Periodic orbits bounded systems

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

The periodic-orbit contribution derived by Gutzwiller is general and applies to different kinds of periodic orbits. However, the applicability of (2.13) rests on the property that the periodic orbits are isolated, that is, they do not belong to a continuous family. This is the case in hyperbolic dynamical systems where all the periodic orbits are linearly unstable. We should emphasize that the Gutzwiller trace formula may apply both to bounded and scattering systems. 

Heller, E.J. (1984). Bound-state eigenfunctions of classically chaotic Hamiltonian systems Scars of periodic orbits, Phys. Rev. Lett. 53, 1515-1518. 

One of the phenomena that is unique in systems with many degrees of freedom is that the state of the system can undergo large changes regardless of the existence of invariant sets that are remnants of periodic/quasi-periodic orbits of the unperturbed system [1], Arnold showed this phenomenon for a specific model (shown in Section IE) and calculated the upper bound of the rate of change in the action variable by linear perturbation theory. This phenomenon is sometimes called Arnold diffusion [2]. 

We have also used the periodic reduction method to predict with good accuracy the 3D structure of vibrationally bonded molecules. It should be stressed though, that in principle it is not necessary to use periodic reduction. As shown in Fig. 9 the RPO s of the IHI system are stable also in 3D, one can find bound quasi-periodic orbits and quantize them semiclassically directly without resorting to periodic reduction. 

The stability that we mentioned before refers to the evolution of the deviation vector ( ) = x (t) — x(l) between the perturbed solution x (t) and the periodic orbit x(t) at the same time t. If ( ) is bounded, then the periodic orbit is stable. In this case two particles, one on the periodic orbit x t) and the other on the perturbed orbit x (t), that start close to each other at t = 0, would always stay close. A necessary condition is that all the eigenvalues of the monodromy matrix be on the unit circle in the complex plane. However, in a Hamiltonian system this condition is not enough for stability, because there is only one eigenvector corresponding to the double unit eigenvalue and consequently a secular term always appears in the general solution, as can be seen from Equation (49). We remark that this secular term appears if the vector of initial deviation (0) = a (0) — s(0) has a component along the direction /2(C)). 

Once the molecular orbitals Eq. (4) are defined, much of the schematic code design is the same whether the systems are periodic or bounded clusters. The KS orbitals cp,-,r(r) are expanded as... 

As a consequence of energy conservation, any bounded individual trajectory of this system will be a periodic orbit. Consider the propagation of a small disk of initial conditions,... 

Since the contraction in the local map can be made arbitrarily strong and the derivative of the global map is bounded, the superposition T = To oTi inherits the contraction of the local map for all small p as well. It then follows from the Banach principle of contracting mappings (Sec. 3,15) that the map T has a unique stable fixed point on So- As this is a map defined along the trajectories of the system, it follows that the system has a stable periodic orbit in V which attracts all trajectories in V. The period of this orbit is the sum of two times the dwelling time t of local transition from Sq to S and the flight time from Si to Sq. The latter is always finite for all small p. It now follows from (12.1.4) that the period of the stable periodic orbit increases asymptotically of order tt/x/a This completes the proof. 

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

It is easy to see, that if C remains bounded away from C = I, then system (13.6.28) has a unique solution for small /i and e. Observe that the symmetry of the system implies yi = 1/2 in this case, which means that the map T cannot have orbits of period two when C is not close to 1. 

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