Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems

We propose a route to spatiotemporal chaos for one-dimensional stationary patterns, which is a natural extension of the quasiperiodicity route for low-dimensional chaos to extended systems. This route is studied through a universal model of pattern formation. The model exhibits a scenario where stationary patterns become spatiotemporally chaotic through two successive bifurcations. First, the pattern undergoes a subcritical Andronov-Hopf bifurcation leading to an oscillatory pattern. Subsequently, a secondary bifurcation gives rise to an oscillation with an incommensurable frequency with respect to the former one. This last bifurcation is responsible for the spatiotemporally chaotic behavior. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.

Figure 1

Spatiotemporal diagram of the field $u(x,t)$ observed in the nonvariational generalization of the Swift-Hohenberg model, Eq. (1), $\eta =-0.04,\epsilon =-0.0921,\nu =-1,$ $b=-3.5$, with a discretization of 512 points, $dx=0.6$, $dt=0.01$, (a) $c=4.5$, (b) $c=5.5$, and (c) $c=10$ show, respectively, stationary, oscillatory, and spatiotemporal chaotic patterns. The lower panels show the temporal evolution of two arbitrary spatial points of the respective spatiotemporal diagram.

Figure 2

spatiotemporal dynamics of one-dimensional pattern of Eq. (1), in the parameter space $\eta =-0.04$, $\epsilon =-0.0921$, $\nu =-1$, $b=-4.0$, and $c=10$ with a discretization of 256 points, $dx=0.6$, $dt=0.01$. (a) Spatiotemporal diagram of $u(x,t)$. (b) Spatiotemporal diagram of the field $\zeta (x,t)$ obtained from a small localized initial difference of order ${10}^{-2}$. The respective spatiotemporal diagrams (c) and (d) correspond to an enlargement of the diagrams (a) and (b). The temporal evolution of the spatial norm of the difference field $\left|\zeta \right(t\left)\right|$ is defined by Eq. (3).

Figure 3

Characterization of extended chaos. (a) Lyapunov spectrum of Eq. (1) for different numbers of integration points, in the parameter space $\eta =-0.04$, $\epsilon =-0.0921$, $\nu =-1$, $b=-3.5$, and $c=10$. (b) Yorke-Kaplan dimension as function of the number of integration points. The linear growth of the Yorke-Kaplan dimension can be fitted by a slope of 0.0924 and intersects $-0.8087$. The spectrum was computed using $\gamma =160$ as stabilization value for Eq. (7).

Figure 4

Bifurcation diagram of Eq. (1). Largest Lyapunov exponent of Eq. (1) as a function of the nonlinear advection parameter $c$ for $\eta =-0.020$, $\epsilon =-0.092$, $\nu =-1.000$, and $b=-3.000$.

Figure 5

Numerical characterization of Andronov-Hopf bifurcation of Eq. (1) with specular boundary condition. (a) Sets of eigenvalues of the operator $\mathcal{L}\left[u_p\left(x\right)\right]$; (b) real part of the marginal mode; (c) periodic solution $u_p\left(x\right)$.

Figure 6

Spatiotemporal quasiperiodicity route for pattern of Eq. (1) as a function of nonlinear advection $c$ with $\eta =-0.020$, $\epsilon =-0.092$, $\nu =-1.000$, and $b=-3.000$. We show in the respective regions the typically observed frequency power spectrum $F\left(f\right)$ of the evolution of the spatial norm of the difference field $\left|\zeta \right(t\left)\right|$ using acquisition frequency $0.1$.

Figure 7

Hysteresis loop of the bifurcated solutions of Eq. (1) with $\eta =-0.020$, $\epsilon =-0.092$, $\nu =-1.000$, and $b=-3.000$. $A$ stands for the modulus of the amplitude of the critical marginal mode with spatial period $u_{2p}\left(x\right)$.

Figure 8

Five largest Lyapunov exponents as functions of the nonlinear advection parameter $c$. The colored areas represent the different regions of dynamic behaviors. The static pattern region is characterized by having only negative Lyapunov exponents. The oscillatory pattern region presents a plateau and its largest Lyapunov exponent is zero. Finally, the region of chaotic spatiotemporal pattern is characterized by the crossing of the horizontal axis by its Lyapunov exponents and thus becoming positive.