Estimating the Dimension of an Inertial Manifold from Unstable Periodic Orbits

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for the Kuramoto-Sivashinsky system and find it to be equal to the “physical dimension” computed previously via the hyperbolicity properties of covariant Lyapunov vectors.

Figure 1

(a) Floquet exponents for ${\overline{\mathrm{ppo}}}_{10.25}$ (circles), ${\overline{\mathrm{rpo}}}_{16.31}$ (squares), and Lyapunov exponents of a chaotic trajectory (crosses). The inset shows a close-up of the eight leading exponents. ${\overline{\mathrm{ppo}}}_{10.25}$ and ${\overline{\mathrm{rpo}}}_{16.31}$ have, respectively, two and one positive Floquet exponents: ${\lambda }_{1,2}=0.033$ and ${\lambda }_1=0.328$. Only one Lyapunov exponent is positive: ${\lambda }_1=0.048$. The number of vanishing exponents is always two. The fourth Lyapunov exponent is small but strictly negative: ${\lambda }_4=-0.003$. (b) Time series of local Floquet exponents ${\lambda }_j\mathbf{(}\mathbf{x}(t)\mathbf{)}$ for ${\overline{\mathrm{ppo}}}_{10.25}$. (c) Close-up of (b) showing the eight leading exponents. Dashed lines indicate degenerate exponent pairs corresponding to complex Floquet multipliers.

Figure 2

A histogram of the principal angles $\varphi$ between $S_n$ (the subspace spanned by the $n$ leading Floquet vectors) and ${\overline{S}}_n$ (the subspace spanned by the remaining $d-n$ Floquet vectors), accumulated over the 400 orbits used in our analysis. (Top panel) For $n=1,2,\dots ,7$ ($S_n$ within the entangled manifold), the angles can be arbitrarily small. (Bottom panel) For $n=8,10,12,\dots ,28$ (in the order of the arrow), for which all entangled modes are contained in $S_n$, the angles are bounded away from unity.

Figure 3

(a) Shadowing event between a chaotic trajectory and ${\overline{\mathrm{ppo}}}_{33.39}$, drawn over $2T_p$. (b) Parametric plot of $\sin {\phi }_n(t)$ vs $\parallel \mathrm{\Delta }\widehat{u}(x,t)\parallel$ during the single shadowing event shown in (a), for $n=6$, 7, 8. (c) The same as (b), but a total of 230 shadowing events of ${\overline{\mathrm{ppo}}}_{33.39}$ are used. (d) Average of $\sin {\phi }_n$ in (c), taken within each bin of the abscissa, for $n=4$, 5, 6, 7, 9, 11, 17, 21, 25 from top to bottom. (e),(f) The same as (c),(d), respectively, but for 217 shadowing events with ${\overline{\mathrm{rpo}}}_{34.64}$. The dashed lines show $\sin {\phi }_n\propto \parallel \mathrm{\Delta }\widehat{u}\parallel$ in all panels.