Lyapunov value

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

An equilibrium state 0(0,0) with one zero chareicteristic exponent and with a non-zero Taylor-series coefficient Z2, called a Lyapunov value or coefficient. The Lyapunov value can be easily calculated from the associated normal form equation... 

If the Lyapunov value h 0 here, then the associated double fixed point corresponds to a double (semi-stable) limit cycle of the original system. 

The basic tools for studying critical cases include the method of reduction to the center manifold and the method of normal forms. The latter allows us to calculate the Lyapunov values that determine the stability of a critical equilibrium state. 

Fig 9.2.1. A saddle-node with different Lyapunov values. See comments in the text. 

If the first non-zero Lyapunov value is negative and has an odd index number, i.e. < 0, fc = 2p+l, then the equilibrium state is stable. All trajectories tend to O as t -foo. Moreover, the trajectories which do lie on the strong stable manifold converge to O along as shown... 

Pig. 9.2.4. A degenerate equilibrium state with hp+i 0. The center manifold is continued here in both directions. Such a bifurcation called a pitchfork is typical for systems where due to symmetry the first non-zero Lyapunov value at a degenerate equilibrium state is always of an odd order. 

Note that in order to calculate the first non-zero Lyapunov value there is no need to reduce the system to the center manifold. If the original system has the form... 

Let us show that formula (9.2.9) does give us the Lyapunov value. Indeed, by definition, the Lyapunov value is the first non-zero coefficient of the expansion... 

It follows from formula (9.2.9) that if the right-hand side of the system (9.2.6) is analytic, and if all Lyapunov values vanish, then g[x,ip x)) = 0. Hence, since y = (p x) is the solution of the system (9.2.7), it follows that the curve y = ip x) is filled out by the equilibrium states of the system (9.2.6). Thus, it is an invariant manifold of this system. Since it is tangent to y = 0 at O, it is the center manifold by definition. It follows that for the case imder consideration, the system has an analytic center manifold W y = (p(x) which consists of equilibrium states as illustrated in Fig. 9.2.5. 

The system (9.3.5) or (9.3.4) is the normal form for the second critical case. The coefficients Lq are called the Lyapunov values. Observe from the above procedure that in order to calculate Lq one needs to know the Taylor expansion of the Eq. (9.3.1) up to order p + g == 2Q + 1. 

Returning to the original high-order system [see (9.1.1)-(9.1.3)], we observe that if the first non-zero Lyapunov value is negative, then the trajectories... 

If the first non-zero Lyapunov value is positive and if all non-critical characteristic exponents (71,...,7n) lie to the left of the imaginary axis in the complex plane, then the equilibrium state is a complex saddle-focus, as shown in Fig. 9.3.2(b). Its stable manifold is and the unstable manifold coincides with the center manifold W, The trajectories lying neither in nor pass nearby the equilibrium state. 

The formula for the first Lyapunov value expressed in terms of the coefficients of the system (9.3.1) was first derived by Bautin [24]. If we write down the system as... 

We note that by rescaling the r-variable in (9.3.13), the value Lk can be made equal to one in absolute value. Meanwhile, it is obvious that the sign (as well as the number) of the first non-zero Lyapunov value is not altered by non-singular changes of variables and time. The sign determines whether the given equilibrium state is stable or not, whereas the number determines the speed of convergence of the trajectories to zero [see (9.3,14)]. 

For the case of all zero Lyapunov values the trajectory behavior can be described only in the analytic case. 

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... 

Fig. 9.3.3. When all Lyapunov values vanish in an anzilytical system, the equilibrium state is a center on W. In its extended neighborhood is foliated by invariant cylinders.

In the case, the origin is not necessarily a center if all Lyapunov values vanish. For example, in the system... 

If all Lyapunov values vanish and the map is analytic, then the center manifold is analytic too and it consists of fixed points (Fig. 10.2.8). Observe that if the map has the form... 

Hence if all Lyapunov values vanish, then it follows from the analyticity of / that... 

If all Lyapunov values are equal to zero and the system is analytic, then the center manifold is also analytic, and all points on it, except O, are periodic of period two. This means that for the system of differential equations there exists a non-orientable center manifold which is a Mobius band with the cycle L as its median and which is filled in by the periodic orbits of periods close to the double period of L (see Fig. 10.3.2). 

We will now present an algorithm for calculating the first non-zero Lyapunov value for maps which are not reduced to the standard form. First we write the map in the form... 

The fixed point O under consideration is called either a complex or weak) stable focus or a complex weak) unstable focus depending on the sign of the Lyapunov value. 

In the case where the Lyapunov value Lk is positive, the fixed point of the original map is a weak saddle-focus. Its stable and unstable manifolds are and respectively, as shown in Fig. 10.4.2. 

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). 

We have seen in the previous sections that the qualitative behavior of a strongly resonant critical fixed point differs essentially from that of a non-resonant or a weakly resonant one. It is therefore natural to ask the question what happens at a strongly resonant point as the frequency varies In particular, in the case of the resonance a = 27t/3 the fixed point is a saddle with six separatrices in general, but when an arbitrarily small detuning is introduced the point becomes a weak focus (stable or unstable, depending on the sign of the first Lyapunov value). The question we seek to answer is how does the dynamics evolve before and after the critical moment ... 

We have seen in Sec. 10.4 that in the case of weak resonance cj = 2nM/N N > the stability of the critical fixed point is, in general, determined by the sign of the first non-zero Lyapunov value. The same situation applies to the critical case of an equilibrium state with a purely imaginary pair of characteristic exponents. However, there is an essential distinction, namely, for a resonant fixed point only a finite number which does not exceed N—3)/2 of the Lyapunov values is defined. The question of the structure of a small neighborhood of the fixed point in the case where all Lyapunov values vanish is difficult, so we do not study it here. Instead, we consider two examples. 

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

Consider first the case where the first Lyapunov value I2 is non-zero. Following the scheme outlined in the preceding section, we first derive the equation of the boundary of the stability region near e = 0. Next we will find the conditions under which is a smooth surface of codimension one. Finally, we will select the governing parameter and investigate the transverse families. 

Indeed, the condition under which the system has an equilibrium state x with one zero exponent and zero Lyapunov values 2 > h-i is given by... 

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