Hyperchaotic Intermittent Convection in a Magnetized Viscous Fluid

We consider a low-dimensional model of convection in a horizontally magnetized layer of a viscous fluid heated from below. We analyze in detail the stability of hydrodynamic convection for a wide range of two control parameters. Namely, when changing the initially applied temperature difference or magnetic field strength, one can see transitions from regular to irregular long-term behavior of the system, switching between chaotic, periodic, and equilibrium asymptotic solutions. It is worth noting that owing to the induced magnetic field a transition to hyperchaotic dynamics is possible for some parameters of the model. We also reveal new features of the generalized Lorenz model, including both type I and III intermittency.

Figure 1

Dependence of the largest Lyapunov exponent ${\lambda }_1$ (color coded) on ${\omega }_0$ and $r$ parameters of the generalized Lorenz model for (a) ${\sigma }_{\mathrm{m}}=0.1$, (b) ${\sigma }_{\mathrm{m}}=1$, and (c) ${\sigma }_{\mathrm{m}}=3$. Other parameters of the system have fixed values: $\sigma =10$, $b=8/3$. Convergence of the solutions of Eqs. (4, 5, 6, 7) to fixed points (${\lambda }_1<0$) is shown in black, to periodic solutions (${\lambda }_1=0$)—in violet/blue color (see the color bar for ${\lambda }_1=0$), to chaotic solutions (${\lambda }_1>0$)—in a color, consistently with the color bar scale, from violet/blue to yellow. Fine structures are shown in the insets.

Figure 2

Dependence of the two largest Lyapunov exponents ${\lambda }_1$ (dashed line) and ${\lambda }_2$ (solid line), ${\lambda }_1>{\lambda }_2$, on parameter $r$.

Figure 3

Distribution of the lengths of laminar phases for $r=256$, ${\omega }_0=3.74$, ${\sigma }_{\mathrm{m}}=1$.

Figure 4

Scaling of the mean length of the laminar phase with control parameter $\epsilon =|{\omega }_0-{\omega }_{0c}|$ for the case shown in Fig. 3, where ${\omega }_{0c}$ is a critical value at which intermittency appears.