Phase plane diagram

Vary the value of A from -1.0 to +1.0 and examine the changing stability of the system as shown on a phase-plane diagram. 

Make suitable changes in the initial conditions of X and S, and plot the phase plane diagram X versus S. By making many runs at a range of initial conditions, the washout region can be identified. 

Figure 4. Oscillation and phase-plane diagrams (see Figure 1) of the externally driven limit cycles for 4 different values of the external field strength F and for a fixed external frequency x. with < o>, w2 (F, < F2 < Fs < F(j.

Figure 5. v(t) — t (amplitude as a junction of time) and v(t) — 8(%) (phase-plane) diagrams of the coherent oscillation model (Model 2) with F0 == 0 = A. The computer plot shows a typical limit cycle oscillation. 

Figure 7. Oscillation and phase-plane diagrams of the externally driven limit cycle of the coherent oscillation model.

The addition of an appropriate amount of substrate at the right phase thus allows the system to switch reversibly from one periodic regime to the other. The same transitions can of course occur in response to an addition of product, as suggested by the phase plane diagram of fig. 3.7. 

Figure 6.6 Phase-plane diagram (Ca — T) for a CSTR exhibiting steady-state multiplicity.

Figure 6.13. Bistability for growth limitation of an inhibiting substrate shown in a phase-plane diagram. The attraction domains of the two steady states are touched by a separatrix.

Fig. 7.7 Phase plane diagram in different lime regimes, (a) Approaching to the phantom attractor , (b), (c) Intermediate regime, (d) Near the true attractor.

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

Figure 8.8(a) shows a typical example of the phase plane for a system with three stationary solutions, chosen such that there are two stable states and the middle saddle point. The trajectories drawn on to the diagram indicate the direction in which the concentrations will vary from a given starting point. In some cases this movement is towards the state of no conversion (ass = 1, j8ss = 0), in others towards the stable non-zero solution. Only two trajectories approach the saddle point these divide the plane into two and separate those initial conditions which move to one stable state from those which move to the other. These two special trajectories are known as the separatrices of the saddle point. 

Uppal, Ray, and Poore did a further study taking residence time as the distinguished parameter10 and the fine structure of the diagrams was explored by other workers11 who added many more varieties of phase plane. It should... 

This is called Hopf bifurcation. Figure 10 (A-2) shows two Hopf bifurcation points with a branch of stable limit cycles connecting them. Figure 13 (A-2) shows a schematic diagram of the phase plane for this case when g = g. In this case a stable limit cycle surrounds an unstable focus and the behavior of the typical trajectories are as shown. Figure 11 (A-2) shows two Hopf bifurcation points in addition to a periodic limit point (PLP) and a branch of unstable limit cycles in addition to the stable limit cycles branch. 

Figure 1. X(t) — t (amplitude as a function of time) and Xftj — X(tj (phase-plane) computer diagram of the generalized Van der Pol oscillator with F0 — 0 = X. for 4 different values of initial conditions. Both the small and large amplitude oscillations are shown.

The phase plane plot of Figure 2 illustrates the behavior of the concentrations of X and Y within this region. Initial concentrations of X and Y corresponding to a point above the broken line will evolve in time to the limit cycle. The broken line represents the separatrix of the middle unstable steady state which has the stability characteristics of a saddle point. Initial values for X and Y corresponding to a point below the separatrix will evolve to the stable state to the right of the diagram. 

The principles of a.c. circuits necessary for the comprehension of some of the ideas and concepts presented here are given in Appendix 2. The impedance is the proportionality factor between potential and current if these have different phases then we can divide the impedance into a resistive part, R, where the voltage and current are in phase, and a reactive part, Xc = l (oCy where the phase difference between current and voltage is 90°. As shown in Appendix 2, it is often easier for posterior calculation and analysis to display the impedance vectorially in complex-plane diagrams. 

It should be pointed out that the wave vectors need not reside in the same plane. The BOXCARS phase-matching diagram could be folded along the dotted line in Fig. 3.6-12. With this folded BOXCARS arrangement (Shirley et al., 1980) a great advantage... 

A thorough analysis of chaotic oscillations for the NH3/O2 reaction over Pt has been performed by Sheintuch and Schmidt (2//). Bifurcation diagrams were presented in great detail, as well as phase plane maps and Fourier spectra. Figure 18 shows a series of oscillation traces obtained for various oxygen concentrations. By extracting a next return map from trace g in Fig. 18, evidence for intermittency could be obtained. 

Fig. 3.7. Birhythmicity. In the phase plane, two stable limit cycles (solid lines) are separated by an unstable limit cycle (dotted line). The situation is that of the bifurcation diagram of fig. 3.6e, with v = 0.255 s, i.e. (qvlk ) = 4.25. Vertical arrows indicate how to switch from one stable cycle to the other, by adding substrate (Moran Goldbeter, 1984).

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