Rest point

To this end we note that the point = l5 i.e., the rest point at which the mass velocity of the gas is zero, moves from left to right, and x1 — xX increases. 

We consider the volume of gas for which the momentum and energy equations are written—the volume contained between the rest point and the shock wave x-y < x < X. 

Because the rest point was chosen as the second boundary of the volume, the momentum of the gas leaving the volume is zero. 

Fig. 1. Examples of w-limit sets, (a) Rest point (b) limit cycle (c) Lorenz attractor [projection on the (Cj, c3) plane, a = 10, r = 30, b = 8/3].

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. 

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

In what way is an unsteady kinetic model investigated to elucidate whether the rest point is locally stable In this case a combination of approaches, which can be called a "rite , is used ... 

The stability of the rest point for eqn. (73) depends on the roots of the characteristic equation. The rest point is asymptotically stable if the real parts of all the roots in eqn. (80) are negative. It is unstable if the real part of at least one of the roots is positive. In the case where some roots in eqn. (80) are purely imaginary and the rest of them have a negative real part, the stability cannot be established by using only linear approximations. In this case the rest point of eqn. (77) is stable but not asymptotic. 

The character of trajectories is illustrated in Fig. 2(c)where v is a straight line specified by the equation = C12/C22. In both cases the rest point is also called a stable node. 

The second Lyapunov method implies that one uses V values which have maxima at the rest point under study whose derivative [eqn. (98)] is not positive (V < 0) in the vicinity of this point and zero values are admitted only at this point. 

From the various versions of this method we will choose only one. Let V < 0 and, only at the rest point under study c, V - 0. Then let Vhave its minimum, V(c) = at the point c and for some e > Vmin the set specified by the inequality V(c0) < e is finite. Therefor any initial conditions c0 from this set the solution of eqn. (73) is c(t, k, c0) - c at t - oo. V(c) is called a Lyapunov function. The arbitrary function whose derivative is negative because of the system is called a Chetaev or sometimes a dissipative function. Physical examples are free energy, negative entropy, mechanical energy in systems with friction, etc. Studies of the dissipative functions can often provide useful information about a given system. A modern representation for the second Lyapunov method, including a method of Lyapunov vector functions, can be found in ref. 20. 

Any dynamic system becomes stable eventually and comes to the rest point, i.e. attains its equilibrium or steady state. For closed systems, a detailed equilibrium is achieved at this point. This is not so simple as it would seem, as substantiated by a principle of the thermodynamics of irreversible processes. At a point of detailed equilibrium not only does the substance concentration remain unchanged (dcjdt = 0), but also the rate of each direct reaction is balanced by that of its associated reverse counterpart... 

So far (Sect. 1) we have discussed only approaches to derive chemical kinetic equations for closed systems, i.e. those having no exchange with the environment. Now let us study their dynamic properties. For this purpose let us formulate the basic property of closed chemical systems expressed by the principle of detailed equilibrium a rest point for the closed system is a point of detailed equilibrium (PDE), i.e. at this point the rate of every step equals zero... 

Not a single steady-state point in kinetic equations cannot be asymptotically stable in Z) if it does not coincide with a point of G minimum. Indeed, let us denote this steady-state point as Na and assume that it is not the point of G minimum. Then in any vicinity of Na there exist points N for which G(N) < G(N0) (otherwise N0 would be a point of local minimum). But a solution of the kinetic equations whose initial values are such values of N, since G(N) < G(N0), at t - oo cannot tend to N0 G(N) can only diminish with time. Consequently, NQ is not an asymptotically stable rest point in D. In its vicinity in D there exists such N points that, coming from these points, solutions for kinetic equations do not tend to Na at t - oo. 

Closed systems. Here a rest point is always a PDE wherein the rate of every direct reaction is equal to that of the reverse reaction. 

This method works well for viscosities from 1 to 103 Pa-s. For somewhat higher viscosities, the torque on the spindle becomes excessive. For these viscosities, a crucible rotation speed is selected so that the spindle lag is near the maximum measurable value. The rotation is then stopped, and the spring loading of the spindle is then allowed to drive the spindle back to its rest point (zero torque). The elapsed time between two selected angles from the rest point is measured during the return. The viscosity is then measured from the following equation [4] ... 

To observe accurately the position of the beam, a scale micrometer microscope is used. In order to determine the rest point of the balance, the micrometer screw is adjusted until the balance beam is deflecting equally about a fixed point on the scale. Reading the amplitudes is necessary where measurements are carried out in high vacuum (less than 10-5 mm. Hg). For measurements involving an appreciable gas concentration, the system is damped and the beam assumes a stable position within a few minutes. Here the pointer is followed directly on the micrometer screw. 

A particularly important class of solutions are the constant ones, which are called steady states, rest points, or equilibrium points. In terms of (3.1), such a solution is a zero of f(y), that is, a vector >> 6 K" such that f(y ) = 0. In the terminology of dynamical systems, a rest point is an element peM such that Tr p,t) = p for all telR. Similarly, a periodic orbit is one that satisfies -K(p,t + T) = -ir(p, t) for all t and for some fixed number T. The corresponding solution of (3.1) will be a periodic function. 

If the omega limit set is particularly simple - a rest point or a periodic orbit - this gives information about the asymptotic behavior of the trajectory. An invariant set which is the omega limit set of a neighborhood of itself is called a (local) attractor. If (3.1) is two-dimensional then the following theorem is very useful, since it severely restricts the structure of possible attractors. 

Theorem (Poincare-Bendixson). //(3.1) is two-dimensional and if 7 (x) remains in a closed and bounded region of the plane without rest points, then either 7 (x) is a periodic orbit (and 7 (x) = a)(x)) or w(x) is a periodic orbit. 

Theorem. Let 7 (> o) be a positive semi-orbit of (3.1) which remains in a closed and bounded subset K of and suppose that K contains only a finite number of rest points. Then one of the following holds ... 

Figure 3.1 illustrates the possibilities. Additionally, if a two-dimensional system has a periodic orbit then it must have a rest point inside that orbit. These simple facts (and their generalizations) play an important role in the analysis presented here. 

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