Effects of spatial frequency distributions on amplitude death in an array of coupled Landau-Stuart oscillators

Heterogeneity induced splay state of amplitude envelope in globally coupled oscillators

Splay states of the amplitude envelope are stably observed as a heterogenous node is introduced into the globally coupled identical oscillators with repulsive coupling. With the increment of the frequency mismatches between the heterogenous nodes and the rest identical globally coupled oscillators, the formal stable splay state based on the time series becomes unstable, while a splay state based on the new-born amplitude envelopes of time series is stably observed among the rest identical oscillators. The characteristics of the splay state based on the amplitude envelope are numerically and theoretically presented for different parameters of the coupling strength ϵ and the frequency mismatches Δω for small coupling strength and large frequency mismatches. We expect that all these results could reveal the generality of splay states in coupled nonidentical oscillators and help to understand the rich dynamics of amplitude envelopes in multidisciplinary fields.

Sinusoidal and nonsinusoidal patterns in amplitude envelope synchronization

In this work, amplitude envelope synchronization (AES), as a general phenomenon characterized with highly correlated amplitude envelope but uncorrelated phases and frequencies in coupled nonidentical nonlinear systems, is investigated theoretically and numerically. Two different types of AES patterns, including sinusoidal and nonsinusoidal, are widely observable in coupled periodic and/or chaotic oscillators. They both come from modulation of phase mismatch on amplitude but show different patterns due to different behaviors of phase mismatch. With increase of frequency mismatch, the system tends to crossover from nonsinusoidal to sinusoidal AES. With the aid of synchronization manifold and transverse stability analyses of the AES state, the physical mechanism and scale relations for the AES are well revealed. We expect that all these results could uncover the generality of AES in coupled nonlinear oscillators and help to understand the rich dynamics of phase and amplitude coupling in multidisciplinary fields.

Frequency and Phase Characteristics of Candle Flame Oscillation

The combustion of candles exhibits a variety of dynamical behaviors. Binding several candles together will result in flickering of candle flames, which is generally described as a nonlinear oscillator. The impact on the frequency of the flame by several factors, such as the arrangement, the number and the asymmetry of the oscillators, is discussed. Experimental results show that the frequency gradually decreases as the number of candles increases in the case of an isolated oscillator, while alternation between the in-phase and the anti-phase synchronization appears in a coupled system of two oscillators. Moreover, envelopes in the amplitude of the oscillatory luminance are displayed when candles are coupled asymmetrically. Since the coupling between oscillators is dominated by thermal radiation, a “overlapped peaks model” is proposed to phenomenologically explain the relationship between temperature distribution, coupling strength and the collective behavior in coupled system of candle oscillators in both symmetric and asymmetric cases.

Stabilization of a spatially uniform steady state in two systems exhibiting Turing patterns

This paper deals with the stabilization of a spatially uniform steady state in two coupled one-dimensional reaction-diffusion systems with Turing instability. This stabilization corresponds to amplitude death that occurs in a coupled system with Turing instability. Stability analysis of the steady state shows that stabilization does not occur if the two reaction-diffusion systems are identical. We derive a sufficient condition for the steady state to be stable for any length of system and any boundary conditions. Our analytical results are supported with numerical examples.

The impact of propagation and processing delays on amplitude and oscillation deaths in the presence of symmetry-breaking coupling

We numerically investigate the impacts of both propagation and processing delays on the emergences of amplitude death (AD) and oscillation death (OD) in one system of two Stuart-Landau oscillators with symmetry-breaking coupling. In either the absence of or the presence of propagation delay, the processing delay destabilizes both AD and OD by revoking the stability of the stable homogenous and inhomogenous steady states. In the AD to OD transition, the processing delay destabilizes first OD from large values of coupling strength until its stable regime completely disappears and then AD from both the upper and lower bounds of the stable coupling interval. Our numerical study sheds new insight lights on the understanding of nontrivial effects of time delays on dynamic activity of coupled nonlinear systems.

