Bistable chaos without symmetry in generalized synchronization

COVID-19 in Africa: Underreporting, demographic effect, chaotic dynamics, and mitigation strategy impact

The epidemic of COVID-19 has shown different developments in Africa compared to the other continents. Three different approaches were used in this study to analyze this situation. In the first part, basic statistics were performed to estimate the contribution of the elderly people to the total numbers of cases and deaths in comparison to the other continents; Similarly, the health systems capacities were analysed to assess the level of underreporting. In the second part, differential equations were reconstructed from the epidemiological time series of cases and deaths (from the John Hopkins University) to analyse the dynamics of COVID-19 in seventeen countries. In the third part, the time evolution of the contact number was reconstructed since the beginning of the outbreak to investigate the effectiveness of the mitigation strategies. Results were compared to the Oxford stringency index and to the mobility indices of the Google Community Mobility Reports.

Compared to Europe, the analyses show that the lower proportion of elderly people in Africa enables to explain the lower total numbers of cases and deaths by a factor of 5.1 on average (from 1.9 to 7.8). It corresponds to a genuine effect. Nevertheless, COVID-19 numbers are effectively largely underestimated in Africa by a factor of 8.5 on average (from 1.7 to 20. and more) due to the weakness of the health systems at country level. Geographically, the models obtained for the dynamics of cases and deaths reveal very diversified dynamics. The dynamics is chaotic in many contexts, including a situation of bistability rarely observed in dynamical systems. Finally, the contact number directly deduced from the epidemiological observations reveals an effective role of the mitigation strategies on the short term. On the long term, control measures have contributed to maintain the epidemic at a low level although the progressive release of the stringency did not produce a clear increase of the contact number. The arrival of the omicron variant is clearly detected and characterised by a quick increase of interpeople contact, for most of the African countries considered in the analysis.

Intermingled attractors in an asymmetrically driven modified Chua oscillator

Understanding the asymptotic behavior of a dynamical system when system parameters are varied remains a key challenge in nonlinear dynamics. We explore the dynamics of a multistable dynamical system (the response) coupled unidirectionally to a chaotic drive. In the absence of coupling, the dynamics of the response system consists of simple attractors, namely, fixed points and periodic orbits, and there could be chaotic motion depending on system parameters. Importantly, the boundaries of the basins of attraction for these attractors are all smooth. When the drive is coupled to the response, the entire dynamics becomes chaotic: distinct multistable chaos and bistable chaos are observed. In both cases, we observe a mixture of synchronous and desynchronous states and a mixture of synchronous states only. The response system displays a much richer, complex dynamics. We describe and analyze the corresponding basins of attraction using the required criteria. Riddled and intermingled structures are revealed.

Emergence of chimeras through induced multistability

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins

We study the multistability that results when a chaotic response system that has an invariant symmetry is driven by another chaotic oscillator. We observe that there is a transition from a desynchronized state to a situation of multistability. In the case considered, there are three coexisting attractors, two of which are synchronized and one is desynchronized. For large coupling, the asynchronous attractor disappears, leaving the system bistable. We study the basins of attraction of the system in the regime of multistability. The three attractor basins are interwoven in a complex manner, with extensive riddling within a sizeable region of (but not the entire) phase space. A quantitative characterization of the riddling behavior is made via the so-called uncertainty exponent, as well as by evaluating the scaling behavior of tongue-like structures emanating from the synchronization manifold.

Synchronization of intermittent behavior in ensembles of multistable dynamical systems

We propose a methodology to analyze synchronization in an ensemble of diffusively coupled multistable systems. First, we study how two bidirectionally coupled multistable oscillators synchronize and demonstrate the high complexity of the basins of attraction of coexisting synchronous states. Then, we propose the use of the master stability function (MSF) for multistable systems to describe synchronizability, even during intermittent behavior, of a network of multistable oscillators, regardless of both the number of coupled oscillators and the interaction structure. In particular, we show that a network of multistable elements is synchronizable for a given range of topology spectra and coupling strengths, irrespective of specific attractor dynamics to which different oscillators are locked, and even in the presence of intermittency. Finally, we experimentally demonstrate the feasibility and robustness of the MSF approach with a network of multistable electronic circuits.

Driving-induced bistability in coupled chaotic attractors

We examine the effects of symmetry-preserving and -breaking interactions in a drive-response system where the response has an invariant symmetry in the absence of the drive. Subsequent to the onset of generalized synchronization, we find that there can be more than one stable attractor. Numerical as well as analytical results establish the presence of phase synchrony in such coexisting attractors. These results are robust to external noise.

