Phase portraits

Theory of Bifurcations.—In the preceding sections we have reviewed a few more important points of the existing topological methods in the theory of oscillation, assuming that the topological configuration or the phase portrait remains fixed. 

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

This particular bifurcation exists in the example from electronics mentioned above the regenerative amplifier has as its phase portrait SUS and after the bifurcation it becomes simply US, that is, an ordinary oscillator. 

Peculiar particle velocity, 19 Pendulum problem, 382 Periodicity conditions, 377 Perturbed solution, 344 Pessimism-optimism rule, 316 Petermann, A., 723 Peterson, W., 212 Phase plane, 323 "Phase portrait, 336 Phase space, 13 Photons, 547... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

Figure 3. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, iv = 0.52 (B) F = 20, iv = 1.04 and (C) F = 10 and u> = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potentiai (see text). 

The results of the quantum simulations for cases A, B and C are shown in the lower two panels in Fig. 3. The corresponding classical phase portraits shown reinforce our inferences from the stability diagram no stabilization for A while larger islands exist for C as compared with B. However, the ionized fraction as calculated from the quantum evolution supports the contrary result that there is more stabilization for A as compared with B. Case C is the most stable which is at least consistent with the classical prediction. What is the origin of this discrepancy ... 

Figures (d) and (e) show transients of C and T with different inlet temperatures. Figure (f) is a plot of T against C and is called a phase portrait.

In order to analyze both systems, some techniques from nonlinear science are burrowed. Firstly, a phase portrait is constructed from delay coordinates, a Poincare map is also computed, FFT is exploited to derive a Power Spectrum Density (PSD) Maximum Lyapunov Exponents (MLE) are also calculated from time series. Although we cannot claim chaos, the evidence in this chapter shows the possible chaotic behavior but, mostly important, it exhibits that the oscillatory behavior is intrinsically linked to the controlled systems. The procedures are briefly described before discuss each study case. 

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... 

Figure 2. Time evolution of intensity (a) and phase portrait for the fundamental mode (b). Solution of Eqs. (7) for the initial conditions = 0.1 + zO.l and CC20 = 0.01 + z 0.01. Quasi-periodic behavior.

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

A more complicated behavior of the MLE is observed for higher values of 7j. Varying the length of the pulse 7j, we observe regions of order and chaos. By way of an example, the phase portrait Reoti versus Imai for a chaotic attractor is shown in Fig. 15. 

Fig. 16a symmetric limit cycles for the second-harmonic mode (GCL) and in Fig. 16b, an nonsymmetric phase portrait example for 7) = 0.5 for BCL. In both cases the phase point settles down into a closed-loop trajectory, although not earlier than about x > 200. An intricate limit cycle is usually related to multiperiod oscillations. For example, the cycle in Fig. 16a corresponds to five-period oscillations of the fundamental and SHG modes intensity, and the phase portrait in Fig. 16b resembles the four-period oscillations (see Fig. 17). Generally, for 7) > 0.5, we observe many different multiperiod (even 12-period) oscillations in intensity and a rich variety of phase portraits.

The two largest Lyapunov exponents versus a duration of pulse 7] are presented in Fig. 18a for the cases of BCL. There are a two regions of hyperchaos. A Typical hyperchaotic phase portrait is presented in Fig. 18b. 

Figure 13. (a) A typical phase portrait in the short-pulse regime for GCL case (b) an... 

Figure 16. Phase portraits for the second-harmonic mode (a) symmetric example for GCL, (b) nonsymetric example for BCL. The parameters are the same as for Fig. 11, and the time is 200 < < 500.

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

As we move along any one of these stationary-state loci, varying rres, the number of solutions, limit cycles, and their relative orientations change, giving rise to corresponding changes in the phase portraits. It is difficult to be sure that we have ever completely counted the number of different phase portraits which occur even for a system as simple as this. Those which have been confirmed for this model (so far) are shown in Fig. 8.14. 

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