Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects gives rise to rich dynamics: this extends from Poincaré-Bendixson–type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections or to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, and chaotic spatiotemporal structures. Suitable low-dimensional phase-space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

Figure 1

The scenario $\omega \left(u_0\right)=\left\{{\varphi }_b\right\}$. Left: Convergence to fixed point $\mathbf{A}$. Right: Fixed point $\mathbf{A}$ as a limit circle of radius $\sqrt{-\gamma /\delta }$.

Figure 2

The scenario $\omega \left(u_0\right)=\left\{{\mathrm{\Phi }}_p\right\}$. Upper panels: Density snapshots at times (a) $t\approx 500$, (b) $t\approx 683$, and (c) $t\approx 700$. Lower panels: Orbital connections $\mathbf{O}\to \mathbf{A}\to \mathbf{B}$ in 2D (left) and 3D (right) spaces.

Figure 3

Dynamics scenario $\omega \left(u_0\right)=\mathbf{L}$, i.e., a space-time periodic traveling wave. Upper panels: Density snapshots at times (a) $t\approx 135$, (b) $t\approx 150$, and (c) $t\approx 180$. Lower panels: Convergence $\mathbf{O}\to \mathbf{A}\to \mathbf{L}$, the limit cycle in 2D (left) and 3D (right) spaces.

Figure 4

Dynamics scenario $\omega \left(u_0\right)=\mathbf{L}$, in the presence of third-order dispersion only, namely, for $\beta =0.02$ and $\mu =\nu ={\sigma }_R=0$. Upper panel: Density snapshots at times (a) $t\approx 400$, (b) $t\approx 420$, and (c) $t\approx 450$, for a single-cw initial condition of $K=15,\epsilon =0.01$, and $\beta =0.02$. Lower panels: Convergence $\mathbf{O}\to \mathbf{A}\to \mathbf{L}$, i.e., the stable limit cycle, in the 2D (left) and 3D (right) spaces.

Figure 5

Birth of a chaotic attractor $\omega \left(u_0\right)=\mathbf{S}$. Transition from the instability of the cw steady state $\mathbf{A}$, to quasiperiodic behavior, and then to chaotic behavior for $t\in [0,330]$.

Figure 6

Upper panels: A chaotic path in $\mathbf{S}$ for $t\in [180,200]$ (left) and projection in 3D space ${\mathcal{P}}_3$ of the invariant torus-like set $\mathbf{Q}$ for $t\in [1800,2000]$ (right). Lower panels: Chaotic waveforms, corresponding to point $\mathbf{P}\mathbf{1}$ at time $t\approx 150$ (left) and point $\mathbf{P}\mathbf{2}$ at time $t\approx 165$ (middle), and a quasiperiodic solution in $\mathbf{Q}$ at time $t\approx 1900$ (right).

Figure 7

From top to bottom. First row: $\beta -{\left|\right|u\left|\right|}_{\infty }$ bifurcation diagram (Diagnostic I) for fixed ${\sigma }_R=\mu =\nu =0.01$ and the cw initial conditions of $\epsilon =0.01$ and $K=5$. Second row: Magnification of the quasiperiodic region DE shown in the preceding panel. Third row: Profiles of the distinct steady states involved in the orbital connection $\mathbf{E}\mathbf{1}\to \mathbf{E}\mathbf{2}\to \mathbf{E}\mathbf{3}$ occurring at $\beta =0.3$ in the metastable region BC. The system is at rest in steady state $\mathbf{E}\mathbf{1}$ for $5\lesssim t\lesssim 35$, in $\mathbf{E}\mathbf{2}$ for $60\lesssim t\lesssim 68$, and in $\mathbf{E}\mathbf{3}$ for $100\lesssim t\lesssim 3000$, which is the end of integration. Fourth row: Transition from an unstable periodic orbit $\left(\mathbf{PO}\right)$ to chaotic oscillations $\left(\mathbf{CH}\right)$ which are eventually damped to steady state $\mathbf{E}\mathbf{3}$. The unstable periodic orbit $\left(\mathbf{PO}\right)$ survives for $62\lesssim t\lesssim 105$, and the chaotic orbit for $120\lesssim t\lesssim 203$. The system is at rest in steady state $\mathbf{E}\mathbf{3}$ for $217\lesssim t\lesssim 3000$, which is the end of integration.

Figure 8

$\beta -\left|\right|u\left(T_{\max }\right){\left|\right|}_{\alpha }^2$ bifurcation diagram (Diagnostic II) for fixed ${\sigma }_R=\mu =\nu =0.01$ and the cw initial conditions of $\epsilon =0.01$ and $K=5$.

Figure 9

Upper panel: ${\sigma }_R-\left|\right|u\left(T_{\max }\right){\left|\right|}_{\alpha }^2$ bifurcation diagram (Diagnostic II) for fixed $\beta =0.52,\mu =\nu =0.01$, and the cw initial conditions of $\epsilon =0.01$ and $K=5$. Lower-left panel: A chaotic path for $t\in [600,650]$, when $\beta =0.53,{\sigma }_R=\mu =\nu =0$, and the initial conditions are as in the upper panel. Lower-right panel: Chaotic waveforms corresponding to points $\mathbf{P}\mathbf{1}$ at $t\approx 600$ (top), $\mathbf{P}\mathbf{2}$ at $t\approx 625$ (middle), and $\mathbf{P}\mathbf{3}$ at $t\approx 650$ (bottom).