Slow driving induced multistability and remote synchronization in chaotic Chua's circuit

TL;DR

The paper investigates how driving a chaotic Chua's circuit with a chaotic signal of variable time-scale affects synchronization and multistability. Using an auxiliary-system framework, it uncovers generalized synchronization and multiple remote synchronization states that arise when the drive is slow compared to the response, including complete, lag, and correlated synchronization forms. The study reveals slow driving induces bistability between single-scroll attractors and enables coexisting RS states, with transitions to GS and CS as the drive frequency approaches the response frequency. These findings enhance understanding of time-scale mismatch in drive–response chaotic systems and hint at rich collective dynamics in networks with hub-like coupling structures.

Abstract

We study the response of Chua's circuit driven by a chaotic signal of variable time-scale. We observe that when the frequency of the drive is significantly lower than that of the response and the driving strength is above a threshold, the Chua's circuit exhibits multiple stable attractors. The features of the attractors change as the driving strength ε increases, for instance the attractors are double-scroll at low ε and are single-scroll when ε is high. We also investigate generalized synchronization(GS) between the drive and the response systems by employing the auxiliary system approach. When the drive is much slower than the response, we observe different scenarios of remote synchronization(RS) between response and auxiliary units. In addition to complete synchrony between response and auxiliary systems indicating GS between drive and response, we notice that the response and auxiliary units can be lag synchronized and can also have correlated trajectories indicating novel forms of RS. The slow drive can induce multistability between these RS states which disappears as the frequency of drive increases and become equivalent to the response Chua's ciruit.

Figures (19)

• Figure 1: (a)Circuit diagram of Chua's oscillator and (b)V-I characteristics of Chua's diode.
• Figure 2: Upper panel shows the bifurcation diagram of Chua's oscillator obtained by plotting the maxima, $V_P$ in the time-series of $V_1$ as $R$ varies showing period-doubling route to chaos and regions of existence of different types of attractors: double-scroll chaotic (A), single-scroll chaotic (B), period-4 (C), period-2 (D), and period-1 (E). The attractors in different regions are shown in the lower panel at $R$ = 1920 $\Omega$ (A), 1950 $\Omega$ (B), 1963.5 $\Omega$ (C), 1970 $\Omega$ (D), and 2000 $\Omega$ (E) for a fixed initial condition icfig3. Please note that we have presented only one variant of the single-scroll attractors in the bistable region(B to E).
• Figure 3: Circuit diagram of Chua's circuit driven by the common base Colpitts oscilator.
• Figure 4: The attractor in the $V'_1-V'_2$ plane for the Colpitts oscillator used to drive the Chua's circuit.
• Figure 5: Time-series of (a) $V_1$ of the response Chua's circuit at $R$ = 1920 $\Omega$, and driving signal ($V'_2$) at different values of time-scale parameter, $\Phi$ obtained from circuit simulation.