Hamiltonian chaos

Because of the prevalence of Hamiltonian chaos, trajectories may cross the dividing surface and than cross it back and forth a certain amount time, even an indefinite one. While this is impossible at the linear level, one has to resort to perturbation theory to overcome this drawback. 

The area of the hysteresis loop increases as Tr 1 [19]. The power a is the same as that of the 1 /fa noise. This is the case of nonstationary Hamiltonian chaos in the mixed phase space [20]. The power of the 1 // is greater than 1, a > 1. We will return this point in Section IV. 

Several of the exercises give a taste of the new phenomena that arise in areapreserving maps. To learn more about the fascinating world of Hamiltonian chaos, see the review articles by Jensen (1987) or Henon (1983), or the books by Tabor (1989) or Lichtenberg and Lieberman (1992). 

Zaslavsky, G.M. 1994. Renormalization group theory of anomalous transport in systems with Hamiltonian chaos. Chaos 4 25-33. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

We stress that the chaos identified here is not merely a formal result - even deep in the quantum regime, the Lyapunov exponent can be obtained from measurements on a real system. Quantum predictions of this type can be tested in the near future, e.g., in cavity QED and nanomechanics experiments (H. Mabuch et.al., 2002 2004). Experimentally, one would use the known measurement record to integrate the SME this provides the time evolution of the mean value of the position. From this fiducial trajectory, given the knowledge of the system Hamiltonian, the Lyapunov exponent can be obtained by following the procedure described above. It is important to keep in mind that these results form only a starting point for the further study of nonlinear quantum dynamics and its theoretical and experimental ramifications. 

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

A. M. Ozorio de Almeida, Hamiltonian Systems Chaos and Quantization, Cambridge University Press, Cambridge, 1988. 

Figure 1. Stability borders for the system Hamiltonian at fixed u = 0.001 and m - 0.4. The lower curve is the usual chaos border eo l/50o>y3 = (mo/u>o)1,1,4/50(a>m 3) / 2. For small wo this border approaches the usual static border eo 0.13. The magic mountain of stability is delimited from below by the stabilization border eo 12u)o/mo and from above by the destabilization border eo = (16f,/x)(u>o/mo)2 with L = ln[v/2eo/(ex)/(< >omo)]. The dashed lines eo = (em/ai)(uo/mo) are drawn at constant e (a) e = 0.0025 (6) e = 0.05 (c) e = 1. The border (3) below which the Kepler map description is valid is given, in the present case, with fixed m and u, by the line eo = 0.2(u>o/mo) (not drawn in the figure). The present picture is drawn at fixed oi and m. If instead we keep no fixed, then the system will always be stable in the region to the right of the dotted vertical line given by u>o">o = 3.

Heller, E.J. (1986). Qualitative properties of eigenfunctions of classically chaotic Hamiltonian systems, in Quantum Chaos and Statistical Nuclear Physics, ed. T.H. Seligman and H. Nishioka (Springer, Berlin). 

G. M. Zaslavsky, Physics of Chaos in Hamiltonian Systems, Imperial College Press, London, 1998. 

Recently, Wiggins et al. [15] provided a firm mathematical foundation of the robust persistence of the invariant of motion associated with the phase-space reaction coordinate in a sea of chaos. The central component in RIT that is, unstable periodic orbits, are naturally generalized in many DOFs systems in terms of so-called normally hyperbolic invariant manifold (NHIM). The fundamental theorem on NHIMs, denoted here by M, ensures [21,53] that NHIMs, if they exist, survive under arbitrary perturbation with the property that the stretching and contraction rates under the linearized dynamics transverse to jM dominate those tangent to M. Note that NHIM only requires that instability in either a forward or backward direction in time transverse to M is much stronger than those tangential directions of M, and hence the concept of NHIM can be applied to any class of continuous dynamical systems. In the case of the vicinity of saddles for Hamiltonian problems with many DOFs, the NHIM is expressed by a set of all (p, q) satisfying both q = p = Q and o(Jb) + En=i (Jb, b) = E, that is. 

On that system were exact TS discovered [39], the importance of mass mismatch between atoms A,B,C underlined and chaos in reactive scattering described [3, 29,40-42]. It must be underlined that studies in atomic physics and celestial dynamics were decisive in a definition of a TS, with less obvious Hamiltonians, see the chapter by Jaffe et al. in this book. 

Figure 16 shows how the amplitude d (a) of the Melnikov integral depends on the frequency co of the external force. Remember that the characteristic time scale of the unperturbed Hamiltonian is 1. Then, the fact that d o)) attains a maximum at co 1 implies that chaos is caused by resonance between the unperturbed system and the external force. 

The random matrix was first introduced by E. P. Wigner as a model to mimic unknown interactions in nuclei, and it has been studied to describe statistical natures of spectral fluctuations in quantum chaos systems [17]. Here, we introduce a random matrix system driven by a time-dependent external field E(t), which is considered as a model of highly excited atoms or molecules under an electromagnetic field. We write the Hamiltonian... 

Here we employ the quantum kicked rotor as a simple model of quanmm chaos systems. The Hamiltonian of a kicked rotor is written as... 

This equation is Schrodinger s wave equation, where h is Planck s constant and H is the Hamiltonian of the system to be investigated. The Schrodinger equation is a deterministic wave equation. This means that once ip t = 0) is given, ip t) can be calculated uniquely. Prom a conceptual point of view the situation is now completely analogous with classical mechanics, where chaos occurs in the deterministic equations of motion. If there is any deterministic quantum chaos, it must be found in the wave function ip. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

The above scheme suggests that all Hamiltonian systems are inte-grable, since all Hamiltonian systems seem to possess a maximal set k = 1,...,2/ of constants of the motion. Indeed, this is the impression conveyed by traditional textbooks on classical mechanics (see, e.g., Landau and Lifschitz (1970), Symon (1971), Goldstein (1976)). But if all Hamiltonian systems have a maximal set of constants of the motion, how can we reconcile this fact with the occurrence of chaos in most Hamiltonian systems ... 

The situation changes drastically, however, if we irradiate the surface state electrons with a sufficiently strong microwave field. Strong fields change the physics of the SSE system profoundly, eventually driving it into chaos. That chaos can indeed occur in the SSE system was first demonstrated by Jensen in 1982. The system he proposed is shown in Fig. 6.1(b). It is an extension of the system shown in Fig. 6.1(a). A microwave field is applied perpendicular to the helium surface S such that the field direction is parallel to the x direction. The resulting classical Hamiltonian of the combined SSE plus microwave field can be written as... 

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