Homoclinic butterfly

When (7 = 10, 6 = 8/3 and r 13.926, both of the one-dimensional unstable separatrices Fi and F2 of the saddle 0(0,0,0) return to the saddle, along the same direction (the positive z semi-axis). They form a geometrical configuration called a homoclinic butterfly (Fig. 13,6.3). Note that a homoclinic butterfly may only occur in with n > 3. 

Fig. 13.6.3. A homoclinic butterfly. Both separatrices return to the saddle tangentially to each other.

In general, the bifurcation of a homoclinic butterfly is of codimension two. However, the Lorenz equation is symmetric with respect to the transformation (x y z) <-)> (—X, —y z). In such systems the existence of one homoclinic loop automatically implies the existence of another loop which is a symmetrical image of the other one. Therefore, the homoclinic butterfly is a codimension-one phenomenon for the systems with symmetry. 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

In order to resolve this problem, it was proposed in [138] to study the homoclinic butterflies in the Cases A, B and C above. Namely, it was established that the bifurcation of a homoclinic butterfly results in the immediate appearance of a Lorenz attractor when... 

Case (a) corresponds to a codimension-three bifurcation, while Cases (b) and (c) are of codimension four. However, if the system exhibits some symmetry, then all of the above three bifurcations reduce to codimension two. It was established in [126, 127, 129] that a symmetric homoclinic butterfly with either a = 0 or A = 0 appears in the so-called extended Lorenz model, and in the Shimizu-Morioka system, as well as in some cases of local bifurcations of codimension three in the presence of certain discrete symmetries [129]. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

Fig. 13.7.6. A bifurcation diagram for an orientable (Ai,2 > 0) homoclinic butterfly of the saddle with a negative saddle value.

It is not hard to conclude from numerical experiments, which reveal the manner in which the separatrices converge to the homoclinic butterfly that A must be within the range (0,1). In this case, when a < 0, everything is simple the homoclinic butterfly splits into either two stable periodic orbits (Fig. C.7.8(g)), or just one stable symmetric periodic orbit (Fig. C.7.8(i)). It follows from Sec. 13.6 that when <7 > 0, two bifurcation curves originate from this codimensiomtwo point. They correspond to the saddle-node bifurcation (Fig. C.7.8(d)) and to the double homoclinic loop (Fig. C.7.8(f)). The... 

So far an important conclusion since there is a homoclinic butterfly with A < 1, the region of the existence of the Lorenz attractor adjoins to the codimension-two point in the parameter space. The interested reader is advised to consult [127, 129, 187] on the bifurcations of Lorenz attractor in the... 

A fragment of its (r, cr) bifurcation diagram is shown in Fig. C.7.14. Detect the points where the path cr = 10 intersect the curve HB of the homoclinic butterfly and the curve LA on which the one-dimensional separatrices of the saddle tend to the saddle periodic orbits. Find the point on the curve LA above which the Lorenz attractor does not arise upon crossing LA towards larger values of r. The dashed line passing through the T-point in Fig. C.7.14 corresponds to the moment of the creation of the hooks in the two-dimensional Poincare map when the separatrix value varishes A — 0 (see discussion on the Shimizu-Morioka model). ... 

Let us complete this section by an illustration corresponding to the homoclinic butterfly of the saddle-focus in the four-dimensional case. Let us consider... 

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