Deterministic coherence and anti-coherence resonances in two coupled Lorenz oscillators: numerical study versus experiment

TL;DR

The study addresses how coherence-like resonances arise in deterministic chaotic systems without noise by examining two bidirectionally coupled identical Lorenz oscillators. It combines numerical simulations with an electronic analog to demonstrate that deterministic coherence resonance (DCR) and deterministic anti-coherence resonance (DACR) occur simultaneously as the coupling strength $K$ increases, within the on-off intermittency regime near the synchronization threshold. Laminar-phase statistics follow $N(\tau) \sim \alpha \tau^{-3/2}$ and $\langle \tau \rangle \sim (K^{crit}_2 - K)^{-1}$, while the correlation time $t_{cor}$ exhibits non-monotonic behavior: DCR peaks for $(x,y)$ at $K^{peak}_1$ and DACR dips for $(z)$ at $K^{peak}_2$, with numerical values $K^{peak}_1 \approx 1.8$ and $2.3$, and experimental values $K^{peak}_1 \approx 2.8$ and $3.8$. The results show good agreement between simulations and experiments and reveal a robust link between resonance phenomena and synchronization dynamics in chaotic systems, even in the absence of full synchronization in the experimental setup.

Abstract

We demonstrate the deterministic coherence and anti-coherence resonance phenomena in two coupled identical chaotic Lorenz oscillators. Both effects are found to occur simultaneously when varying the coupling strength. In particular, the occurrence of deterministic coherence resonance is revealed by analysing time realizations $x(t)$ and $y(t)$ of both oscillators, whereas the anti-coherence resonance is identified when considering oscillations $z(t)$ at the same parameter values. Both resonances are observed when the coupling strength does not exceed a threshold value corresponding to complete synchronization of the interacting chaotic oscillators. In such a case, the coupled oscillators exhibit the hyperchaotic dynamics associated with the on-off intermittency. The highlighted effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence and disclosing thereby the robustness of the observed phenomena.

Figures (6)

• Figure 1: Electronic model of two bidirectionally coupled chaotic Lorenz oscillators (see Eqs.(\ref{['eq:2']})): (a) Schematic illustration of the experimental setup where the oscillators' attractors are illustrated by projections in phase plane ($X$,$Y$); (b) Electronic circuit of each oscillator (both oscillators are assumed to be identical). Operational amplifiers are TL072CP. Analog integrator elements are $C=100$ nF and $R=10$ k$\Omega$; (c)-(d) Projections of the experimentally obtained single oscillator phase portrait (see Eqs. (\ref{['eq:2']}) at $K\equiv0$, $\sigma=10$, $\beta=8/3$ and $P=2.3$) in planes ($X$,$Y$) (panel (c)) and ($X$,$Z$) (panel (d)). Since oscillators 1 and 2 are assumed to be identical, indexes 1 and 2 in panels (c) and (d) are not specified.
• Figure 2: Evolution of the oscillatory dynamics in numerical model (\ref{['eq:1']}) when increasing the coupling strength: $K=0.6$ (panels (a) and (b)), $K=2.2$ (panels (c) and (d)) and $K=4.0$ (panels (e) and (f)). The dynamics evolution is illustrated by using time realizations $x_{1,2}(t)$ (left panels) and trajectories in phase plane ($x_{1}$, $x_2$) (right panels). The time periods corresponding to the laminar phase in panels (a) and (c) are coloured in grey. The oscillators' parameters are $\sigma=10$, $\rho=28$, $\beta=8/3$. Random initial conditions within the range [-1:1] are used.
• Figure 3: Evolution of the Lyapunov exponent spectrum in numerical model (\ref{['eq:1']}): panel (a) illustrates changes caused by increasing the coupling strength in range $K\in [0:10]$, panel (b) describes the dependencies $\lambda_{1,3,4}(K)$ in the areas delineated by the red dashed rectangles in panel (a). The range $K\in(K^{\text{crit}}_{1}:K^{\text{crit}}_{2})$ corresponds to the occurrence of the on-off intermittency and is coloured in grey. The oscillators' parameters are $\sigma=10$, $\rho=28$, $\beta=8/3$. Random initial conditions within the range [-1:1] are used.
• Figure 4: Evolution of the oscillatory regimes in electronic setup (\ref{['eq:2']}) when increasing the coupling strength: $K=0.6$ (panels (a) and (b)), $K=2.2$ (panels (c) and (d)) and $K=10.0$ (panels (e) and (f)). The dynamics evolution is illustrated by using time realizations $x_{1,2}(t)$ and trajectories in phase plane ($x_{1}$, $x_2$) similarly to Fig. \ref{['fig2']}. The time periods corresponding to the laminar phase in panels (a) and (c) are coloured in grey. The oscillators' parameters are $\sigma=10$, $P=2.3$ and $\beta=8/3$.
• Figure 5: Statistical characteristics of the on-off intermittency exhibited by numerical model (\ref{['eq:1']}) (panels (a) and (b)) and electronic setup (\ref{['eq:2']}) (panels (c) and (d)). Panels (a) and (c) illustrate the distribution of laminar phase lengths $N(\tau)$ at fixed coupling strength $K=2.2$, whereas panels (b) and (d) depict the dependencies of the mean laminar phase duration $<\tau>$ on the critical onset parameter $K_2^{\text{crit}}-K$. Since the mathematical model and experimental setup have different timescales, $\tau$ and $<\tau>$ in panels (c) and (d) are rescalled by $(RC)^{-1}$. Panels (a)-(d) contain red solid lines which represent the results of curve-fitting using the functions noted in the legends. The parameters estimated by means of curve-fitting are: $\alpha=0.4216$ (panel (a)), $\alpha=0.03663$ (panel (b)), $\alpha=0.2902$ (panel (c)) and $\alpha=0.03566$ (panel (d)). The parameter values of Eqs. (\ref{['eq:1']}) and (\ref{['eq:2']}) are $\sigma=10$, $\beta=8/3$, $\rho=28$ (numerical simulation), $P=2.3$ (physical experiment). Random initial conditions within the range [-1:1] are used for numerical simulations.