The classical dynamics of an impulsively driven Morse oscillator

James Franklin Heagy

doi:10.17918/00000970

We investigate the classical dynamics of a one-dimensional Morse oscillator subjected to periodic impulsive (delta function) forcing. This system serves as a rough model for the interaction of diatomic molecule (or single bond of a polyatomic molecule) with an intense laser field. The dynamics of the system are shown to be governed by an area-preserving map (period 1 return map) of the flow generated by the underlying Hamiltonian. The properties and behavior of this map are the main focus of study. The map is shown to be reversible and we obtain an explicit factorization of the map into a product of two orientation-reversing involutions. The invariant sets (symmetry lines) of these involutions are then determined analytically. The so called dominant symmetry line is identified as one of the symmetry lines. Many of the structures within phase space are organized by the symmetry lines and their iterates under the map. Among these structures are the (Poincare-Birkhoff) periodic orbits, resonance islands, and (primary) homoclinic orbits. We classify all of the Poincare-Birkhoff orbits according to the symmetry lines which they visit. The resonance islands, which are 'built' around these orbits, are similarly organized. We also give a similar symmetry classification for the primary homoclinic orbits. We extend these observations by proving that the homoclinic orbits in all reversible area-preserving maps fall on symmetry lines. Bifurcation behavior is examined in light of the symmetry line structure of phase space. The generation (or destruction) of period one orbits with winding numbers n/1 (n [greater than or equal to] 1), via tangent bifurcations, is found to be governed by systematic rules. These rules relate to the crossing of two symmetry lines as the control parameters of the map (amplitude and period of the driving term) are varied. We generalize some of the symmetry observations to a larger class of 'kicked' Hamiltonian systems. We provide evidence that the dominant symmetry line for this class of systems is determined solely by the form of the coupling function to the periodic forcing term, and we give an expression for the dominant symmetry line. We conjecture that the dominant symmetry line for all maps within this class is given by this expression. Finally, we examine the dissociation behavior of the Morse oscillator as a function of the control parameters. This study results in a boundary in parameter space which separates parameters which lead to dissociation from those which do not. A series of numerical dissociation experiments carried out over successively smaller regions of parameter space reveals that this boundary has a considerable amount of structure, even at very fine scales, and may possibly be fractal.

The classical dynamics of an impulsively driven Morse oscillator

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The classical dynamics of an impulsively driven Morse oscillator

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