Phase planes

This chapter is mainly about fixed points. The next two chapters will discuss closed orbits and bifurcations in two-dimensional systems. 

The general form of a vector f ield on the phase plane is 

Furthermore, the entire phase plane is filled with trajectories, since each point can play the role of an initial condition. 

Some of the most salient features of any phase portrait are  

The arrangement of trajectories near the fixed points and closed orbits. For example, the flow pattern near A and C is similar, and different from that near B. 

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The state variables are the smallest number of states that are required to describe the dynamic nature of the system, and it is not a necessary constraint that they are measurable. The manner in which the state variables change as a function of time may be thought of as a trajectory in n dimensional space, called the state-space. Two-dimensional state-space is sometimes referred to as the phase-plane when one state is the derivative of the other. 

Fig. 4.10 6 — phase-plane contours of equal energy see text. 

We see that the phase-plane is broken up into a sequence of fixed points and a series of both open and closed constant-energy curves. The origin (0= =0) and its periodic equivalents (0 27rn, = 0), are stable fixed points (or elliptic... 

Phase Plane Singular Points.—We shall define the plane of the variables (x,y = x) as the phase plane and investigate the behavior of integral curves (or characteristics) in that plane by means of Eq. (6-2). In case we wish to associate with these curves the motion of the representative point R(x,y), we shall rather speak of them as trajectories and in this case one has to use Eq. (6-1). 

We recall that in this terminology the center is the singular point (the state of rest) for simple harmonic motion represented in the phase plane by a circle (or by an ellipse). The trajectories in this case axe closed curves not having any tendency to approach the singular point (the center). 

As a closed trajectory in the phase plane means obviously a periodic phenomenon, the discovery of limit cycles was fundamental for the new theory of self-excited oscillations. 

In the old (Galileo) theory of oscillations the pattern of a periodic motion was assumed to be the closed trajectory around a center. As is well known a trajectory of this kind is determined by its initial conditions—a point (x0, y0) in the phase plane. If the initial conditions are changed, there will be another closed trajectory and so on. One has, thus, a continuous family of dosed trajectories, each of which can be realized by means of proper initial conditions. 

In some cases there also occur semistable limit cycles (in this discussion the single term cycle is used wherever it is unambiguous or if no confusion is to be feared) characterized by stability on one side and instability on the other side. Figure 6-5(a), (b), and (c) illustrate these definitions. Physically, only stable cycles are of interest the unstable cycles play the role of separating the zones of attraction of stable cycles in the case when there are several cycles. It is seen from this definition that, instead of an infinity of closed trajectories, we have now only one such trajectory determined by the differential equation itself and the initial conditions do not play any part. In fact, the term initial conditions means just one point (x0,y0) of the phase plane as a spiral trajectory O passes through that point and ultimately winds itself onto the cycle 0, it is clear that the initial conditions have nothing to do with this ultimate closed trajectory C—the stable [Pg.329]

Given Eq. (6-1), this criterion states if dPjdx + 8Q/8y does not change its sign (or vanishes identically) in a region D (of the phase plane), no dosed trajectory can exist in D. 

Nonanalytic Cycles.—In recent years the concept of limit cycles was enlarged so as to include cycles that have widespread application in connection with the description of oscillatory phenomena whose stationary states are not describable in the phase plane by a trajectory that is analytic. 

In the phase plane the trajectory of the torsional pendulum between... 

Instead of the ordinary (xjc) phase plane, Li6nard introduces the phase plane of the variables (x, y) where... 

The principle is to consider a differential equation as an operator which after some time (which will be assumed to be 2n) transfers the representative point JR from a point A of the phase plane to some other point B. This process can be written conventionally as... 

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 types of system behaviour predicted, by the above analysis are depicted in Figs. 3.16 and 3.17. The phase-plane plots of Fig. 3.17 give the relation of the dependant variables C and T. Detained explanation of phase-plane plots is given in control textbooks (e.g., Stephanopoulos, 1984). Linearisation of the reactor model equations is used in the simulation example, HOMPOLY. 

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 5.8. Phase-plane plot showing variation of R2 and Rg versus R. ...

Program THERM solves the dynamic model equations. The initial values of concentration and temperature in the reactor can be changed after each run using the ISIM interactive commands. The plot statement causes a composite phase-plane graph of concentration versus temperature to be drawn. Note that for comparison both programs should be used with the same parameter values. 

CSTR WITH EXOTHERMIC REACTION AND JACKET COOLING Dynamic solution for phase-plane plots Located steady-states with THERMPLO and use same parameters. 

Using THERMPLO, locate the steady states. With the same parameters, verify the steady states using THERM. To do this, change the initial conditions (A and TR) using VAL and GO. Plot as a phase-plane and also as concentration and temperature versus time. 

Choose a multiple steady-state case and try to upset the reactor by changing AO, F, TO or TJ interactively during a run. Only very small changes are required to cause the reactor to move to the other steady state. Plot as time and phase-plane graphs. 

Modify the program to generate a dimensionless, phase-plane display of BDIM versus ADIM, repeating the studies of Exercise 1. Note that the oscillatory behaviour tends to form stable limit cycles in which the average yield of B can be increased over steady-state operation. 

Figure 5.51. A phase-plane plot of dimensionless A versus dimensionless temperature for the conditions of Fig. 5.49.

Program REFRIG2 calculates the dynamic behaviour and generates a phase-plane plot for a range of reactor concentrations and temperatures. 

Figure 5.55. Plot of the concentration-temperature phase-plane, corresponding to the run in Fig. 5.54 using REFRIG2.

Show that in the absence of control and feed disturbances (u = v = 0), the system has a singular, stable, steady-state solution of C = 0.1654 and T = 550. This can best be done by carrying out runs with different initial conditions (CO and TEMPO) and plotting the results as a phase-plane, TEMP versus C. 

Figure 5.56. A phase-plane plot of oxcillations of C and TEMP without control for amplitude of feed temperature disturbance, A =1.

Figure 5.155. Phase-plane of liquid height in tank 1 versus flow rate. Note the reversal in the flow direction.

As noted by Walas, the solution exhibits random behaviour and the solution jumps erratically between two critical point regions, as shown very clearly by plotting the variables as a phase-plane plot. 

Program the equations in ISIM and study the resulting phase-plane response. 

Figure 5.259, The chaotic behaviour is Figure 5.260. The run of Fig. 5.259 is apparent. plotted as a phase-plane of XI versus X2.

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. 

Vary A, Al, B, and B1 individually to see the influence of each. Plot the variables versus time and as phase planes. 

The orbit described by the representative point in the phase plane (pq) is given by the relationship between q and p. From... 

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