Shimizu-Morioka model

Let us consider next a few examples. We will focus our consideration on the Lorenz equation, the Chua s circuit, the Shimizu-Morioka model and some others. 

The system (C.2.24) is the asymptotic normal form appearing in the study [129] of local codimension-three bifurcations of equilibria and periodic orbits of systems with a symmetry (see Sec. C.4). When H = 0, system (C.2.24) is the Shimizu-Morioka model [127], [191]... 

The Shimizu-Morioka model has three equilibria when 6 > 0. The origin 0(0,0,0) is a saddle of type (2,1) with the characteristic exponents... 

The characteristic equation at the non-trivial equilibria Oi 2(=bVb) 0 1) of the Shimizu-Morioka model is given by... 

Fig. C.7.9. The Lorenz-like attractor in the Shimizu-Morioka model near the point

Shimizu-Morioka model, and [114, 115, 117, 161] for the original Lorenz and some other Lorenz-like equations. ... 

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,... 

Fig. C.7.13. Homoclinic doubling cascade in the Shimizu-Morioka model, as the parameter a varies (6 = 0.40). Using the shooting approach, find the corresponding values of parameter a.

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) ... 

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). ... 

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]. 

Shilnikov, A. L, [1993] Bifurcations of the Lorenz attractors in the Morioka-Shimizu model, Physica D62, 338-346. 

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