Poincare period

Conclusions, (i) For irreversible emission to occur it is necessary that the frequencies kn of the bath oscillators are dense.9 0 If there had been only a few oscillators the energy would shuttle back and forth between them and the main oscillator. The return time (or Poincare period ) would be of the order of the reciprocal of the distance between the kn. 

C3.6.1(a )), from right to left. Suppose that at time the trajectory intersects this Poincare surface at a point (c (tg), C3 (Sq)), at time it makes its next or so-called first reium to the surface at point (c (tj), c 3 (t )). This process continues for times t, .. the difference being the period of the th first-return trajectory segment. The... 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

This relationship was derived by Poincare and defines the range of frequencies, where the earth or any planet is not broken. The remarkable feature of this inequality is the fact that it is independent of the dimensions of the planet, and only the density defines the maximal permissible frequency. Introducing the period T, we represent Equation (2.102) as... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

In order to identify the periodic orbits (POs) of the problem, we need to extract the periodic points (or fixed points) from the Poincare map. Adopting the energy F = 0.65 eV, Fig. 31 displays the periodic points associated with some representative POs of the mapped two-state system. The properties of the orbits are collected in Table VI. The orbits are labeled by a Roman numeral that indicates how often trajectory intersects the surfaces of section during a cycle of the periodic orbit. For example, the two orbits that intersect only a single time are labeled la and lb and are referred to as orbits of period 1. The corresponding periodic points are located on the p = 0 axis at x = 3.330 and x = —2.725, respectively. Generally speaking, most of the short POs are stable and located in... 

Figure 31. Periodic points of the Poincare map at the energy E = 0.65 eV. The Roman numerals indicate how often the corresponding orbit intersects the surface of section. Panel (b) shows an enlargement of the main regular island around x,p) = (3.3,0). The thin hnes represent various tori of the system.

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Figure 5 shows the 3-dimensional reconstructed attractors and their projections on canonical planes. The reconstructed phase portraits do not exhibit a defined structure, i.e., it is not toroidal or periodic. As matter of fact, the oscillatory structure is only observed in the Poincare map. The Poincare map is often used to observe the oscillatory structure in dynamical systems. The... 

Fig. 6. Poincare maps. The section was chosen I (z) = zs = 0 and the crosses indicate no periodic oscillation. Once again, the smallest attractor corresponds to experiment E2.b. zi,Z2,zs are also dimensionless.

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

A technique for distinguishing between phase-locked and quasi-periodic responses, and which is particularly useful when m and n are large numbers, is that of the stroboscopic map. This is essentially a special case of the Poincare map discussed in the appendix of chapter 5. Instead of taking the whole time series 0p(r), for all t, we ask only for the value of this concentration at the end of each forcing period. Thus at times t = 2kn/a>, with k = 1, 2, etc., we measure the surface concentrations of one of our species. If the system is phase locked on to a closed path with a>/a>0 = m/n, then the stroboscopic map will show the measured values moving in a sequence between m points, as in Fig. 13.12(a). If the system is quasi-periodic, the iterates of 0p will never repeat and, eventually, will draw out a closed cycle (Fig. 13.12(b)) in the... 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

Such Hamiltonian mappings are generated by a Poincare surface of section transverse to the orbits of the flow. Thus, v(q) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. 

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

The cases of hyperbolic-without-reflection and hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincare section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration. 

In particular, the periodic orbits are in correspondence with finite sequences such as )W2 Up of period p. The periodicity occurring in the symbol sequences translates into the periodicity of the corresponding trajectory crossing the Poincare surface of section. 

Figure 15. Three-branch Smale horseshoe in the 2F collinear model of Hgl2 dissociation at the energy E = 600 cm 1 above the saddle in a planar Poincare surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

The phase plane has to give place to the three-dimensional state-space when there are three dimensions. If we have a stable periodic solution such as is shown in Fig. 27, all trajectories in its neighborhood tend toward it, but this is harder to show in three dimensions than it is in two and showing that x(T + t) = x(t) when you do not know T is virtually impossible. Poincare solved this difficulty by setting up a plane transversal to the limit cycle, as in Fig. 28. The limit cycle penetrates this surface at P0 and, if a nearby trajectory penetrates it at a succession of points that, as shown, converge on P0, then... 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

Figure 14. The basins of attraction of the SC (shaded) and CA (white) for a Poincare cross section with

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

Period doubling occurs at Dpo, as shown in the Poincare diagram in Figure 7.31(A) and the period versus D diagram in Figure 7.31(B). 

Fig. 7.8 Poincare sections after 2000 cycles. Initially nine marker points were placed along the y axis and six along the x axis. The dimensionless amplitude was 0.5, as in Fig. 7.7. The parameter was the dimensionless period (a) 0.05 (h) 0.10 (c) 0.125 (d) 0.15 (e) 0.20 (f) 0.35 (g) 0.50 (h) 1.0 (i) 1.5. For the smallest values of the time period we see that the virtual marker points fall on smooth curves. The general shape of these curves would he the streamlines of two fixed continuously operating agitators. As the time period increases the virtual marker particles fall erratically and the regions indicate chaotic flow. With increasing time periods larger and larger areas become chaotic. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).]...

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