Heteroclinic connection

Here, we limit our argument to a system with a homoclinic connection—that is, a separatrix connecting a saddle with itself. The following argument can be straightforwardly extended to a system with a heteroclinic connection— that is, a separatrix connecting different saddles. 

In the phase plane p,q),dL front corresponds to trajectory that connects two steady states of (4.4). Such a trajectory is know as a heteroclinic orbit or a heteroclinic connection. The steady states of (4.4) are given by (p, 0), where F(p) = 0. The phase plane trajectories or orbits of (4.4) are the solutions of... 

V < 2V. The state (1,0) is always a saddle point. To be physically acceptable, a front must always be nonnegative. Consequently, only nonnegative heteroclinic orbits are acceptable. Such orbits can only exist if (0,0) is a stable node. In other words, fronts only exist for v > 2 /z5r. Since there exists a heteroclinic connection or front for each value of v with v > 2 /Dt, this analysis does not yield a unique propagating velocity. In fact, the front velocity v depends on the initial condition, specifically on the tail of the initial condition. 

Similarly, if there were a separatrix loop to a saddle at // = 0, it would be split for some non-zero /i, as shown in Fig. 7.1.2. We see that an arbitrarily small smooth perturbation of the vector field will modify the phase portrait of a system with a homoclinic loop or a heteroclinic connection this obviously means that such a system is non-rough. 

Fig 7.5.2. A structurally stable heteroclinic connection between two saddles in... 

Fig. 8.1.2. A structurally unstable heteroclinic connection between a saddle-node Oi and a saddle O2.

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). 

The violation of structural stability in Morse-Smale systems is caused by the bifurcations of equilibrium states, or periodic orbits, by the appearance of homoclinic trajectories and heteroclinic cycles, and by the breakdown of transversality condition for heteroclinic connections. However, we remark that some of these situations may lead us out from the Morse-Smale class moreover, some of them, under rather simple assumptions, may inevitably cause complex dynamics, thereby indicating that the system is already away from the set of Morse-Smale systems. 

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

Structurally unstable heteroclinic connections in systems of higher-dimension may require new moduli besides known ones. Moreover, even structurally unstable diffeomorphisms with simple dynamics may require infinitely many moduli for their description. The conditions under which they have either a finite or an infinite number of moduli are presented in [43]. 

Fig. 8.4.1. (a) A heteroclinic connection between a saddle O2 and saddle-focus Oi (b) a heteroclinic connection between two saddle-foci Oi,2 (c) a homoclinic figure-eight to a saddle-focxis. 

Fig. 10.6.3. Two ways of how a heteroclinic connection may be broken down. See the discussion in the text.

In the general case the picture may be more involved due to the appearance of multi-circuit heteroclinic connections. 

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... 

Fig. 13.7.16. Three types of heteroclinic cycles (a) Ai 2 > 0 (b) A 2 < 0 (c) A semiori-entable heteroclinic connection between two saddles on a Mobius band A < 0, A2 > 0.

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

Fig. 13.7.22. The bifurcation diagram for the heteroclinic connection in Fig. 13.7.12 when A > 0, A2 < 0, i/i > 1, 1/2 < 1 and 1/11/2 > 1. The system has one simple periodic orbit in regions 1, 2, 3 and 5, two periodic orbits (one simple and one of double period) in region 4, and no periodic orbits elsewhere. The stable periodic orbit loses stability on the curve PD corresponding to a period-doubling (flip) bifurcation. The unstable limit cycle of double period becomes a double-circuit separatrix loop on L. The stable simple limit cycle terminates on Li.

The saddles have a closed symmetric heteroclinic connection at the level i/ = 1/4. The equations of the trajectories connecting the saddles can be found explicitly, and for the upper one it is given (verify this) by... 

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