Distinguishing between folding and tearing mechanisms in strange attractors

We establish conditions for distinguishing between two topologically identical strange attractors that are enclosed by identical bounding tori, one of which is generated by a flow restricted to that torus, the other of which is generated by a flow in a different bounding torus and either imaged or lifted into the first bounding torus.

Figure 1

First return map for the Rössler attractor. Parameter values: $(a,b,c)=(0.432,2.0,4.0)$.

Figure 2

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 components of the Poincaré section shows the branched manifold has four branchs. Tearing occurs. Parameter values: $(R,\sigma ,b)=(28.0,10.0,8∕3)$.

Figure 3

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 components of the Poincaré section shows the branched manifold has six branches. Tearing and folding occur. Parameter values: $(R,\sigma ,b)=(65.584,13.0,2.4167)$.

Figure 4

Projection of the Lorenz attractor onto (a) the $x\text{−}z$ plane and (b) the $x\text{−}y$ plane. Parameter values: $(R,\sigma ,b)=(278.56,30.0,1.0)$.

Figure 5

First return maps for the Lorenz attractor shown in Fig. 4a. The map onto the two components $L$ and $R$ shows that one component suffices. (b) The return map on a single component shows that stretching and folding is the operative mechanism. Both maps show the flow has four branches.

Figure 6

(a) Projection of the $2\mapsto 1$ image of the Lorenz attractor onto the $v\text{−}w$ plane. The global Poincaré section has only the one component shown. First-return map on the Poincaré section shows the branched manifold has two branches. The return map is not differentiable everywhere, showing this is the image of a strange attractor where tearing occurs. Parameter values: $(R,\sigma ,b)=(28.0,10.0,8∕3)$.

Figure 7

(a) Projection of the $2\mapsto 1$ image of the Lorenz attractor onto the $v\text{−}w$ plane. The global Poincaré section has only the one component shown. First-return map on the Poincaré section shows the branched manifold has three branches. The return map is not differentiable everywhere, showing this is the image of a strange attractor where tearing occurs. The differentiable minimum shows that folding also occurs in the covering attractor. Parameter values: $(R,\sigma ,b)=(65.584,13,2.4167)$.

Figure 8

(a) Projection of the $2\mapsto 1$ image of the Lorenz attractor onto the $v\text{−}w$ plane. The global Poincaré section has only the one component shown. First-return map on the Poincaré section shows the branched manifold has two branches. The return map is differentiable everywhere, showing this is a strange attractor where folding occurs. Parameter values: $(R,\sigma ,b)=(278.56,30.0,1.0)$.

Figure 9

Bifurcation diagram for the image of the Lorenz attractor, with $\mid x\mid$ plotted as a function of $\rho$.

Figure 10

(a) This double cover of the Rösler attractor can be enclosed in a genus-3 bounding torus. The Poincaré section has two components. (b) First-return map on the Poincaré section has four panels. The discontinuity in the return map shows that this strange attractor is generated by stretching and tearing. The equality of the slopes at the discontinuity shows that it is the symmetric lift of a strange attractor generated in a bounding torus of genus 1 and that the two components of the Poincaré section are related by this symmetry. This strange attractor is described by a branched manifold with four branches. Parameter values: $(a,b,c)=(0.432,2.0,4.0)$.

Figure 11

(a) Another double cover of the Rösler attractor. This can also be enclosed in a genus-3 bounding torus. (b) The discontinuity in the return map shows that this strange attractor is generated by stretching and tearing. The differentiable maxima show that folding also takes place. The equality of the slopes at the discontinuity shows that it is the symmetric lift of a strange attractor generated in a bounding torus of genus 1. This strange attractor is described by a branched manifold with six branches. Parameter values: $(a,b,c)=(0.432,2.0,4.0)$.

Figure 12

(a) One double cover of the Rössler attractor that can be enclosed in a genus-1 bounding torus. The Poincaré section has a single component. (b) First-return map on the Poincaré section. This strange attractor is generated by stretching and folding, and is described by a branched manifold with four branches. Parameter values: $(a,b,c)=(0.432,2.0,4.0)$.