When are projections also embeddings?

We study an autonomous four-dimensional dynamical system used to model certain geophysical processes. This system generates a chaotic attractor that is strongly contracting, with four Lyapunov exponents ${\lambda }_i$ that satisfy ${\lambda }_1+{\lambda }_2+{\lambda }_3<0$, so the Lyapunov dimension is $D_L=2+\mid {\lambda }_3\mid ∕{\lambda }_1<3$ in the range of coupling parameter values studied. As a result, it should be possible to find three-dimensional spaces in which the attractors can be embedded so that topological analyses can be carried out to determine which stretching and squeezing mechanisms generate chaotic behavior. We study mappings into $R^3$ to determine which can be used as embeddings to reconstruct the dynamics. We find dramatically different behavior in the two simplest mappings: projections from $R^4$ to $R^3$. In one case the one-parameter family of attractors studied remains topologically unchanged for all coupling parameter values. In the other case, during an intermediate range of parameter values the projection undergoes self-intersections, while the embedded attractors at the two ends of this range are topologically mirror images of each other.

Figure 1

Plot of the four Lyapunov exponents ${\lambda }_j$ and the Lyapunov dimension $D_L$ for the chaotic attractor generated by Eq. (1) as a function of $\beta$. For these values the attractor is effectively three dimensional. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.

Figure 2

$X\text{−}Y$ plane projection of the chaotic attractors generated by the EMR equations Eq. (1) for different $\beta$ values. 0.3 (a), 3.8 (b), and 7.9 (c). Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.

Figure 3

Two linked orbits projected onto a two-dimensional surface. Positive crossings are indicated by + and the negative crossing by a ●. The linking number of the orbits is half the sum of signed crossings: $L(RLLR,RLL)=(+5-1)∕2=+2$. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$ and $\beta =0.3$.

Figure 4

Branched manifolds for the Lorenz attractor with (a) rotation and (b) inversion symmetry. Branched manifolds describe the topological organization of all the unstable periodic orbits in a chaotic attractor. The two branched manifolds shown are group continuations of one another [10].

Figure 5

Chaotic attractor projected from $R^4(X,Y,Z,U)$ into $R^3(X,Y,Z)$ for three values of the parameter $\beta =\left(\mathrm{a}\right)$ 0.3, (b) 3.8, and (c) 7.9. Self-intersections are evident for $\beta =3.8$. The two projected attractors for $\beta =0.3$ and $\beta =7.9$ are described by branched manifolds that are mirror images of each other.

Figure 6

Differential embeddings based on the odd variables $X$, $Y$, and $U$. Projections on the $X\text{−}\dot{X}$ (a), $Y\text{−}\dot{Y}$ (b), and $U\text{−}\dot{U}$ (c) planes. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.

Figure 7

$Z\text{−}\dot{Z}$ plane projection of the differential embedding based on the even variable $Z$. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$ and $\beta =3.8$.

Figure 8

Plane projections of the image attractors $(X,Y,Z,U)\to (u,v,Z)$ with $u=X^2-Y^2$, $v=2XY$, for three values of $\beta =\left(\mathrm{a}\right)$ 7.9, (b) 3.8, and (c) 0.3. A great deal of twisting takes place in the neighborhood of the origin. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.

Figure 9

Graph showing the couplings between the dynamical variables of the EMR dynamo equations.

Figure 10

Unfolded schematic view of the variables reached when the first and second derivatives are computed. A nonlinearity in the first derivative (a dashed arrow between the observable and its first derivative) induces a more serious lack of observability than when a nonlinearity occurs in the second derivative (a dashed arrow between the first and the second derivatives of the observable).

Figure 11

(a) Intersections of the chaotic attractor with the two components of the global Poincaré surface of section $X=Y=\pm 9.5$, $\beta =3.8$. (b) First-return plot for the intersections of the attractor with the global Poincaré surface of section. (c) Return map based on parametrization of the intersection shown in (b), with $s$ parametrizing the left-hand intersection and $t$ parametrizing the right-hand intersection. Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.

Figure 12

For the image attractor shown in Fig. 8 with $\beta =7.9$: (a) Intersection with a Poincaré section (b) Return map based on coordinate $u_n$. The separation between the two cusps increases with $\beta$. (c) Return map based on parametrization of the intersection shown in (a). Parameter values: $(\sigma ,\nu ,R,\Lambda )=(10,8∕3,74.667,3.2)$.