Lyapunov theorem

If the real parts of all eigenvalues e, Ree, <0 are negative, according to the Lyapunov theorem [14, 15] the stationary point is asymptotically stable... 

A general formulation of the Lyapunov theorem concerning the stability of stationary points of autonomous systems may now be given. Let the system... 

The phase space is partitioned into cells co of diameter 5. In the limit of an arbitrarily fine partition, the entropy per unit time tends to the Kolmogorov-Sinai entropy per unit time which is equal to the sum of positive Lyapunov exponents by Pesin theorem [16] ... 

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

Let V be a region in space bounded by a closed surface S (of Lyapunov-type [24, 50]), and f (x) be a vector field acting on this region. A Lyapunov-type surface is one that is smooth. The divergence (Gauss) theorem establishes that the total flux of the vector field across the closed surface must be equal to the volume integral of the divergence of the vector (see Theorem 10.1.1). 

The stationary states that faU into fragment 1 of the curve (Figure 3.5) are stable at minor deviations of a from (1q in virtue of the theorem on the minimal rates of entropy production in these states. On further run ning away from the point a = aq, we may faU outside the region of applicability of nonlinear thermodynamics while remaining in the thermodynamic branch, which is described, for example, by a station ary state functional as a kind of positively defined Lyapunov function... 

The existence of Lyapunov exponents is proved, under a general condition, by the multiplicative ergodic theorem of Oseledec [8], However, the convergence of the exponents is found to be quite slow (algebraically) in time for a generic dynamical system [9], due to its nonhyperbolicity. 

The transition of spectra from structured to unstructured ones is not described by a theorem, but has been studied numerically in Guzzo, Lega and Froeschle (2002) by comparing the geometry of resonances of a given system computed with the Fast Lyapunov Indicator with the structure of the spectra of an observable computed on well selected chaotic solutions. More precisely, in Froeschle et al. (2000) we estimated with the FLI method that the transition between Littlewood and Chirikov regime for the Hamiltonian system ... 

The Oseledec theorem (Eckmann and Ruelle, 1985) implies that under rather general conditions the eigenvectors of M)r(xo)Mt(x0) converge, as t —> oo, to a set of Lyapunov vectors v. Since MfMt is symmetric and positive definite the Lyapunov vectors form an orthonormal set and the eigenvalues A are all positive. If the initial separation is chosen to be oriented along one of the Lyapunov vectors, So = with <5o distance between the particles at a later time t is... 

The proportion of fluid elements experiencing a particular anomalous value of the Lyapunov exponent A / A°° decreases in time as exp(—G(X)t). In the infinite-time limit, in agreement with the Os-eledec theorem, they are limited to regions of zero measure that occupy zero volume (or area in two dimensions), but with a complicated geometrical structure of fractal character, to which one can associate a non-integer fractional dimension. Despite their rarity, we will see that the presence of these sets of untypical Lyapunov exponents may have consequences on measurable quantities. Thus we proceed to provide some characterization for their geometry. 

Equation 3.33 is a simple quadratic equation, known as the characteristic equation, with two unique solutions (Xi and I2), where li and I2 are known as the eigenvalues. It is the nature of these eigenvalues that allow us to identify nodes, according to Lyapunov s theorem. These nodes can be classified as follows [10] ... 

According to Fisher s theorem, the mean fitness w increases steadily along the trajectory of (7.18). Technically it means that w is Lyapunov function. 

Lyapunov s theorem in the case of thermodynamic steady-state... 

Let us now assume that, due to the stress applied along the selected polar axis, a partial coorientation of the particles has taken place (i.e., the system acquired texture ). As a result, the value of the particle size projections mean on that axis has increased. To estimate the probability of this state, we will use the approach developed by Yushenko on the basis of suggestions offered by Kolmogorov. Since the projection z of each particle is a random number, the Lyapunov central limit theorem can be applied to the system. Then the distribution around the mean value of the projection, is distributed normally with a mean, p = 8/2, and a dispersion, = a /Af = 8 /12/V. The probability of a deviation of a given mean of a projection on a given axis from the most probable value of the mean, 5/2, in the absence of coorientation is given by the function... 

Oseledek, V. I. (1968). A multiplicative ergodic theorem characteristic Lyapunov exponents of dynamical systems, Trudy Mask. Mat. (Obsc.), 19, 179-210. 

In this context, the following question raised by Andronov and Vitt is remarkable Which P-trajectory is stable in the sense of Lyapunov Its answer is settled by the following theorem. 

Theorem 7.6. (Markov) If a Poisson-stable trajectory is uniformly stable in the sense of Lyapunov then it is almost-periodic. 

Now, let the equilibrium state x = 0 of the reduced system (9.1.2) be stable in the sense of Lyapunov. By definition, this means that for the system (9.1.2 and 9.1.3) in the standard form, the x-coordinate remains small in the norm for all positive times, for any trajectory which starts sufficiently close to O, provided y remains small. At the same time, the smallness of x implies the inequality (9.1.4) for the y-coordinate, i.e. y t) converges exponentially to zero. Thus, we have the following theorem. 

Theorem 9.1. If the equilibrium state is Lyapunov stable in the center manifold then the equilibrium state of the original system (9.1.1) is Lyapunov stable as well Moreover if the equilibrium state is asymptotically stable in the center manifold, then the equilibrium state of the original system is also asymptotically stable. 

Remark. The Lyapunov function is a universal tool of stability theory. Typically, a proof concerning stability consists of either constructing a Lyapunov function, or proving its existence. Moreover, its applicability is not limited to critical equilibria for example, in our analysis of studying the structurally stable equilibria (Theorem 2.4), we have implicitly shown that the norm of a vector in a Jordan basis is a Lyapunov function. 

Theorem 9.3. If all Lyapunov values are equal to zero, then the associated analytic system has an analytic invariant (center) manifold which is filled with closed trajectories around the origin, as shown in Fig. 9.3.3. On the center manifold the system has a holomorphic integral of the type... 

Formula (10.4.20) is similar to the formula (10.4.14) for the non-resonant case and the only difference is that in. the case of a weak resonance only a finite number of the Lyapunov values Li,..., Lp is defined (for example, only L is defined when N = b). If at least one of these Lyapunov values is non-zero, then Theorem 10.3 holds i.e. depending on the sign of the first non-zero Lyapunov value the fixed point is either a stable complex focus or an unstable complex focus (a complex saddle-focus in the multi-dimensional case). 

The existence of a Lyapunov function implies the asymptotic stability, i.e. in this case the theorem holds. One can construct a Lyapunov function for an enlarged parameter region > 1, L < 0. To do this let us... 

Theorem 11.1. If the first Lyapunov value Li in (11.5.3) is negative, then for small /i < 0, the equilibrium state O is stable and all trajectories in some neighborhood U of the origin tend to O. When fx > 0, the equilibrium state becomes unstable and a stable periodic orbit of diameter y/Ji emerges see Fig. 11.5.1) such that all trajectories from U, excepting O, tend to it. 

We have already established in the last section that when the first Lyapunov value does not vanish, the passage over the stability boundary 9Jl p e) = 0 is accompanied by the appearance of an invariant two-dimensional torus (in the associated Poincare map this corresponds to the appearance of an invariant closed curve). If we are not interested in the behavior of the trajectories on the torus, we can restrict our consideration to the study of one-parameter families transverse to 9Jl, In this case Theorem 11.4 in Sec. 11.6, gives a complete description of the bifurcation structure. In order to examine the... 

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