Design strategies for generalized synchronization

We describe a general procedure to couple two dynamical systems so as to guide their joint dynamics onto a specific transversally stable invariant submanifold in the phase space. This method can thus be viewed as a means of constraining the dynamics, with the coupling functions providing the forces of constraint, which results in the coupled systems being in generalized synchronization. The required coupling functions are, however, not uniquely defined and can therefore be chosen in order to satisfy a desired design criterion.

Figure 1

Schematic of our general strategy. The surface represents the target synchronization manifold $\mathcal{M}\left(\mathrm{\Phi }\right)$. The points ${\mathbf{X}}_0,{\mathbf{X}}_A$ are on $\mathcal{M}\left(\mathrm{\Phi }\right)$, and also shown is ${\mathbf{N}}_i$, a normal to the surface. The coupling is designed to bring nearby trajectories (from ${\mathbf{X}}_B$, say) onto this submanifold in the phase space. Subsequently, the additional terms in the coupling keep the flow on $\mathcal{M}\left(\mathrm{\Phi }\right)$ (as indicated at ${\mathbf{X}}_A$) and, furthermore, constrain it to this surface.

Figure 2

Projective synchrony in bidirectionally coupled Lorenz systems for $a=2.6, b=1.8, c=3.6$. (These values are arbitrary: any scale factors can be achieved with this form of the coupling and the desired $a,b$, and $c$.) The $\mathbf{x}$ dynamics is shown in black, while the $\mathbf{y}$ dynamics is in blue. The axes are labeled only by the $\mathbf{x}$ variables.

Figure 3

Master-slave coupling in two Lorenz systems achieves the desired functional relationship, $y_3=x_3^2$, between the master and slave subsystems. The form of the coupling is, however, quite complex. As in Fig. 2, the axes are labeled by the $\mathbf{x}$ variables whose trajectory is shown in black, while the $\mathbf{y}$ dynamics is in green. Note the logarithmic scale on the vertical axis.

Figure 4

Trajectory on the synchronization submanifold. The parabolic dependence of $y_3$ on $x_3$ is quite evident.

Figure 5

Flipped synchrony, with $x_1=y_2$ and $y_1=x_2$, which automatically constrains $x_3$ to be identical to $y_3$. Trajectories are shown in projection, with the axes being labeled by the $\mathbf{x}$ variables whose trajectory is shown in black, while the $\mathbf{y}$ dynamics is in green.