Saddle equilibrium state

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

An analogous picture takes place in the case of three-dimensional flows possessing the chain Qi < Q < Q2 where Q denotes a saddle periodic orbit, and Qi and Q2 stand for either saddle equilibrium states or periodic orbits. 

By the results of Secs. 5.2 and 5.3, to prove the existence and uniqueness of the stable separatrix of the saddle equilibrium state of the non-autonomous system it is sufficient to check that in a small neighborhood of the equilibrium, for all positive times, the non-linearities remain small along with all derivatives. Thus, we must check that the functions i,2(-R> i R ))/R 3. (x 1) the right-hand side of (10.5.20) are small along with all derivatives, provided that for some small <5... 

The saddle equilibrium states are the saddle fixed points of the shift map, and respectively, their separatrices are the invariant manifolds. Returning to the original (non-rescaled) variables we find that the fixed points must lie apart from the origin at some distance of order e. If the third iteration (10.6.2) of the map (10.6.1) were the shift map of the reduced system (10.6.5), then the above theorem would follow from our arguments because the fixed points Oi, O2,03 of the third iterations correspond to the cycle of period three of the original map. 

Fig. 12.1.7. The boundary of the stability region of the periodic orbit born at the bifurcation of the on>edge homoclinic loop to a degenerate saddle-node corresponds to the homoclinic loop of the border saddle equilibrium state Oi.

This is a one-dimensional system which may have stable and unstable equilibrium states corresponding to stable and saddle equilibrium states of the entire system (12.4.6) or (12.4.7). The evolution along Meq is either limited to one of the stable points, or it reaches a small neighborhood of the critical values of X. Recall, that we consider x as a governing parameter for the fast system and critical values of x are those ones which correspond to bifurcations of the fast system. In particular, at some x two equilibrium states (stable and saddle) of the fast system may coalesce into a saddle-node. This corresponds to a maximum (or a minimum) of x on Meq, so the value of x cannot further... 

BIFURCATIONS OF HOMOCLINIC LOOPS OF SADDLE EQUILIBRIUM STATES... 

Consider a continuous one-parameter family of C -smooth (r > 1) systems on a plane which have a saddle equilibrium state O. Suppose that at /i = 0, the system has a separatrix loop of the saddle i.e. the separatrix T [ coincides with the separatrix at /i = 0. 

Consider an (n + l)-dimensional C -smooth (r > 4) system with a saddle equilibrium state O. Let O have only one positive characteristic exponent 7 > 0 the other characteristic exponents Ai, A2,..., A are assumed to have negative real parts. Moreover, we want the leading stable exponent Ai to be real ... 

Estimates of the behavior of trajectories near a saddle equilibrium state... 

In this section, we prove our estimates of the solutions near a saddle equilibrium state which we used throughout this chapter. 

Consider a C -smooth (r > 3) system in a neighborhood of a saddle equilibrium state with m-dimensional stable and n-dimensional unstable invariant manifolds. 

A periodic orbit merges with a homoclinic loop r e) to a saddle equilibrium state Oe whose characteristic exponents Ai(6r),..., An( ) satisfy the following conditions ... 

The last comment on the Chua circuit concerns the bifurcations along the path 6 = 6 (see Fig. C.7.4). Notice that this sequence is very typical for many synunetric systems with saddle equilibrium states. We follow the stable periodic orbit starting from the super-critical Andronov-Hopf bifurcation of the non-trivial equilibrium states at a 3.908. As a increases, both separatrices tend to the stable periodic orbits. The last ones go through the pitch-fork bifurcations at a 4.496 and change into saddle type. Their size increases and at a 5.111, they merge with the homoclinic-8. This, as well as subsequent bifurcations, lead to the appearance of the strange attractor known as the double-scroll Chua s attractor in the Chua circuit. ... 

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