Trajectories never intersect

The existence and uniqueness theorem has an important corollary different trajectories never intersect. If two trajectories did intersect, then there would be two solutions starting from the same point (the crossing point), and this would violate the uniqueness part of the theorem. In more intuitive language, a trajectory can t move in two directions at once. 

Once the initial state x(f = 0) of the system is specified, future states, x(t), are uniquely defined for all times t . Moreover, the uniqueness theorem of the solutions of ordinary differential equations guarantees that trajectories originating from different initial points never intersect. 

Consider a trajectory S of a differential system issuing for t = tQ from a point close to the origin. It is dear that 8 will never intersect a surface F = Fc from inside to outside as dVfdt is negative. 

In the above definitions, 9 represents a set of parameters of the system, having constant values. These parameters are also called control parameters. The set of the system s variables forms a representation space called the phase space [32]. A point in the phase space represents a unique state of the dynamic system. Thus, the evolution of the system in time is represented by a curve in the phase space called trajectory or orbit for the flow or the map, respectively. The number of variables needed to describe the system s state, which is the number of initial conditions needed to determine a unique trajectory, is the dimension of the system. There are also dynamic systems that have infinite dimension. In these cases, the processes are usually described by differential equations with partial derivatives or time-delay differential equations, which can be considered as a set of infinite in number ordinary differential equations. The fundamental property of the phase space is that trajectories can never intersect themselves or each other. The phase space is a valuable tool in dynamic systems analysis since it is easier to analyze the properties of a dynamic system by determining... 

We claimed that different trajectories can never intersect. But in many phase portraits, different trajectories appear to intersect at a fixed point. Is there a contradiction here ... 

The second possibility is that the slope is irrational (Figure 8.6.6).Then the flow is said to be quasiperiodic. Every trajectory winds around endlessly on the torus, never intersecting itself and yet never quite closing. 

We adopt here the convention that the direction of a trajectory points-out from lower to higher temperatures. For example, on the segment representing the binary mixture the trajectory is oriented from the light to the heavy component. The shape of the residue curves depends on the relative volatility of components. It may be observed that in Fig. 9.2 all the residue curves have the same vertices as origin and terminus. More specifically, all the trajectories emanate from the benzene comer and terminate in the ethyl-benzene vertex. These species are the lowest and the highest boilers, respectively. The vertex of toluene is only an intermediate destination, where the residue curves enter from the benzene and leave to the ethyl-benzene. Note that the residue curves do never intersect each other. 

Figure 4.12 (a) PFR trajectories can never intersect each other, and (b) this phenomenon would imply that multiple rate vectors exist at the intersection point. 

Trajectories can never intersect themselves. Proof the same logic that was used in the first observation can be used to prove this. 

Figure 10. A representative reactive trajectory intersecting two Poincare sections Ea and Eg defined in both the regions of A and B through the reactive island at q = 0. Note that the passing through the interior of Iljj — IIa will never occur in that direct back reaction on Ej after passing through the interior of 11a, although it is depicted so for the sake of simplicity. [Reprinted with permission from A. M. Ozorio de Almeida, N. De Leon, M. A. Mehta, and C. C. Marston, Physica D 46, 265 (1990). Copyright 1990, Elsevier Science Publishers, North-Holland.]...

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