Homoclinic loop to a saddle focus

The spiral-like shape of this attractor follows from the shape of homoclinic loops to a saddle-focus (2, 1) which appear to form its skeleton. Its wildness is due to the simultaneous existence of saddle periodic orbits of different topological type and both rough and non-rough Poincare homoclinic orbits. 

We end this section with a consideration of the homoclinic loop to a saddle-focus whose unstable manifold is one-dimensional. It is shown that when the saddle value is positive, infinitely many saddle periodic orbits coexist near such a homoclinic loop of the saddle-focus (Theorem 13.8). 

The existence of complex dynamics near a homoclinic loop to a saddle-focus was discovered by L. Shilnikov for the three-dimensional case in [131]. Subsequently, the four-dimensional case was considered in [132] and the general case in [136]. 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

The simplest example of such a situation is a homoclinic loop to a saddle-focus in the three-dimensional system... 

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

As an example, let us consider the codimension-one bifurcation of three-dimensional systems with a homoclinic loop to a saddle-focus with the negative... 

We should, however, stress that such a reduction to the two-dimensional case is not always possible. In particular, it cannot be performed when the equilibrium state is a saddle-focus. Moreover, under certain conditions, we run into an important new phenomenon when infinitely many saddle periodic orbits coexist in a neighborhood of a homoclinic loop to a saddle-focus. Hence, the problem of finding the stability boundaries of periodic orbits in multidimensional systems requires a complete and incisive analysis of all cases of homoclinic loops of codimension one, both with simple and complex dynamics. This problem was solved by L. Shilnikov in the sixties. 

Remark. Note that the problem on the fixed points of the Poincare map near the homoclinic loop to a saddle-focus is reduced to the study of the fixed point... 

Fig. C.7.6. Dependence of period T of the periodic orbit generating via a super-critical Andronov-Hopf bifurcation on the parameter a (6 = 6) as the cycle approaches the homoclinic loop to a saddle-focus with tr > 0.

C.7. 86.1 Assume there is a homoclinic loop to a saddle-focus in the Shimizu-Morioka model (like a T-point). Without computing the characteristic exponent of the saddle-focus, what can we say about the local structure is it trivial (one periodic orbit), or complex (infinitely many periodic orbits) ... 

We have seen that homoclinic bifurcations in symmetric systems have much in common. Let us describe next the universal scenario of the formation of a homoclinic loop to a saddle-focus in a typical system. In particular, this mechanism works adequately in the Rossler model, in the new Lorenz models, in the normal form (C.2.27), and many others. 

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

Fig. 13.4.15. The structure of the Poincare map near a homoclinic loop to the saddle-focus.

This theorem is a part of a more general assertion [including also the case of a multi-dimensional unstable manifold as well as saddle-foci of types (2,1) and (2, 2)] on complex dynamics near the homoclinic loop of a saddle-focus [136]. Condition p < 1 also known as the Shilnikov condition is very important here, because the structure of the phase space near the homoclinic loop is drastically changed in comparison to the case p > 1 covered by Theorem 13.6. The main bifurcations in the boundary case p = 1, when a small perturbation trigging the system into a homoclinic explosion from simple dynamics (p > 1) to complex dynamics (p < 1) were first considered in [29]. 

The situation becomes different when one considers normal forms of higher dimensions. Three- (and higher) dimensional asymptotic normal forms may exhibit non-trivial dynamics by themselves. For example, a homoclinic loop to the saddle-focus was found in the asymptotic normal form... 

The point NSF a = 0 corresponds to a neutral saddle-focus. At this codimension-two point the dynamics of the trajectories near the homoclinic loops to the saddle-focus becomes chaotic. This bifurcation indeed proceeds the origin of the chaotic double scroll attractor in Chua s circuit. In the general case, this bifurcation was first considered in [29]. The complete unfolding of... 

Thus, in a neighborhood of the homoclinic loop to the saddle-focus with < 1, there may exist structurally unstable periodic orbits, in particular saddle-nodes. This gives rise to the question does the saddle-node bifurcations of periodic orbits result in the appearance of stable ones ... 

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

Another example is the system shown in Fig. 8.2.1 containing a homoclinic loop r to a saddle-focus. If the saddle index... 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

Of special consideration are systems with symmetry where both separatrix loops approach together the saddle point. Such a situation is rather trivial namely when the loops split inwards, each gives the birth to a single stable limit cycle, in view of Theorem 13.4.1. When the loops split outwards, the stability migrates to a large-amplitude symmetric stable periodic orbit that bifurcates from the homoclinic-8 as shown in Fig. 13.7.2. And that is it. This is the reason why the theory below focuses primarily on non-symmetric systems. 

The limit points of this process correspond to the existence of a homoclinic loop which is the o -limit set for a separatrix of the other saddle-focus. 

To answer it, one must examine the two-dimensional Poincare map instead of the one-dimensional one, and evaluate the Jacobian of the former map. If its absolute value is larger than one, the map has no stable periodic points, and hence there are no stable orbits in a neighborhood of the homoclinic trar jectory because the product of the multipliers of the fixed point is equal to the determinant of the Jacobian matrix of the map. One can see from formula (13.4.2) that the value of the Jacobian is directly related to whether — 1 >0 or 2i/ — 1 < 0, or, equivalently, i/ > 1/2 or z/ < 1/2. Rephrasing in terms of the characteristic exponents of the saddle-focus, the above condition translates into whether the second saddle value o-q = Ai + 2ReA is positive or negative. It can be shown [100] that if <7 > 0 but ct2 < 0 (a < 6 in Fig. C.7.4), there may be stable periodic orbits near the loop, along with saddle ones. However, when (72 > 0 > O5 automatically), totally imstable periodic orbits emerge... 

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