Phase space unstable

A typical trajectory has nonzero values of both P and Q. It is part of neither the NHIM itself nor the NHIM s stable or unstable manifolds. As illustrated in Fig. la, these typical trajectories fall into four distinct classes. Some trajectories cross the barrier from the reactant side q < 0 to the product side q > 0 (reactive) or from the product side to the reactant side (backward reactive). Other trajectories approach the barrier from either the reactant or the product side but do not cross it. They return on the side from which they approached (nonreactive trajectories). The boundaries or separatrices between regions of reactive and nonreactive trajectories in phase space are formed by the stable and unstable manifolds of the NHIM. Thus once these manifolds are known, one can predict the fate of a trajectory that approaches the barrier with certainty, without having to follow the trajectory until it leaves the barrier region again. This predictive value of the invariant manifolds constitutes the power of the geometric approach to TST, and when we are discussing driven systems, we mainly strive to construct time-dependent analogues of these manifolds. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

The important word in this sentence is predict. It is important, in my opinion, to make a distinction between existence and predictability. Prigogine himself said (much later, in La Fin des Certitudes, LG.7) Every dynamical system must, of course, follow a trajectory, solution of its equations, independently of the fact that we may or may not construct it. Thus, a trajectory exists but cannot be predicted. The impossibility of prediction is therefore related to the impossibility of defining an instantaneous state (in the framework of classical mechanics) as a limit of a finite region of phase space (thus a limit of a result of a set of measurements). For an unstable system, such a region will be deformed and will end up covering almost all of phase space. The necessity of introducing statistical methods appears to me to be due to the practical (rather than theoretical) impossibility of determining a mathematical point as an initial condition. 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The idea of Pollicott-Ruelle resonances relies on this mechanism of spontaneous breaking of the time-reversal symmetry [20, 21]. The Polhcott-Ruelle resonances are generalized eigenvalues sj of LiouviUian operator associated with decaying eigenstates which are singular in the stable phase-space directions but smooth in the unstable ones ... 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

The single Kerr anharmonic oscillator has one more interesting feature. It is obvious that for Cj = 0 and y- = 0, the Kerr oscillator becomes a simple linear oscillator that in the case of a resonance 00, = (Do manifests a primitive instability in the phase space the phase point draws an expanding spiral. On adding the Kerr nonlinearity, the linear unstable system becomes highly chaotic. For example, putting A t = 200, (D (Dq 1, i = 0.1 and yj = 0, the spectrum of Lyapunov exponents for the first oscillator is 0.20,0, —0.20 1. However, the system does not remain chaotic if we add a small damping. For example, if yj = 0.05, then the spectrum of Lyapunov exponents has the form 0.00, 0.03, 0.12 1, which indicates a limit cycle. 

This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space [56]. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled... 

In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. All the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet... 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the unstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

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