Characteristic exponents

It can be shown that A both exists and is finite. Moreover, we can always find a set of n tangent-space basis vectors, c (i = 1,... n), such that Ax = Sxi,..., Sx ) — "The divergence (or contraction) along a given basis direction, e, is then measured by the j Lyapunov characteristic exponent, A. These n (possibly... 

There are nonetheless a variety of sophisticated techniques available we will content ourselves with mentioning the pullback method introduced by Benettin, et.al. which is frequently used to obtain estimates for the lai gest characteristic exponent, Ai. ... 

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

If the characteristic exponents of (6-42) have negative real parts, the identically zero solution is asymptotically stable. 

If one characteristic exponent has a positive real part, it is unstable. The signs of ht are determined by means of the Hurwitz theorem. 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

The problem (6-126) is much simpler than (6-112) particularly because to be able to ascertain the stability of the periodic solution of Equation (6-112) it is necessary to calculate the characteristic exponents (Section 6.12) which is generally a very difficult problem. In the case of Eq. (6-125) this reduces to ascertaining the stability of the singular point, which does not present any difficulty. 

L. Mandelstam and N. Papalexi were first to establish the theory of the subharmonic resonance based directly on the theory of Poinear6 (Section 6.18). The derivation of this theory, together with the details of the electronic circuits, is given in25, or in an abridged version in6 (pages 464-473). The difficulty of the problem is due to the fact that this case is nonautonomous so that conditions of stability are determined in terms of the characteristic exponents, which always leads to rather long calculations. 

G. Benettin, L. Galgani, and J-M. Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems a method for computing all of them. Meccanica, 15 9-20, 1980. 

This value is a measure of the mean exponential rate of divergence (convergence) of two initially very close trajectories, i.e. when (xq,0) 0. The values from (59) are the so called Lyapunov characteristic exponents, which can be ordered by size ... 

The Belousov-Zhabotinskii reaction in an isothermal CSTR can undergo a series of transitions among periodic and chaotic states. One segment of this series of transitions is investigated in detail. Liapunov characteristic exponents are calculated for both the periodic and chaotic regions. In addition, the effect of external disturbances on the periodic behavior is investigated with the aid of a mathematical model. 

Figure 5. Liapunov characteristic exponent as a function of reciprocal residence...

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. 

While the Gr formulation is easier to employ for the constant-heat-flux case, we see that the characteristic exponents fit nicely into the scheme which is presented for the isothermal surface correlations. 

Table 7-3 lists values of the constants C, n, and m for a number of physical circumstances. These values may be used for design purposes in the absence of specific data for the geometry or fluid being studied. We should remark that some of the data correlations represented by Table 7-3 have been artifically adjusted by Holman [74] to give the characteristic exponents of J and i for the laminar and turbulent regimes of free convection. However, it appears that the error introduced by this adjustment is not significantly greater than the disa-... 

CASE 3 Consider the case of shear only of same fluid in both the domain i.e. p = 1. The characteristic exponents then simplify to. 

In general, Orr-Sommerfeld equation is a fourth order ODE and thus, will have four fundamental solutions whose asymptotic variation for y —> 00, is given by the characteristic exponents of (2.4.3) i.e. 

The nature of the intramolecular motion may also be identified by studying the way the separation of two trajectories evolves with time [353]. If the motion is regular (quasi-periodic) the separation is linear with time, but exponential if the motion is irregular (chaotic). If the separation is exponential, the rate of the separation — called the Lyapunov characteristic exponent — provides qualitative information concerning the IVR rate for the chaotic trajectories. This type of analysis has been reported, for example, for NO2 [271] and the Cl CHsBr complex [354]. 

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