Geometry of Inertial Manifolds Probed via a Lyapunov Projection Method

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a possible way to also determine experimentally inertial manifolds and their geometric characteristics.

Figure 1

The largest Lyapunov exponents (of 128) arranged in descending order for the Kuramoto-Sivashinsky equation, with $L=22$. The largest exponent has the value ${\lambda }^{(1)}=0.048$ [15]; the Kaplan-Yorke dimension of the attractor is about 5.2. The inset shows the distribution $p(\theta )$ of the manifold angle for several cutting dimensions $n$.

Figure 2

Scatter plots of the normalized projection errors $\widehat{s}(n,r)$ for varying distance $r$ between reference and recurrent states and for $n=7$, 8, 9, and 11 (superimposed), respectively. A line with slope 1 is shown to guide the eye.

Figure 3

The normalized average projection error $\widehat{S}(n,r)$ versus the distance to the reference state $r$ with $n$ from 4 to 11 (from top to bottom). Linear fits of the data for $n<M=9$ confirm the saturation of $\widehat{S}(n,r)$ to a nonzero value for $r\to 0$, in contrast to the case $n\ge M$.

Figure 4

Distributions of the projection error $\widehat{s}(n,r)$ for different $r$ (sampled equidistantly in the regime of Fig. 2) and two cases (a) $n=5<M$ and (b) $n=9=M$. (c) The normalized quantities $p(\widehat{s})/p({\widehat{s}}_{\mathrm{peak}})$ versus $\widehat{s}(n,r)/{\widehat{s}}_{\mathrm{peak}}$, with $n=5$ and 9. The data collapse for different $r$ in the small-$\widehat{s}$ regime indicates a common source of the projection errors represented by the small-$\widehat{s}$ peak appearing in all cases with different $n$. (d) The normalized quantities $p(\widehat{s})/p({\widehat{s}}_0)$ versus $1-\widehat{s}(n,r)$ for $n=5$. The value $s_0$ is arbitrarily chosen in the regime close to 1. The data collapse to a power law in the regime $\widehat{s}(n,r)\simeq 1$ indicates another common source of the projection error for the cases with $n<M$. The value of $r$ increases as indicated by the arrow.