Liapunov stable

Finally we comment briefly on weakly elastic fluids. (See [26] and section 3.2.) We assume that the given Newtonian solution v satisfies ] v i < (ciRe), where Ci is some constant depending only on the domain of the flow this condition ensures that v is asymptotically Liapunov stable. (See e.g., [67].) Then, the viscoelastic solution (v ,Tt) close to (v, 0) is linearly (asymptotically) stable for c > 0 small enough. 

In a recent paper [70] Renardy has investigated the nonlinear stability of flows of Jeffreys-type fluids at low Weissenberg numbers. More precisely, assuming the existence of a steady flow (v, r), he proves that this flow is linearly and Liapunov stable provided the spectrum of the linearized operator lies entirely in the open plane 3 A < 0 and that the following quantity is sufficiently small... 

However, in practice the two types of stability often occur together. If a fixed point is both Liapunov stable and attracting, we ll call it stable, or sometimes asymptotically stable. 

Finally, x is unstable in Figure 5.1.5e, because it is neither attracting nor Liapunov stable. 

A graphical convention we ll use open dots to denote unstable fixed points, and solid black dots to denote Liapunov stable fixed points. This convention is consistent with that used in previous chapters. 

Attracting and Liapunov stable) Here are the official definitions of the various types of stability. Consider a fixed point x of a system x = f(x). 

In contrast, Liapunov stability requires that nearby trajectories remain close for all time. We say that x is Liapunov stable if for each e > 0, there is a S > 0 such that II x(z) - X II < e whenever r > 0 and x(0) - x < S. Thus, trajectories that start within 5 of x remain within e of x for all positive time (Figure 1). 

Finally, x is asymptotically stable if it is both attracting and Liapunov stable. 

Consider the system r = r(1 - r ), 6 = 1 - cos d, where r, 6 represent polar coordinates. Sketch the phase portrait and thereby show that the fixed point r = 1, d 0 is attracting but not Liapunov stable. 

Until further notice we use stable for what is technically called asymptotically stable in the sense of Lyapunov see, e.g., J. La Salle and S. Lefshetz, Stability by Liapunov s Direct Method (Academic Press, New York 1961). 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

Hence E<0, with equality only if e = 0. Therefore is a Liapunov function, and so e = 0 is globally asymptotically stable. ... 

Suppose that f has a stable p-cycle containing the point. Show that the Liapunov exponent A < 0. If the cycle is superstable, show that A = —oo. ... 

The question of stable operation of ordinary chemical reactors apparently originated with van Heerden (H6). Analyses of the stability question for such systems were given by Bilous and Amundson (B8) and Aris and Amundson (A2). Analysis depends on a purely mathematical result obtained by Liapunov (L3). This result is known as Liapunov s theorem, or the stability theorem, and a proof may be found in Davis book (Dl). A more advanced treatment of the stability problem is given by Hahn (HI). The analysis given below is essentially that of Spicer (S5). 

FfgHK 10 Critkality as a tangency condition between integral curves (0,y) and a target curve TC derived from a Liapunov Junction or arbitrary inflection criterion curve (a) satisfies the criterion at all times and would be deemed stable, curve (b) represents the critical integral solution, and curve (c) violates the criterion for some period of time... 

The Liapunov number can be used as a quantitative measure for chaos. The connection between chaos and the Liapunov number is through attractors. An attractor is a set of points S such that for nearly any point surrounding S, the dynamics will approach S as the time approaches infinity. The steady state of a fluid flow can be termed an attractor with dimension zero and a stable limit cycle dimension one. There are attractors that do not have integer dimensirms and are often called strange attractors. There is no tmiversally acceptable definition for strange attractors. The Liapunov number is determined by the principle axes of the ellipsoidal in the phase space, which originates from a ball of points in the phase space. The relatitaiship between the Liapunov number and the characterization of chaos is not universal and is an area of intensive research. 

Rigorous analysis of stability was performed by means of Liapunov s theorem, which was written concisely as the inequalities Eq. (1.19). This analysis showed that the initial and end points A and C are stable nodes while the middle B is usually an instable saddle point. This saddle point can sometimes transform into an unstable node or focus, thus allowing for birth of a limit cycle and self-oscillation behaviour. Unexpectedly, the mathematic analysis showed also the possibility of a stable intermediate state in some narrow region of parameters values (such that the maximum in Fig. 5.15 is not very far from the straight line W = ). This result differs from the intuitive physical considerations above. 

Energy methods can also be applied to systems not initially at equilibrium. A Liapunov function must be found which (a) vanishes for the initial state, (b) is positive for all other states, and (c) decreases in value as perturbation amplitude decreases. The initial state is stable if the function s value decreases continuously during system response to all possible perturbations. As with the thermodynamic energy method, the Liapimov method is generally easier to use than perturbation of the governing equations, especially for perturbations of large amplitude. However, a Liapunov function must first be foimd. Further information on this approach may be found in Denn (1975), Dussan (1975), and Joseph (1976). 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

As is the cases in earlier chapters, the function in (4.17) is zero at stationary states, increases on removal from stable stationary states and decreases from any initial given state on its approach to the nearest stable stationary state along a deterministic kinetic trajectory. These specifications make a Liapunov function in the vicinity of stable stationary states, which indicates the direction of the deterministic motion. Hence for every variation from a stable stationary state we have... 

The right hand side of (5.34) is negative semidefinite, so that the system tends towards the minimum of stable stationary state. Thus the function is a Liapunov function of the system. Further, satisfies the stationary solution of the master equation in the thermodynamic limit. All these properties assure that the function provides nessecary and sufficient conditions for the existence and stability of stationary states. 

The function is the driving force (frequently so-called, in fact a potential) toward stable stationary states. is lower bounded and is a Liapunov function with the derivative of with respect to time... 

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