Anticipated synchronization in coupled inertial ratchets with time-delayed feedback: A numerical study

We investigate anticipated synchronization between two periodically driven deterministic, dissipative inertial ratchets that are able to exhibit directed transport with a finite velocity. The two ratchets interact through a unidirectional delay coupling, one is acting as a master system while the other one represents the slave system. Each of the two dissipative deterministic ratchets is driven externally by a common periodic force. The delay coupling involves two parameters, the coupling strength and the (positive-valued) delay time. We study the synchronization features for the unbounded, current carrying trajectories of the master and the slave, respectively, for four different strengths of the driving amplitude. These in turn characterize differing phase space dynamics of the transporting ratchet dynamics, regular, intermittent and a chaotic transport regime. We find that the slave ratchet can respond in exactly the same way as the master will respond in the future, thereby anticipating the nonlinear directed transport.

Figure 1

The dimensionless periodic ratchet potential used in our simulations.

Figure 2

(Color online) Trajectories for the master [gray (red)] and slave (black) ratchets in a typical synchronization scenario. The magnified part of the trajectory reveals the anticipation effect. The parameters are $a=0.08$, $b=0.1$, $\omega =0.67$, $K=0.6$, $\tau =1.3$.

Figure 3

Parameter space indicating the stability region for the anticipated synchronization to take place. We depict four cases, one with intermittent, directed transport dynamics at a driving amplitude of $a=0.0823$, one with a chaotic transport behavior at $a=0.08$, and two regular situations with $a=0.074$ and $a=0.081$, respectively. The remaining parameters are as follows: $b=0.1$, $\omega =0.67$. Within the white area the slave exhibits exponentially growing oscillations. The gray scale represents the quality parameter $p$ (defined in the text). Four gray tones from light to dark gray correspond to the ranges of $p$, 0, …, 0.25; 0.25, …, 0.5; 0.5, …, 0.75; and 0.75, …, 1, respectively. Each point in this plot was obtained by averaging the ratio $p$ over 40 trajectories of the system (5) with random initial conditions. Each trajectory has a length of 20 000 time units. Every plot consumed about 200 h of CPU on a typical workstation. The bold line exhibits the necessary stability condition $K\tau =\pi ∕2$ resulting from the linear part of the delay equation, see also the text.

Figure 4

(Color online) Regular regime at the driving amplitude $a=0.074$. (a) Trajectories for the master (light gray) and slave [dark gray (red)] ratchet dynamics. (b) The logarithm of the absolute value of the difference between the future position of the master and the present one of the slave. In the depicted scenario the slave already synchronizes to the transient of the master; later both systems together reach the growing, regular (period-2) orbit, see text. The parameters are $a=0.074$, $b=0.1$, $\omega =0.67$, $K=0.2$, $\tau =1.8$.

Figure 5

(Color online) Regular regime for a driving amplitude $a=0.081$. (a) Trajectories for the master [light gray (green)] and slave [dark gray (red)] ratchets. (b) The logarithm of the absolute value of the difference between the respective positions. In this case the master has already reached the regular orbit. The slave occasionally synchronizes and desynchronizes again. The parameters are $a=0.081$, $b=0.1$, $\omega =0.67$, $K=0.0981$, $\tau =3.4$. The symbol $v$ stands for the average velocity of the ratchet dynamics. The trajectories are depicted in the frame moving with velocity $v\simeq -0.025$ in order to pronounce the oscillations.

Figure 6

(Color online) Directed transport in the chaotic regime with a driving strength of $a=0.08$. Depicted are the trajectories for the master and slave (upper curves) light gray and dark gray (red) lines, respectively, and the logarithm of the absolute value of the difference between the two positions (lower curves). The slave occasionally synchronizes with, and subsequently desynchronizes again from the master. The distance $\ln \mid X(t+\tau )-x\left(t\right)\mid$ exhibits a random-walk-like pattern. The magnification in the upper panel depicts a short desynchronization event at $t=500$. The parameters are $a=0.08$, $b=0.1$, $\omega =0.67$, $K=0.305$, $\tau =1.6$.

Figure 7

(Color online) Intermittent transport regime for the driving strength $a=0.0823$. (a) Trajectories for the master (black line) and the slave [gray (red) line]. (b) The logarithm of the absolute value of the difference between the two positions. The slave occasionally desynchronizes from the master. The parameters are $a=0.0823$, $b=0.1$, $\omega =0.67$, $K=0.305$, $\tau =1.6$.