Isolated closed trajectory

The limit cycle O is an isolated closed trajectory having the property that dll other trajectories O in its neighborhood are certain spirals winding themselves onto C either for t - oo in which case C is called a stable limit cycle) or for t- - — ao (an unstable limit cycle). 

A limit cycle is an isolated closed trajectory. Isolated means that neighboring trajectories are not closed they spiral either toward or away from the limit cycle (Figure 7.0.1). 

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. 

It must be noted that all these closed trajectories are due to boundary effects. Therefore they are not of the same nature as the motion of the core of an isolated spiral seen in excitable media [12] or in oscillatory media [13]. In an infinite medium, with no boundaries, this latter motion would persist, whereas the motion we report here would not be present. In fact, the boundary conditions, which are here of the zero-flux type, create a mirror-image of the isolated spiral beyond the boundary. This image plays the role of a virtual spiral which interacts with the real one in a similar way as in a pair of spirals. Therefore, the motions reported here could be compared, in first approximation, to the motion of a pair of spirals [13-16]. 

Figures 4.44 and 4.45, best viewed in color, show a benign complication of the problem caused by the Lewis numbers. If, however, we reduce the Lewis number LeA further to 0.07, the system trajectories indicate periodic explosions of the underlying system throughout all time, and the trajectories do not converge to the steady state at all, even with what we thought to be proper feedback. The trajectory that these curves settle at is called a periodic attractor of the system in contradistinction to the earlier encountered point attractor of Figures 4.43 or 4.44, for example. A point attractor, or more accurately a fixed-point attractor, is a more commonly encountered steady state in chemical and biological engineering systems. It could be called a stationary nonequilibrium state to distinguish it from the stationary equilibrium states associated with closed or isolated batch processes.

Unimolecular reactants are energized by a variety of experimental techniques including collisional and chemical activation, internal conversion and intersystem crossing transitions between electronic states, and different photo-activation techniques, which include excitation of isolated resonance states for reactants with a low density of states (see also Sect. 3). Trajectory simulations usually begin with the preparation of an ensemble of trajectories, whose initial coordinates and momenta resemble — as close as possible — those realized in a particular experiment [20,329]. 

Theorem 6.5.1 (Nonlinear centers for conservative systems) Consider the system x = f(x), where x = (x,y)GR and f is continuously differentiable. Suppose there exists a conserved quantity F(x) and suppose that x is an isolated fixed point (i.e., there are no other fixed points in a small neighborhood surrounding X ). If X is a local minimum of E, then all trajectories sufficiently close to X are closed. 

Ideas behind the proof Since E is constant on trajectories, each trajectory is contained in some contour of E. Near a local maximum or minimum, the contours are closed. (We won t prove this, but Figure 6.5.3 should make it seem obvious.) The only remaining question is whether the trajectory actually goes all the way around the contour or whether it stops at a fixed point on the contour. But because we re assuming that x is an isolated fixed point, there cannot be any fixed points on contours sufficiently close to x. Hence all trajectories in a sufficiently small neighborhood of x are closed orbits, and therefore x is a center. ... 

As a result of the appearance of the new frequency /i at p = p2, the director motion becomes quasi-periodic characterized by the two frequencies /o and /i. This is illustrated in Fig. 9(a), where the trajectory of the director in the (nx,Uy) plane is plotted for p = 1.55 at some z inside the layer. This trajectory is not closed in the laboratory frame indicating quasi-periodicity of the director. In fact, the two independent motions, namely the precession (/o) and the nutation (/i), can be isolated by transforming to a frame that rotates with frequency /o. In the rotating frame, the director performs a simple periodic motion with frequency /i as is seen in Fig. 9(b) with the arrow indicating the sense of rotation for the case where the incident light is left circularly polarized. (The sense of rotation is always opposite to that of the underlying precession [43].) As is seen from Fig. 9(c,d) starting from initial conditions near the unstable UPl solution or the UPS one, the director eventually settles on the NUP solution, which is represented by a simple limit cycle. 

A single isolated molecule of mass tn would just travel forever on a linear trajectory with constant velocity v and kinetic energy E (kin) = / mv dv = 1/2 mv. When two molecules are close to one another, they interact with a potential such as shown in Fig. 4.4, which will here be denoted simply by (pot, inter). This potential may have... 

Gravity is a force of infinite range, and it is impossible for any pair of objects to be truly isolated and subject to a point mass central field. The closed form solution of the two-body problem thus represents an idealized orbit. The departures from this trajectory are treated by perturbation theory. The action of any additional mass in a system can... 

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