The method of point-to-point mappings has been receiving increasing attention in recent years. In this paper we discuss instead dynamical systems governed by cell-to-cell mappings. The justifications of considering such mappings come from the unavoidable accuracy limitations of both physical measurements and numerical evaluation. Because of these limitations one is not really able to treat a state variable as a continuum of points but rather only as a collection of very small intervals. The introduction of the idea of cell-to-cell mappings has led to an algorithm which is found to be potentially a very powerful tool for global analysis of dynamical systems. In this paper an introductory theory of cell-to-cell mappings is offered. The theory provides a basis for the algorithm presented in [14]. In the first half of the paper we discuss the analysis of cell-to-cell mappings in their own right. In the second half the cell-to-cell mappings which are obtained from point-to-point mappings by discretization are examined in order to see what properties of the point mapping systems are preserved in the discretization process.

A dynamical system having multiple degrees of freedom and under parametric excitation is governed by a system of ordinary differential equations with periodic coefficients. In this paper a first-approximation analysis is carried out and criteria for instability are derived explicitly.

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When a chain molecule can be viewed as a collection of overlapping spherical groups. the effect of a solvent on the conformational structure of the chain molecule is described by the distribution function for the cavity particles associated with the spherical groups. This article discusses the calculation of the cavity distribution function for n -butane disolved in various apolar solvents: the liquids carbon tetrachloride. n· butane. and n -hexane. We consider the common picture where the CH 3 and CH 2 groups in n ·butane are simple spheres. For that model. the cavity distribution function is a four·point correlation function. We find that the superposition approximation for the four·point function. while qualitatively correct. overestimates the effects of the solvent. An alternative scheme. which is called the two cavity model. yields results that agree quantitatively with a computer simulation study of liquid n ·butane. We find that a solvent medium produces significant shifts in the conformational equilibrium of n ·butane from that found in the gas phase. This phenomenon is the result of the nature of the local packing of solvent molecules neighboring the solute species under investigation. The conformational equilibrium is sensitive to this packing. The bulk packing fractions (molecular density times the volume of the molecule) of the liquids CCl 4 and C 4 H lO are nearly identical. Even so there are noticeable differences between the intramolecular structure of n ·butane in liquid carbon tetrachloride and in the neat liquid. Previous ideas on conformational equilibria have ignored the importance of steric (i.e .• liquid packing) effects, and have assumed that solvent shifts in conformational structures can be attributed entirely to dielectric effects. Our calculations show that this assumption is wrong. The n ·butane molecule contains no significantly polar groups, yet solvent media produce substantial shifts. For example, the trans-gauche equilibrium constant, xglx" for n·butane in the gas phase at room temperature is roughly 0.5, while we find it is about 1.0 when n -butane is dissolved in liquid CCl 4 at the same temperature and 1 atm pressure. We discuss why the phenomenon has been overlooked, and suggest experiments to document its existence.

Interaction between molecular impurities trapped in rare gas crystals. I. Effective potential calculations Molecular dynamics technique has been used to study the effect of the interaction potential on crystal nucleation and the symmetry of the nucleated phase. Four systems, namely rubidium, Lennard-Jones, rubidium-truncated, and Lennard-Jones-truncated, have been studied each at reduced densitx 0.95. Two types of calculations were performed. Firstly, starting from a liquid state, each system was quenched rapidly to a reduced temperature of -0.1. The nucleation process for these systems was monitored by studying the time dependence of temperature and the pair correlation function, and the resulting crystalline structure analyzed using among other properties the Voronoi polyhedra. Only in the case of rubidium was a b.c.c. structure nucleated. In the other three cases we obtained f.c.c. ordering. Secondly, we have studied the effect of changing the interaction potential in a system which has already achieved an ordered state under the action of some other potential. After establishing a b.c.c. structure in a rubidium system, the change in the symmetry of the system was studied when the pair potential was modified to one of the other three forms. The results from both types of calculations are consistent: the rubidium potential leads to a b.c.c. structure while the other three potentials give an f.c.c. structure. Metastable disordered structures were not obtained in any of the calculations. However, the time elapse between the moment when the system is quick-quenched and the moment when nucleation occurs appears to depend upon the potential of interaction. 4974

A new method is offered here for global analysis of nonlinear dynamical systems. It is based upon the idea of constructing the associated cell-to-cell mappings for dynamical systems governed by point mappings or governed by ordinary differential equations. The method uses an algorithm which allows us to determine in a very effective manner the equilibrium states, periodic motions and their domains of attraction when they are asymptotically stable. The theoretic base and the detail of the method are discussed in the paper and the great potential of the method is demonstrated by several examples of application.

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A class of systems of coupled Hill’s equations is studied. It is shown that for systems of equations in this class, completely decoupled Hill’s equations may be obtained and the stability of the solutions may be studied in a very much simpler manner. Several physically significant problems are treated to demonstrate the nature of the method.