Comparison between covariant and orthogonal Lyapunov vectors

Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover, the previous classification of two universality classes works for CLVs as well, i.e., the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems, respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.

Figure 1

(Color online) Contour plot of the static CLV structure factors for [(a) and (c)] Hamiltonian and [(b) and (d)] dissipative coupled map lattices. The local dynamics used are [(a) and (b)] the skewed tent map and [(c) and (d)] the sinusoidal map, respectively. Other parameters are $ϵ=1.3$ and $r=0.15$. A ridge structure can be clearly seen in the small $\left(\lambda ,k\right)$ regime which indicates the existence of HLMs.

Figure 2

(Color online) Dispersion relations $\lambda \text{−}{k}_{max}$ obtained from CLVs and OLVs for (a) dissipative and (b) Hamiltonian systems, respectively. The local map is the skewed tent map with $ϵ=1.3$ and $r=0.15$. Note the perfect agreement between data from CLVs and OLVs for the highly hyperbolic cases shown here.

Figure 3

(Color online) Similar to Fig. 2 but the local map is the sinusoidal map and $ϵ=1.3$. Note that for the nonhyperbolic cases shown in this figure CLVs and OLVs still have the same asymptotic behavior.

Figure 4

(Color online) Dominant wave number $k_{max}$ and the significance measure $S\left(k_{max}\right)$ of CLVs and OLVs for [(a) and (b)] dissipative and [(c) and (d)] Hamiltonian systems, respectively. The local map is the skewed tent map with $ϵ=1.3$ and $r=0.15$. The long-wavelength structure is as significant in CLVs as in OLVs for the highly hyperbolic cases shown.

Figure 5

(Color online) Similar to Fig. 4 but the local map is the sinusoidal map and $ϵ=1.3$. The significance of HLM is strongly reduced in CLVs of nonhyperbolic dissipative systems, whereas the HLM significance is comparable for CLVs and OLVs in nonhyperbolic Hamiltonian systems.

Figure 6

(Color online) Influence of the hyperbolicity variation on the significance measure $S\left(k_{max}\right)$ of HLMs in the dissipative system. The parameter $r$ is (a) 0.2, (b) 0.3, and (c) 0.4, respectively, which corresponds to a decreasing hyperbolicity. The local map is the skewed tent map and $ϵ=1.3$.

Figure 7

(Color online) Influence of the hyperbolicity variation on the significance measure $S\left(k_{max}\right)$ of HLMs in the Hamiltonian system. The parameter $r$ is (a) 0.2, (b) 0.3, and (c) 0.4, respectively, which corresponds to a decreasing hyperbolicity. The local map is the skewed tent map and $ϵ=1.3$.

Figure 8

(Color online) Static LV structure factor corresponding to the smallest positive Lyapunov exponent of the Hamiltonian case. The local map is the sinusoidal map. The large difference between these specific CLVs and OLVs is due to the asymmetric location of the largest value of $S\left(k_{max}\right)$ of OLVs as shown in Fig. 5b.

Figure 9

(Color online) Variation of the significance measure $S\left(k_{max}\right)$ of HLMs with the system size $N$ for the Hamiltonian case with the sinusoidal map. The peak moves gradually to the spectral center point $\alpha /2N=0.5$ as increasing $N$.

Figure 10

Instantaneous profiles of two conjugate pairs of CLVs and OLVs for the Hamiltonian case. The local map is the skewed tent map with $ϵ=1.3$ and $r=0.15$. The system size used is $N=128$. Note that OLVs and CLVs have different symmetry relations for the conjugate pair.

Figure 11

(Color online) Comparison of the dominant wave number $k_{max}$ and $S^{\left(\alpha \right)}\left(k_{max}\right)$ obtained from the coordinate and momentum parts of [(a) and (b)] OLVs and [(c) and (d)] CLVs, respectively. The local map of the studied Hamiltonian system is the skewed tent map with $ϵ=1.3$ and $r=0.15$.

Figure 12

(Color online) Similar to Fig. 11 but the local map is the sinusoidal map and $ϵ=1.3$.

Figure 13

(Color online) (a) Contour plot of $⟨\text{cos}\left(\theta \right)⟩$ of CLVs for the Hamiltonian case. The local map is the skewed tent map with $ϵ=1.3$ and $r=0.15$. (b) shows the enlargement of the central part of (a). Note that in the regime $\lambda \sim 0$ CLVs are nearly mutual orthogonal besides that the conjugate pair of CLVs has a very small angle.

Figure 14

(Color online) The quantity $⟨\text{cos}\left(\theta \right)⟩$ of angles between the conjugate pair (square) and neighboring CLVs (circle) of the Hamiltonian system. The local map is (a) the skewed tent map with $r=0.15$ and (b) the sinusoidal map. The coupling strength is $ϵ=1.3$. Switching to the nonhyperbolic case the orthogonality between neighbors CLVs in the regime $\lambda \sim 0$ is broken.

Figure 15

(Color online) Instantaneous profiles of CLVs of two systems related via variable transformation (B15). As shown in (d) CLVs of the two systems are related via the relation given in Eq. (B17).

Figure 16

Difference between finite-time Lyapunov exponents of two systems related via variable transformation (B15).