Experimental study of firing death in a network of chaotic FitzHugh-Nagumo neurons

The FitzHugh-Nagumo neurons driven by a periodic forcing undergo a period-doubling route to chaos and a transition to mixed-mode oscillations. When coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems.

The development of generalized synchronization on complex networks

In this paper, we numerically investigate the development of generalized synchronization (GS) on typical complex networks, such as scale-free networks, small-world networks, random networks, and modular networks. By adopting the auxiliary-system approach to networks, we observe that GS generally takes place in oscillator networks with both heterogeneous and homogeneous degree distributions, regardless of whether the coupled chaotic oscillators are identical or nonidentical. We show that several factors, such as the network topology, the local dynamics, and the specific coupling strategies, can affect the development of GS on complex networks.

Synchronization of semiconductor lasers with coexisting attractors

We study synchronization of unidirectionally coupled optical bistable systems. In particular, we consider two semiconductor lasers with an external cavity, which exhibit, when isolated, coexistence of two different attractors: fixed point and chaos, fixed point and one periodic orbit, and two periodic orbits with different periods. The analysis is performed with a cross-correlation function between the master and slave laser oscillations calculated with model equations based on the Lang-Kobayashi approach. Depending on both the laser operating point and the coupling strength, different bifurcations (Hopf, period doubling, saddle node, torus, and crisis) and diverse dynamical regimes (steady state, periodicity, quasiperiodicity, bistability, and chaos) occur in the route from asynchronous motion to complete synchronization. We show some similarities and differences between synchronization of monostable and bistable lasers.

Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control

In this paper, an adaptive fuzzy sliding mode control (AFSMC) scheme is proposed for the synchronization of two chaotic nonlinear systems in the presence of uncertainties and external disturbance. To design the reaching phase of the sliding mode control (SMC), a fuzzy controller is used. This will reduce the chattering and improve the robustness. An AFSMC is used (as an equivalent control part of the SMC) to approximate the unknown parts of the uncertain chaotic systems. Although the above schemes have been proposed in the past as separate stand-alone control schemes, in this paper, we integrate these methods to propose an effective control scheme having the benefits of each. The stability analysis for the proposed control scheme is provided and simulation examples are presented to verify the effectiveness of the method.

Synchronization of chaotic systems with coexisting attractors

Synchronization of coupled oscillators exhibiting the coexistence of chaotic attractors is investigated, both numerically and experimentally. The route from the asynchronous motion to a completely synchronized state is characterized by the sequence of type-I and on-off intermittencies, intermittent phase synchronization, anticipated synchronization, and period-doubling phase synchronization.

Chaotic synchronization through coupling strategies

Usually, complete synchronization (CS) is regarded as the form of synchronization proper of identical chaotic systems, while generalized synchronization (GS) extends CS in nonidentical systems. However, this generally accepted view ignores the role that the coupling plays in determining the type of synchronization. In this work, we show that by choosing appropriate coupling strategies, CS can be observed in coupled chaotic systems with parameter mismatch, and GS can also be achieved in coupled identical systems. Numerical examples are provided to demonstrate these findings. Moreover, experimental verification based on electronic circuits has been carried out to support the numerical results. Our work provides a method to obtain robust CS in synchronization-based chaos communications.

Effect of noise on generalized chaotic synchronization

When two characteristically different chaotic oscillators are coupled, generalized synchronization can occur. Motivated by the phenomena that common noise can induce and enhance complete synchronization or phase synchronization in chaotic systems, we investigate the effect of noise on generalized chaotic synchronization. We develop a phase-space analysis, which suggests that the effect can be system dependent in that common noise can either induce/enhance or destroy generalized synchronization. A prototype model consisting of a Lorenz oscillator coupled with a dynamo system is used to illustrate these phenomena.

Phase synchronization between two essentially different chaotic systems

In this paper, we numerically investigate phase synchronization between two coupled essentially different chaotic oscillators in drive-response configuration. It is shown that phase synchronization can be observed between two coupled systems despite the difference and the large frequency detuning between them. Moreover, the relation between phase synchronization and generalized synchronization is compared with that in coupled parametrically different systems. In the systems studied, it is found that phase synchronization occurs after generalized synchronization in coupled essentially different chaotic systems.