-1791 (1987) |
| [34] | T. Tél, private communication (1988).; T. Tél, private communication (1988). |
| [35] | Gutzwiller, M. C., Physica D, 7, 341-355 (1983) |
| [36] | Blümel, R.; Smilansky, U., Phys. Rev. Lett., 60, 477 (1988) |
| [37] | Ziman, J. M., Principles of the Theory of Solids (1972), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0121.44801 |
| [38] | Goldstein, H., Classical Mechanics (1957), Addison-Wesley: Addison-Wesley Reading, MA |
| [39] | Devancy, R. L., An Introduction to Chaotical Dynamical Systems (1986), Benjamin-Cummings: Benjamin-Cummings New York · Zbl 0632.58005 |
| [40] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcation of vector fields, (Appl. Math. Sci., 42 (1986), Springer: Springer New York) · Zbl 0515.34001 |
| [41] | Grassberger, P., Information flow and maximum entropy measures for 1-D maps, Physics D, 14, 365-373 (1985) · Zbl 0584.94008 |
| [42] | Firth, W. J., Phys. Rev. Lett., 61, 329-332 (1988), in the context of optical memory |
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