Sensitivity and stability of long periodic orbits of chaotic systems

The relationship between the stability of unstable periodic orbits of dissipative chaotic systems and the sensitivity of period averages to parameter perturbations is clarified. Using Floquet theory, we show that period averages vary most rapidly when structural perturbations of the equations of motion are well expressed by adjoint Floquet eigenfunctions associated to Floquet exponents of small magnitude, along invariant subspaces closest to marginality. Building on this analysis, we then construct a large inventory of periodic orbits of the Lorenz equations and of the Kuramoto-Sivashinsky system in a minimal-domain configuration, focusing on long periodic orbits spanning a large fraction of the chaotic attractor. We examine how the statistical distributions of Floquet exponents, period averages and their sensitivities to parameter perturbations vary as the period $T$ increases. We show that, at least for these two systems, the distribution of these quantities converges to a Dirac delta function as $T\rightarrow\infty$. We observe that the limiting value of period averages and Floquet exponents tend to the same asymptotic value obtained on long chaotic trajectories. Conversely, the limiting value of the sensitivity is not consistent with the response of long-time averages to finite-amplitude parameter perturbations, due to the lack of a linear response for non-hyperbolic dynamics.