Phase space portraits

Figure 1. Phase-space portrait of the dynamics described by the linearized Hamiltonian in Eq. (5), projected onto (a) the reactive mode and (b) a bath mode.

Figure 13. The phase-space portrait of A -DOF saddle Hamiltonian.

Fig. 5.4(c) shows the phase-space portrait for Fl = 2 > Kc- Indeed, as expected, all the sealing lines are now broken and the angular momentum is firee to roam in the I direction. In this case we expect that the molecule is able to absorb energy out of the external field. This is verified by Fig. 5.5(b), which shows En for K = 2. In this case the energy grows linearly in time and without limit. 

In order to develop the criterion more quantitatively, consider the sequence of phase-space portraits shown in Figs. 5.4(a) - (d). This sequence suggests that, as the control parameter K increases, the diameter of the resonance islands at Z = 0 mod 27t grows in action. In order to predict the touching point of the resonances, we need the widths of the resonances as a function of K. The width of the resonances is derived on the basis of the Hamiltonian (5.2.1). Since the dynamics induced by H is equivalent to the chaotic mapping (5.1.6), the Hamiltonian H itself cannot be treated analytically and has to be simplified. One way is to consider only the average effect of the periodic 6 kicks in (5.2.1). The average perturbation... 

Fig. 6.3. Phase-space portrait of unperturbed surface state electrons.

The SSE phase-space portrait shown in Fig. 6.5 reminds us of the phase-space portraits of the kicked rotor presented in Chapter 5. In Fig. 6.5 we can identify resonances and sealing invariant curves. In Chapter 5 we saw that resonance overlap in the standard mapping defines a sudden percolation transition when for K > Kc the seahng invariant... 

Fig. 7.5. Phase-space portraits of the driven one-dimensional hydrogen atom for / = 9.92GHz and — (a) 330V/cm, (b) 350V/cm and (c) 380V/cm.

On the basis of the phase-space portraits presented in Fig. 7.5, we conclude that the ionization thresholds displayed in Fig. 7.3 do indeed correlate with the onset of chaos. The equation ionization thresholds = chaos thresholds is therefore justified. 

There is a Cantor set of trapped trajectories which show up in the deflection functions or in a phase space portrait of the scattering time delays or survival times. This is indicated in Fig. 8 showing the phase space structure at a held strength = 2, where there are no islands of stability. The initial conditions of those trajectories with exactly two zeros are marked black, the white regions in between correspond to trajectories with three or more zeros. A break-down of these regions according to the number of zeros reveals self-similar fractal structure [62]. 

In two-dimensional Hamiltonian systems, the trajectories can be visualized by means of the Poincare surface of section plot. It is also possible to study two-dimensional Hamiltonian systems using the two-dimensional symplectic mapping. A typical phase space portrait of generic nonhyperbolic phase space is... 

Figure 8. Typical episodes of the electrical activity of the human brain as recorded from the electroencephalogram (EEG) time series together with the corresponding phase space portraits. The portraits are two-dimensional projections of the actual attractors. (From Babloyantz and Destexhe [43] with permission.)...

Here e, a, and b are real, positive parameters, e is chosen small in order to guarantee a clear timescale separation between the the fast x-variable (activator) and the slow y-variable (inhibitor). The variables a and b determine the position of the so-called nullclines, the two functions y x) that are determined by setting time derivatives dx/dt = 0 and dy/dt = 0. Depending on the parameters the FHN system has different dynamical regimes. Fig. 1.2 shows phase space portraits together with the nullclines and timeseries for three qualitatively different cases. 

The main difference between the Hamiltonian and dissipative systems arises from the conservation condition that applies to the former. In Hamiltonian systems, the total energy is fixed. A trajectory with a given initial condition and energy will continue with that same energy for the remainder of the trajectory. In the phase space representation, this will result in a stable trajectory that does not pull in toward an attractor. A periodic trajectory in a Hamiltonian system will have an amplitude and position in the phase space that is determined by the initial conditions. In fact, the phase space representation of a Hamiltonian system often includes many choices of initial conditions in the same phase space portrait. The Poincare section, to be described below, likewise contains many choices of initial conditions in one diagram. 

In dissipative systems, many trajectories with different choices ot initial conditions will be attracted to the same region in phase space and end up, asymptotically, on the same attractor. The phase space portrait, then, will usually consist only of the asymptotic state, that is, a trajectory that represents the final state for many different initial conditions. This asymptotic trajectory traces over the attractor and reveals its shape. Figure 22(a) shows a steady state attractor, whereas Figure 22(b) shows a limit cycle attractor. 

A Poincare surface of section is defined for dissipative systems in the same way as for Hamiltonian ones but, again, will look somewhat different because the phase space trajeaory consists only of the asymptotic state, a single attractor. To construct the Poincare seaion, the phase space portrait is cut with a surface to create a cross-sectional view of the attractor. Hence, for a simple limit cycle attractor which is a single loop in phase space, the Poincare section consists of a single point. For a more complex attractor, the Poincare section will be more elaborate, as we will see. 

One way to see the effect of the reduction in dimensionality that occurs as one goes from time series to phase space portrait to Poincare section is to consider the stability of a system as it is reflected in the stability of points in the Poincare section. The single point, which corresponds to the Poincare section of a simple limit cycle, can be treated in the same way an equilibrium point (or steady state point) is treated, even though, here, we are considering the stability of the periodic state, that is, the limit cycle. A small perturbation added to this point in the cross section will be found to decay back toward the point itself, if the limit cycle is stable, or to evolve away from the point, if the cycle is unstable. The stability properties of the point in the Poincare section are the same as the stability properties of the limit cycle to which it corresponds. Hence, a stable point in the section means that the limit cycle is stable, and an unstable one means that the limit cycle is unstable. And, furthermore, any bifurcations which occur for the point in the cross section also correspond to bifurcations which the limit cycle undergoes as a parameter is varied. 

So, the time-delay reconstruction technique yields a phase space portrait that is equivalent to that defined in the usual way (variables and their time derivatives). It is now considered standard to treat the time-delay reconstructed phase portrait as the actual phase space portrait. 

Because the Grassberger-Procaccia algorithm is the most widely used for the calculation of the correlation dimension, we briefly review it here. As discussed above, a phase space portrait must first be reconstructed from the measured data. The resulting attractor consists of a set of ordered points, uj, U2,.. ., U ,.. . , where u,- = x(tj), y(tj), z(tj)) is the fth point in the trajectory from which the attractor has been reconstructed. The correlation dimension is calculated by comparing the distance between any two points on the attractor (see Figure 38) with a given small distance, e, which will be varied. The number of pairs of points, (u u,), that both fall inside a ball of diameter e are counted. The correlation sum C(e) is the average number of point pairs inside balls of diameter e spread over the attractor... 

It is also able to draw complex mathematical figures, including many fractals. Dynamics Solver is a powerful tool for studying differential equations, (eontinuous and diserete) nonlinear dynamieal systems, deterministic chaos, mechanics, and so forth. For instance, you can draw phase space portraits (including an optional direction field). Poincare maps, Liapunov exponents, histograms, bifurcation diagrams, attraction basis, and so forth. The results can be watehed (in perspective or not) from any direction and particular subspaces ean be analyzed. 

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