Amplitude death in a pair of one-dimensional complex Ginzburg-Landau systems coupled by diffusive connections

This paper shows that, in a pair of one-dimensional complex Ginzburg-Landau (CGL) systems, diffusive connections can induce amplitude death. Stability analysis of a spatially uniform steady state in coupled CGL systems reveals that amplitude death never occurs in a pair of identical CGL systems coupled by no-delay connection, but can occur in the case of delay connection. Moreover, amplitude death never occurs in coupled identical CGL systems with zero nominal frequency. Based on these analytical results, we propose a procedure for designing the connection delay time and the coupling strength to induce spatial-robust stabilization, that is, a stabilization of the steady state for any system size and any boundary condition. Numerical simulations are performed to confirm the analytical results.

Experimental and modeling analysis of asymmetrical on-off oscillation in coupled non-identical inverted bottle oscillators

Upside-down bottles containing water which are common in our daily life exhibit rich vibration dynamics. Rich dynamic regimes are observed in bottle oscillators by directly measuring the pressure difference between inside and outside of a bottle with the aid of pressure sensors. We observe experimentally that an asymmetrical oscillation process between the outflow of water and the inflow of air is formed in a single bottle oscillator and, in addition, a kind of 2:1 frequency synchronization occurs in a coupled system of two non-identical bottle oscillators. The peak values of the oscillation of pressure differences between inside and outside of the bottle decease as the height of the liquid surface steps down, while the oscillation period increases gradually. The theoretical model of the oscillator is amended to understand the regimes in the experiment by introducing time-dependent parameters related to the asymmetrical oscillation processes. Our numerical results based on the model fit well with the experimental ones.

Reviving oscillation with optimal spatial period of frequency distribution in coupled oscillators

The spatial distributions of system's frequencies have significant influences on the critical coupling strengths for amplitude death (AD) in coupled oscillators. We find that the left and right critical coupling strengths for AD have quite different relations to the increasing spatial period m of the frequency distribution in coupled oscillators. The left one has a negative linear relationship with m in log-log axis for small initial frequency mismatches while remains constant for large initial frequency mismatches. The right one is in quadratic function relation with spatial period m of the frequency distribution in log-log axis. There is an optimal spatial period m₀ of frequency distribution with which the coupled system has a minimal critical strength to transit from an AD regime to reviving oscillation. Moreover, the optimal spatial period m₀ of the frequency distribution is found to be related to the system size N. Numerical examples are explored to reveal the inner regimes of effects of the spatial frequency distribution on AD.

Oscillator death induced by amplitude-dependent coupling in repulsively coupled oscillators

The effects of amplitude-dependent coupling on oscillator death (OD) are investigated for two repulsively coupled Lorenz oscillators. Based on numerical simulations, it is shown that as constraint strengths on the amplitude-dependent coupling change, an oscillatory state may undergo a transition to an OD state. The parameter regimes of the OD domain are theoretically determined, which coincide well with the numerical results. An electronic circuit is set up to exhibit the transition process to the OD state with an amplitude-dependent coupling. These findings may have practical importance on chaos control and oscillation depression.

Mobility and density induced amplitude death in metapopulation networks of coupled oscillators

We investigate the effects of mobility and density on the amplitude death of coupled Landau-Stuart oscillators and Brusselators in metapopulation networks, wherein each node represents a subpopulation occupied any number of mobile individuals. By numerical simulations in scale-free topology, we find that the systems undergo phase transitions from incoherent state to amplitude death, and then to frequency synchronization with increasing the mobility rate or density of oscillators. Especially, there exists an extent of intermediate mobility rate and density that can lead to global oscillator death. Furthermore, we show that such nontrivial phenomena are robust to diverse network topologies. Our findings may invoke further efforts and attentions to explore the underlying mechanism of collective behaviors in coupled metapopulation systems.