Large-scale structural reorganization of strange attractors

Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems that contains the Lorenz system.

Figure 1

Canonical form for the genus-3 bounding torus consists of an outer disk boundary and three interior holes. The four singularities on the boundary are confined to the central interior hole. Locations of the two disks $(L,R)$ comprising the Poincaré surface of section are shown.

Figure 2

(a) Branched manifold compatible with the bounding torus shown in Fig. 1. There are two branch lines and six branches, three from $L$ labeled 0,1,2 and three from $R$ labeled 0,1,2. (b) Return map for this branched manifold. Diagonal parts describe the $L\to L$ and $R\to R$ flows and off-diagonal parts describe the $L\to R$ and $R\to L$ flows. The local torsion (0,1,0) is shown associated with each branch.

Figure 3

One perestroika that can occur for the genus-3 bounding torus when the two interior flow tubes are restricted.

Figure 4

One perestroika that can occur for the genus-3 bounding torus when the two exterior flow tubes are restricted.

Figure 5

Projection of the Lorenz attractor onto (a) the $X\text{−}Z$ plane and (b) the $X\text{−}Y$ plane. The two components of the global Poincaré section are shown. (c) Return map on the two branch lines shows that tearing occurs. Parameter values $(R,\sigma ,b)=(28.0,10.0,8∕3)$.

Figure 6

Projection of the Lorenz attractor onto (a) the $X\text{−}Z$ plane and (b) the $X\text{−}Y$ plane. (c) First-return map to a single component Poincaré section shows that only folding occurs. Parameter values $(R,\sigma ,b)=(278.56,30.0,1.0)$.

Figure 7

Chaotic attractors for the Burke and Shaw system. (a) Attractor with a tearing mechanism occurs around the upper part of the $Z$ axis which has the transverse stability of saddles. (b) Attractor with a folding mechanism occurs around the lower part of the $Z$ axis which has the transverse stability of foci.

Figure 8

Chaotic attractors for the Chen and Ueta system. (a) Attractor with a tearing mechanism contained in a bounding torus of genus-3 and (b) attractor with a folding mechanism contained in a bounding torus of genus-1.

Figure 9

Chaotic attractors for the Shimizu and Morioka system with $\mu =0.810$. (a) Tearing mechanism within a torus of genus-3 and (b) folding mechanism within a genus-1 torus.

Figure 10

As control parameters vary the attractor flows past the $Z$ axis in different regions. Folding occurs when the transverse stability is that of a focus and tearing occurs when the transverse stability is that of a saddle. Both occur in the transition region.

Figure 11

Lorenz attractor after the saddle fails to split the flow, deflecting it to a single region in the state space. The attractor can be bounded by a genus-1 torus, so only one branch line is necessary. (c) The local torsion can be determined by an isotopy that exchanges writhe for twist. Parameter values $(R,\sigma ,b)=(142.245,10,8∕3)$.

Figure 12

(Color online) (a) Two linked genus-1 strange attractors generated by the Lorenz equations and (b) their return maps. Parameter values $(R,\sigma ,b)=(207,10,8∕3)$.

Figure 13

(Color online) Lyapunov exponents for Lorenz attractor with $(\sigma ,b)=(10.0,8∕3)$. The plots show no evidence that a global bifurcation occurs at $R=142.026$.