Phase trajectories

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

Simple analysis of the motion of mapping point through the phase trajectory (Fig. 2.13) indicates that X represents the stationary concentration of chemisorbed radicals corresponding to a given external conditions. 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

In the case of a chemical clock, the asymptotic (f -A oo) solution depends on time, there are not only singular points but also singular trajectories. An example is the stable limit cycle - Fig. 2.4, i.e., a closed trajectory to which all phase trajectories existing in its vicinity strive. 

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane (iVa, iVb). It should be reminded that in its classical formulation the trajectory of the Lotka-Volterra model is a closed curve - Fig. 2.3. In Fig. 8.1 a change of the phase trajectories is presented for d = 3 when varying the diffusion parameter k. (For better understanding logarithms of concentrations are plotted there.)... 

The top two graphs depict the phase trajectories of y versus x a and of a ,41 versus xb for the first tank. The middle two graphs do the same for the second tank, while the bottom two plots describe the dynamics in tank 3 for the chosen initial conditions 2/3 = 1.3, and various 3 43(0) and x 7/3(0) values. ... 

Figure 6.36 gives a single plot with another set of altered heat and concentration initial values spelled out in detail in the figure s title line. Recall that all phase trajectories start at a small o mark and end at a small mark when r =Tend. 

Since rest points are particular cases of the phase trajectories < ( , k, < ) = c0, the above definitions of stability according to Lyapunov are also valid for them. A rest point is stable according to Lyapunov if, for any e > 0 there exists values of 3 > 0 such that after a deviation from this point within 3, the system remains close to it (within the value) for a long period of time. A rest point is asymptotically stable if it is stable according to Lyapunov and there exists values of S > 0 such that after the deviation from this point within 3 the system tends to approach it at t - oo. 

Phase trajectories extend far from the singular point. It is an unstable node... 

Closed trajectories around the whirl-type non-rough points cannot be mathematical models for sustained self-oscillations since there exists a wide range over which neither amplitude nor self-oscillation period depends on both initial conditions and system parameters. According to Andronov et al., the stable limit cycles are a mathematical model for self-oscillations. These are isolated closed-phase trajectories with inner and outer sides approached by spiral-shape phase trajectories. The literature lacks general approaches to finding limit cycles. 

For example, if the bulk concentration of oxygen is low (or simply lower than its steady-state value), the movement along the phase trajectory towards the increasing concentrations of surface oxygen is retarded by diffusion. As one goes towards the decreasing concentrations, the movement is... 

A plot of [X] against [Y] is called a phase trajectory diagram or phase portrait39. Nonlinear, but non-oscillatory reaction schemes have phase trajectories that are open curve segments,whose form depends on initial conditions and rate constants37. The phase... 

The behavior of complex dynamical systems can be analyzed and represented in a number of ways. Figure 1 represents one such approach, a constraint-response plot. A constraint, in this case [A], is any variable which the experimenter can control directly. A response, [X]ss in this case, is a measurable property of the system which depends upon the constraint values. The constraints are the external variables, e.g., the temperature of the bath surrounding the reactor or the reservoir concentrations, while the responses are the internal variables, e.g., the temperature or concentration of species in the reactor. The phase trajectory diagram of Fig. 4 is one type of response-response plot. Obviously, in a complex system, there will be several constraints and responses subject to independent (or coupled) variation. 

Fig. 4. Phase trajectory diagram for the Lotka mechanism of Table 2.

Fig. 2.14. They - x phase plane in the late diffusional stage of formation of two compound layers. The nodal lines 1 and 2 separate the phase plane into three regions. In regions I and HI the thickness of one of the layers increases, while that of the other decreases. In region II both layers grow simultaneously. The arrows at phase trajectories indicate the direction of variation of the layer thicknesses with increasing time.

Figure 2.4 presents the trajectories of the two concentrations in the phase plane, revealing the presence of the equilibrium manifold the phase trajectories starting from any initial condition (aq o,ay o) [0,1] x [0,1] approach the horizontal line a 2 = 0, followed by convergence towards the equilibrium point (0,0). 

Figure 2.4 Phase trajectories of the reacting system starting from different initial...

Figure 2.6 Phase trajectories of the reacting system with fci = fc2 = 1 s 1 from different initial conditions (xi,o,X2,o) e [0,1] x [0,1].

Figure 2.7 reveals the presence of the equilibrium manifold phase trajectories approach the Ti=T2 line and converge along this line to the equilibrium point T1=T2=Te = 273 K. 

If all phase trajectories by means of continuous deformation can be tranformed into one another, then the summation in Eq. (11) can be extended to all possible paths. However, if the phase space consists of topological non-isomorphic classes C, the summation in Eq. (11) extends only to paths from a given class and the problem of determination of the topological invariant arises. 

The complex flow within an impactor can be studied by using the concept of phase trajectory analysis where the paths of particles with different initial locations and velocities are determined. By analyzing these paths, conclusions can be drawn about a particle s fate as it travels through an impactor. Because in this analysis ideal streamline flow conditions are assumed (which actually may not be the case), phase trajectory analysis helps show how predictions from ideal assumptions may be modified by real-world conditions. A fairly simple case is chosen to illustrate the method. 

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