Saddle connection bifurcation

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

At p = p3, the system jumps to a new state of uniform precession of the director (UP2) with large reorientation (0 74 ) and slow precession rate. As displayed in Fig. 7, starting from the stable UP2 branch above pa and lowering the excitation intensity, one finds a large and rather complicated hysteretic cycle, which eventually flips back to the UPl solution at pg = 1.09. This part of the UP2 branch consists of alternatively stable and unstable regions exhibiting a series of saddle-node bifurcations. Eventually, this branch connects with the UPS one which makes a loop and connects with the UPl branch. 

The two branches of the nontrivial steady states are not connected to the branch of the trivial steady state. They form a so-called isola and appear and disappear via a saddle-node bifurcation. 

Van Strien, S. J. [1982] On Parameter Families of Vector Fields. Bifurcations Near Saddle-connections, Ph.D. Thesis, Utrecht University. 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

For 0.53 < X < 0.72, one has the sequence U —> D —> O —> PR as before [see Fig. 13(b)], however there is an additional bifurcation between PR states. In fact, the limit cycle amplitude of the PR regime, now labeled PRi [curve 2 in Fig. 13(b)], abruptly increases. This results in another periodic rotating regime labeled PR2 with higher reorientation amplitude [curve 3 in Fig. 13(b)]. This is a hysteric transition connected to a double saddle-node structure with the (unstable) saddle separating the PRi and PR2 branches as already found... 

An analogous situation occurs when the system has a separatrix loop to a non-resonant saddle (i.e. its saddle value cr = Ai + A2 0) which is the a -limit of a separatrix of another saddle Oi (see condition (E) and Fig. 8.1.5). In this case, the bifurcation surface is also unattainable from one side, where close nonrough systems may have a heteroclinic connection, as shown in Fig. 8.1.6(b). 

As already mentioned, problems of this nature had appeared as early as in the twenties in connection with the phenomenon of transition from synchronization to an amplitude modulation regime. A rigorous study of this bifurcation was initiated in [3], under the assumption that the dynamical system with the saddle-node is either non-autonomous and periodically depending on time, or autonomous but possessing a global cross-section (at least in that part of the phase space which is under consideration). Thus, the problem was reduced to the study of a one-parameter family of C -diffeomorphisms (r > 2) on the cross-section, which has a saddle-node fixed point O at = 0 such that all orbits of the unstable set of the saddle-node come back to it as the number of iterations tends to -hoo (see Fig. 12.2.1(a) and (b)). 

We will study the case m = 0 in Sec. 12.4 in connection with the problem of the blue sky catastrophe . In the case m > 2, infinitely many saddle periodic orbits are born (see Theorem 12.5) when the saddle-node disappears moreover, even hyperbolic attractors may arise here (see [139]). We do not discuss such kind of bifurcations in this book. 

C12 and C21 (fc = 1,. ) such that at /i the unstable separatrix Fi of Oi i = 12 ) makes k complete rotations along U and enters the saddle Qj i) thereby forming a heteroclinic connection. The curves are defined by the equations pj = hkij p>i) where hkij is some smooth function defined on an open subset of the positive /ii-axis such that the first derivative of hkij tends uniformly (with respect to k) to zero as /ii 0. The exact structure of the bifurcation set corresponding to heteroclinic connections is quite different depending on whether the equilibria Oi are saddles or saddle-foci. 

The bifurcation diagrams for the case where both Ox and O2 are saddles are shown in Figs. 13.7.12-13.7.15. Here, if both the separatrix values are positive, the only possible heteroclinic connections are the original ones which exist at... 

The bifurcation diagram for the case where both Oi and O2 are saddle-foci is shown in Fig. 13.7.17. Here, the curves L and L2 which correspond to the homoclinic loops intersect the curves C21 / i = 0 and C 2 a 2 = 0 infinitely many times. Next, for each A = 0,1,2,..., for any two neighboring points of intersection of L with a connected component of L2 with a connected component of C12) such that the inequality hi fjL2) > hki2 fJ>2) (respectively, /i2(mi) > hk2i f )) holds between these points, there is a component of the curve (respectively C i ) which connects these points. In turn,... 

Fig. 13.7.17. (a) A one-dimensional two-way heteroclinic connection between a pair of saddle-foci. (b) The corresponding bifurcation diagram. 

Afraimovich, V, S. and Shilnikov, L. P. [1974] On some global bifurcations connected with the disappearance of fixed point of a saddle-node type, Soviet Math. Dokl. 15, 1761-1765. 

---