First 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. 

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 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 ... 

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. 

Generically, the first Lyapunov value l differs from zero, i.e. the function G at = 0 starts with the cubic terms... 

The next bifurcation that we will now focus on occurs when the first Lyapunov value vanishes. Here, after getting rid of terms of second and fourth order (the smoothness r of the map is assumed to be not less than five) the map may be reduced to the form... 

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. 

If the first Lyapunov value L is positive, then for small fJL>0, the equilibrium state O is unstable and any other trajectory leaves a small neighborhood U of the origin. When fx <0, the equilibrium state is stable. Its attraction basin is bounded by an unstable periodic orbit of diameter /i which contracts... 

We have seen that bifurcations in one-parameter families transverse to the stability boundary OT may develop in completely different ways depending on the sign of the first Lyapunov value. If the value L vanishes at e = 0, at the very least we have to consider two-parameter families. To explore such a situation let us reduce the system on the center manifold to the normal form up to the terms of fifth order... 

If the first Lyapunov value Li > 0, then the fixed point of the map (11.6.6) is unstable for sufficiently small /x > 0. When /x < 0 the fixed point is stable its attraction basin is the inner domain of an unstable smooth invariant curve of the form (11.6.7). As p —0, the curve collapses into the fixed point see Fig. 11.6.2). 

We postpone our study of bifurcations on the torus until the next paragraph, and consider first what happens if the first Lyapunov value L vanishes. 

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... 

Proof. Let us suppose, for definiteness, that the first Lyapunov value Li is negative. Then, the invariant curve exists when /x > 0. The resonance zone adjoining at the point /jL = = a o) corresponds to periodic orbits of period-... 

Li corresponds to a period-doubling bifurcation, i.e. to a structurally unstable limit cycle with one multiplier equal to —1 and a positive first Lyapunov value ... 

In this case, the topological limit of the bifurcating periodic motion as —> — 0 contains no equilibrium point, but a periodic trajectory of the saddle-node type which disappears when e < 0. The trajectory is a simple saddle-node in the sense that it has only one multiplier, equal to 1, and the first Lyapunov value is not equal to zero. 

The stability of the bifurcating equilibria Oi 2 at the critical moment R = 0 is determined by the first Lyapunov value L. We will derive its analytical expression in Sec. C.5. 

C.5. 62. us give a general formula for the first Lyapunov value at a weak focus of the three-dimensional system... 

The first Lyapunov value computed by the above algorithm is... 

C.5. 65. Find the expression for the first Lyapunov value in the Shimizu-... 

Note that the first Lyapunov value is always negative for a flip-bifurcation of any periodic orbit in the logistic map. Indeed, the Schwarzian derivative ... 

Taking the real part of the right-hand side we arrive at the following formula for the first Lyapunov value L [184]... 

Thus, when a and are small, the sign of the first Lyapunov value equals the sign of the difference (/ — 2a). If it is negative, the stable invariant curve is born through the super-critical Andronov-Hopf bifurcation when crossing the curve AH towards larger (3. ... 

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