Nontrivial Equilibrium Points

To investigate the existence of the nontrivial equilibrium points (i.e., a 0), we divide (6.43) by a and set the result to 0 which gives 

From (6.42), we know that the definite integral in (6.52) is positive for a 0 and 0 for = 0. Since the first term is linear in a with a negative slope, one concludes a a) 0 for 0 a 1, and there are no other nontrivial equilibrium points. A typical plot of amplitude equation, (6.40), for this case is shown in Fig. 6.4b. 

A schematic plot of the variation of the coefficient of friction for this case is shown in Fig. 6.5a. First, we notice that for small velocities satisfying 0 Q where cob = maximum of the friction curve), (6.49) 

Forfi cOb, from (6.49) the trivial solution is stableifc (—/ig r2)iiQri) 0 and it is unstable otherwise. In the case of stable trivial equilibrium point, once again all the coefficients of (6.51) are nonpositive, which imphes that no other solutions are possible. 

49) is not satisfied, bo is positive while the rest of the coefficients, b [given by (6.45)], remain less than or equal to 0  

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As we observed in Section IV it is frequently the case that the linearized approximation of, for example, Eq. (24), with/(t) = 0, exhibits an unstable equilibrium point x = 0 although one of the boundedness criteria in Section IV applies. In such a case a nontrivial solution x(t) of Eq. (24) will obviously be oscillatory. 

The averaged amplitude equations have two equilibrium points A trivial solution, that is, a = 0, corresponding to the steady-sliding equilibrium and a nontrivial solution (i.e., a hmit cycle) is given by... 

If the trivial equilibrium point is unstable, i.e., — c + /q > 0, according to (4.19) there is nontrivial solution a = ai (limit cycle in the original system s phase plane). The stability of this solution is assessed by evaluating... 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

It is convenient to introduce a dimensionless distance a = k/ in describing equilibrium behaviors in the neighborhood k 0) of the critical point. As we shall find out from the ensuing discussions, the convenience arises from the fact that k turns out to be the only nontrivial distance-reducing parameter for all terms occurring in the expansion of %(/ ) which do not contain or the higher terms such as In terms of this new variable x and a new dimensionless function defined by... 

Prom the practical point of view, stability in the sense of Lyapunov is less important than asymptotic stability. In particular, it follows from simple continuity arguments that if a critical equilibrium state is asymptotically stable, then the trajectories of any nearby system will also converge to a small neighborhood of the origin where they will stay forever. The behavior of trajectories in this small neighborhood may be rather nontrivial. Nevertheless, any deviations from zero of trajectories of a nearby system must remain small because the equilibrium state is asymptotically stable at the critical parameter value. 

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