Bifurcation of periodic orbits with multiplier

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

It should be noted that constructing the versal families is realistic only in these simple cases, and in a few special cases. For example, a finite-parameter versal family cannot be constructed for the bifurcation of a periodic orbit with one pair of complex multipliers Nevertheless, this problem does... 

Observe that if the iV-th iteration of the circle map is an identity, then all points on the circle are structurally unstable with a multiplier equal to one. Moreover, all Lyapunov values of each point are equal to zero. This is an infinitely degenerate case. We saw in Sec. 11.3 that to investigate the bifurcations of structurally unstable periodic orbits with A — 1 first zero Lyapunov values it is necessary to consider at least -parameter families. It is now clear that to study bifurcations in this case one has to introduce infinitely many pa rameters. Moreover, it is seen from the proof of Theorem 11.5 that such maps can be obtained by applying a small perturbation to an arbitrary circle map... 

Theorem 12.8. For all p> 0, the system on the Klein bottle has exactly two periodic orbits with negative multipliers. If fo (p) does not vanish identically, each of these two periodic orbits undergoes a period-doubling bifurcation infinitely many times as +0. 

The bifurcation of a periodic orbit with three multipliers +1. On the center manifold we introduce the coordinates (x y z jp) where is the angular coordinate and (x, y, z) are the normal coordinates (see Sec. 3.10), Assuming that the system is invariant under the transformation x,y) —> (—X, — y), the normal form truncated up to second order terms is given by... 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

Thus, the bifurcation of a fixed point with one multiplier equal to —1, and with k — 1) zero Lyapunov values, cannot produce more than k orbits of period two. Moreover, it is easy to specify the precise parameter values for which Eq. (11.4.21) has a prescribed number of positive roots, within the range from 0 to k. This implies that in the parameter space of the map (11.4.16), there are regions where the family has any prescribed number (from 0 to k) of period-two orbits. 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

In the previous section we have reduced the problem of period-two orbits which were spawned from a fixed point with a multiplier —1 to a study of the mapping (11.4.17) analogous to the mapping (11.5.16). Therefore, the bifurcation diagram in this case is the same as in the period-doubling bifurcation with (fc — 1) zero Lyapunov values this consists of a union of the plane /io = 0 on which the equilibrium state at the origin loses its stability and that half of... 

Since A < 0, the Poincare map is decreasing. The new feature in this case is that such maps may have orbits of period two, which correspond to the so-called double limit cycles. They may appear via a period-doubling bifurcation (a fixed point with a multiplier equal to —1) or via a bifurcation of a double homoclinic loop. The latter corresponds to the period-two point of the Poincare map at 2/ = 0 (see Fig. 13.3.2). 

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