Stability Liapunov

A. A. Martynyuk, Stability by Liapunov s Matrix Function Method with Applications... 

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). 

It has been shown that the transition from the two peak periodic oscillation to the chaotic behavior occurs with a loss of stability of the periodic oscillation an unstable two peak oscillation is embedded In the chaotic region. During this transition the Liapunov characteristic exponent changes sign from negative to positive. Furthermore, calculations indicate that a small amplitude regular disturbance does not have a significant effect on the character of the oscillations. 

Remark 4.5 Corollary 4.1 obviously implies the stability of the rest state (the zero solution) for e not to close to 1. Using a different approach, Renardy [42] has recently removed this restriction and showed the Liapunov stability of the rest state for all e s, 0 < e < 1. This fact is not known for Maxwell-type models (e = 1). 

Concerning the Liapunov (nonlinear) stability of the Couette flow under one dimensional perturbations, we have for instance the following result [47]. 

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... 

A brief mathematical digression on Liapunov stability theory will set the stage for the results of this chapter. Those familiar with the LaSalle corollary to Liapunov stability theory, also called the invariance principle by some authors, can skip immediately to Section 3. 

The following theorem is a minor variation of the LaSalle corollary of Liapunov stability theory, taken from [WLu] (see also [H2], where F is required to be continuous on the closure of G). It also holds under less restrictive hypotheses than are required here. 

The stability exponents labeled u in Table 1 of Wintgen et al. [52] are actually the Liapunov exponents of the periodic orbits multiplied by their respective periods, u = 2T.]... 

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. 

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). 

Stability proofs) Prove that your answers to 5.1.10 are correct, using the definitions of the different types of stability. (You must produce a suitable 0 to prove that the origin is attracting, or a suitable S( ) to prove Liapunov stability.)... 

Stability of singular points via Liapunov s second method is also discussed. 

HID) 1965 Gurel, O., Lapidus, L. Liapunov Stability Analysis of Systems with Limit Cycles, Chem. Eng. Symposium Seties, vol. 61, no. 55, 78—87... 

We are beginning to understand chaotic structure in a way not seen before. Numerical methods of measuring chaotic and regular behaviour such as Fast Liapunov Indicators, sup-maps, twist-angles, Frequency Map Analysis, fourier spectal analysis are providing lucid maps of the global dynamical behaviour of multidimensional systems. Fourier spectral analysis of orbits looks to be a powerful tool for the study of Nekhoroshev type stability. Identification of the main resonances and measures of the diffusion of trajectories can be found easily and quickly. Applied to the full N-body problem without simplification, use of these tools is beginning to explain the observed behaviour of real physical systems. 

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). 

Stability analysis by Liapunov s theorem shows that ... 

M> W. Hahn, (a) Theory and Application of Liapunov s Direct Method , Prentice-Hall, London, 1963 (h) Stability of Motion , Springer-Verlag, Berlin, 1967. 

The parallel between the insensitivity criterion of Hahn and the Liapunov stability method is clear. Although these methods are more restrictive they do offer rigorous conditions under which the absence of ignition may be guaranteed and provide a general approach to more complex reacting systems. 

The Liapunov definition of stability has the great virtue that it provides a method of analysis that by-passes the arduous task of integrating the reactor equations. Whereas local stability of the equilibrium states (c.,T.) is easily established via equations, use of Liapunov function V(c,T) establishes a region of stability (Figure 13). This is the region enclosed by the contour V c,T) = constant, which touches c,T) = 0 tangentially. 

The derivative dVjdt is easily found by ditferentiating equation (40) and using the reactor equations (37) for dc/dr and dr/dt. Different values of coefiBcients Oil. a,i, and Oji will provide different r ons of asymptotic stability. The union of all these (UAS) is a more extensive r on of asymptotic stability than any individual. An investigation of the CSTR using this approach was carried out by Berger and Perlmutter "" typical results are own in Figure 14 in terms of the UAS obtained from three choices of Liapunov functions. 

Stability studies of the steady-state solutions of the TRAM are in general very complex and beyond the scope of this Report. The more successful approaches have involved the use of Liapunov functions, " of collocation methods, or of topological fixed-point methods. The genoation of r ons of stability inevitably involves a considerable amount of computational effort. For a full discussion of these methods the reader is again referred to Perlmutter s book. ... 

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. 

Other stability analysis techniques are discussed by Ramirez and Turner (1969). They include the eigenvalues of a linear analysis which give results that only apply to a small region about a steady-state value, and Liapunov s direct method via Kravoskii s theorem which accurately predicted regions of stability with and without feedback control for this nonlinear problem. 

Now, the condition dL/dt 0 as required for the Liapunov-stability of the equilibrium is reduced to the condition that the matrix of the linear phenomenological coefficients is positive definite. This latter property, however, is a direct consequence of the second law of thermodynamics as we have shown in (3.82). With this conclusion we have reconfirmed our preliminary result of the preceding section in a formally precise way a sufficient condition for the stability of the equilibrium is a) a positive-definite capacitance matrix such that L 0 and b) the second law of thermodynamics such that dL/dt 0. Let us emphasize once more the significance of the equivalence between dL/dt < 0 and the second law in the form of (3.82). This equivalence, however, is valid only in the range of validity of the linear relations in (3.81). If the fluxes I were some nonlinear functions of the forces F as will be the case in situations far from the thermodynamic equilibrium, dL/dt 0 is no longer guaranteed by the second law and possibly may no longer be valid at all. 

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