Reflection-antisymmetric spatiotemporal chaos under field-translational invariance

We propose a route to spatiotemporal chaos, in which the system is assumed to have spatial reflection antisymmetry and field-translation symmetry. The lowest-order nonlinear equation that satisfies these symmetries is explored with the weak nonlinear analysis around the bifurcation point. We conclude that the nonlinear term ${\partial }_x^{2}u{\partial }_x^{3}u$ is important to make a nontrivial dynamics, and show that the nonlinear dynamical equation having this term produces a turbulent dynamics.

Figure 1

Transition of the eigenvalue spectrum, $\lambda =-0.3{\epsilon }^{\nu }{q}^2+\epsilon q^4-0.01q^6$. Here, the value of the exponent $\nu$ is set to $\nu =0.3$. Solid lines are eigenvalue spectrums for $\epsilon =0.15,0.13,0.11$, and $0.09$, in which the darker line is obtained when the value of $\epsilon$ is smaller.

Figure 2

Spatio-temporal dynamics of the solution of Eq. (7). (a) Steady wave ($L=18.0$). (b) Traveling wave ($L=12.5$). (c) Heteroclinic oscillation ($L=13.7$). (d) Heteroclinic oscillation ($L=29.5$). (e) Turbulized dynamics ($L=21.5$). Here, we use the spectral algorithm with a fourth order exponential time differencing Runge-Kutta method (ETDRK4) [29], where the whole mode number is $N=512$.

Figure 3

(a) Time-averaged power spectrum. The spectrum higher than the $n=10$ mode continues to exponentially decrease (not shown here). (b) Frequency spectrum of low-wavenumber modes: $n=1,3,5,7$, and 9. These are obtained by numerically solving Eq. (7) with the same parameter values as those used in Fig. 2e.