Nonlinear resonant system

Fano, U., and Racah, G. (1959), Irreducible Tensorial Sets, Academic Press, N.Y. Farrelly, D. (1986), Lie Algebraic Approach to Quantization of Nonseparable Systems with Internal Nonlinear Resonance, J. Chem. Phys. 85, 2119. 

The Duffing Equation 14.4 seems to be a model in order to describe the nonlinear behavior of the resonant system. A better agreement between experimentally recorded and calculated phase portraits can be obtained by consideration of nonlinear effects of higher order in the dielectric properties and of nonlinear losses (e.g. [6], [7]). In order to construct the effective thermodynamic potential near the structural phase transition the phase portraits were recorded at different temperatures above and below the phase transition. The coefficients in the Duffing Equation 14.4 were derived by the fitted computer simulation. Figure 14.6 shows the effective thermodynamic potential of a TGS-crystal with the transition from a one minimum potential to a double-well potential. So the tools of the nonlinear dynamics provide a new approach to the study of structural phase transitions. 

In the resonance region the system oscillates with the external frequency X and with an increased amplitude (entrainment region). Far away from resonance, the internal free oscillations are present. This behaviour is completely absent in systems, where no self-sustained oscillations can exist. A typical example of such a system is a nonlinear conservative system. The resonance diagram has been drawn in Figure 2 for both, the small and the large ampli tude oscillation. 

Interaction with External Fields. The models considered exhibit cooperative behaviour through nonlinear internal oscillations (models 1, 2, 4) or through nonlinear resonances (model 3). This makes plausible the existence of effects, when the system is driven by weak external fields of appropriate frequency. 

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

To the contrary, for systems of N degrees of freedom with N >3, N-dimensional tori no longer separate the equi-energy surface. This implies that going around tori would be possible. One of the possible mechanisms for such behavior is movement along nonlinear resonances. 

In his celebrated article, Arnold actually constructed a model where these movements take place [15]. In this model, orbits move along a nonlinear resonance under the influence of other resonances. From the results of this study, it is found that the dynamics on the network of nonlinear resonances is characteristic for systems of N degrees of freedom with N >3. The network is called the Arnold web [16,17]. 

Nonlinear resonances are important factors in reaction processes of systems with many degrees of freedom. The contributions of Konishi and of Honjo and Kaneko discuss this problem. Konishi analyzes, by elaborate numerical calculations, the so-called Arnold diffusion, a slow movement along a single resonance under the influence of other resonances. Here, he casts doubt on the usage of the term diffusion. In other words, Arnold diffusion is a dynamics completely different from random behavior in fully chaotic regions where most of the invariant structures are lost. Hence, understanding Arnold diffusion is essential when we go beyond the conventional statistical theory of reaction dynamics. The contribution of Honjo and Kaneko discusses dynamics on the network of nonlinear resonances (i.e., the Arnold web), and stresses the importance of resonance intersections since they play the role of the hub there. 

Table III. Adlabatlcally obtained semlclasslcal resonance energies for the H-C-C system of Ref. (O. Results of BR (Borondo, Reinhardt (26)) are compared with EBK results of Hase (O. As discussed In the text, the adiabatic method quantizes the classical resonances, where, as Indicated by Hase was unable to obtain results due to nonlinear resonance.

Figure 1 Individual mode energies, Hn from Eq. (14) for the coupled Morse oscillator system in Eq. (13) as functions of time measured in harmonic vibrational periods. Each figure represents a single trajectory. The total energy in each trajectory is D, the dissociation energy of a single mode, (a) Localization of the excitation. The two modes are not in nonlinear resonance, (b) Quasi-periodic exchange of the excitation. The two modes are in nonlinear resonance.

For systems of more than two degrees of freedom, however, their role is not so obvious because their dimension is not large enough to work as barriers. This problem is closely related to the phenomenon of Arnold diffusion. In other words, dynamical processes along nonlinear resonances may create a way to go around these tori. In particular, intersections of resonances would play a dominant role in IVR since chaotic diffusion... 

In Figure 3.19, a section of the Arnold web of acetylene is displayed where nonlinear resonances up to seventh order are estimated. The value of the energy is chosen to be 18,797 cm. Since, for systems of more than three degrees of freedom, we cannot directly display the whole web, we have to take a section. Here, the section is defined by (uj = 0, V2, = 0, v, Vs)... 

See, for example, the following and references contained therein E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, /. Chem. Phys., 81, 1115 (1984). Intramolecular Vibrational Relaxation and Spectra of CH and CD Overtones in Benzene and Perdeuterobenzene. S. P. Neshyba and N. De Leon,. Chem. Phys., 86, 6295 (1987). Qassical Resonances, Fermi Resonances, and Canonical Transformations for Three Nonlinearly Coupled Oscillators. S. P. Neshyba and N. De Leon,. Chem. Phys., 91, 7772 (1989). Projection Operator Formalism for the Characterization of Molecular Eigenstates Application to a 3 4 R nant System. G. S. Ezra, ]. Chem. Phys., 104, 26 (1996). Periodic Orbit Analysis of Molecular Vibrational Spectra Spectral Patterns and Dynamical Bifurcations in Fermi Resonant Systems. Also see Ref. 6. 

Another typical example of the stochastic resonance system is the nonlinear bistable doublewell dynamic system, which describes the overdamped motion of a Brownian jjartide in a symmetric double-well potential in the presence of noise and periodic forcing as shown in... 

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