Attraction region

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

Because the Coulomb potential plays a central role in the model outlined above, it seems important to examine the full electrostatic potential experienced by an electron as it approaches the C-S bond region. In Fig. 10 we show the electrostatic potential for Me-S-Me molecule in the absence of any positive charges, with blue denoting attractive regions and red labeling repulsive regions. 

Molecules possessing distinct hydrophobic (water-repelling) and hydrophilic (water-attracting) regions. The terms surface-active agents , detergents , and amphiphiles are often used synonymously with the term surfactants . 

As a consequence of the attractive region of the interaction, several bound state solutions of the radial Schrodinger equation generally exist whose wavefunctions are localized in the well region. The bound state eigenenergies are negative, discrete and will be subscripted with the vibrational and rotational quantum numbers, v and the normalization... 

PDEs form a surface in the multitude of positive vectors. A multitude formed by the inequality G(c) < S is a certain vicinity of this surface narrowing towards it at 5 -> 0. At first, with sufficiently small e values, the solution of eqns. (131) behaves like a closed system. For a finite period of time it gets into a small vicinity of the PDE surface, but at the same time remains close to the solution of closed systems. In this vicinity motion is controlled by the substance exchange with the environment and under our assumptions it can be rather complicated. The solution, however, will never leave this area if is sufficiently small. Here we proceed from the suggestion that closed systems have no boundary equilibrium points. But if they do exist, then by opening a system they can be made stable. The area of their attraction region tends to zero at - 0. Hence, the presence of boundary points can also be a source of bifurcation when "opening a system. 

Let us now examine the behaviour of the solutions for the dynamic system (20) in time and analyze the system trajectories in the phase pattern. This analysis permits us to characterize peculiarities of the unsteady-state behaviour (in particular to establish whether the steady state is stable or unstable), to determine its type (focus, node, saddle, etc.) and to find attraction regions for stable steady states, singular lines, etc. 

Fig. 12. Qualitative peculiarities for the dependences of relaxation times t, and t3 on PB. (a) (xa, y0) eVj (b) (x, v0)cV2 (c) (x0Iy0)eV3. V, V2, and V3 are the attraction regions determined by separatrices of saddle-node points of various steady states.

Furthermore, the attractive region occurs only when the free polymer exceeds a certain minimum chain length. If the free polymer chain length is lower than this value, then the force is purely repulsive. For Ng = 101, pgoz = 0.1, and pjtt3 = 0.6, this critical value of Nf lies somewhere between 50 and 100, as shown in Figure 7. 

Figure 5 A steering dominated dissociation event. The purple shows a surface of constant wavepacket probability approaching a PES having very attractive regions into which the wavefunction is drawn (lower panel). The green surface shows die zero of potential.

High-resolution spectroscopic experiments provide a detailed experimental information on the shape of the intermolecular potential in the attractive regions. Recent improvements in supersonic beams and new laser techniques increased dramatically the sensitivity and resolution in the near-infrared region and opened to high-precision measurements the difficult far-infrared region. The latter development made it possible to investigate directly intermolecular vibration bands which are very sensitive probes of the shape of intermolecular potentials. The new spectroscopic techniques provide a lot of accurate data on interaction potentials for atom-molecule complexes, as well as on more complicated systems such as the HF, ammonia or water dimers. 

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