Clear evidence of rogue waves in a multistable system is revealed by experiments with an erbium-doped fiber laser driven by harmonic pump modulation. The mechanism for the rogue wave formation lies in the interplay of stochastic processes with multistable deterministic dynamics. Low-frequency noise applied to a diode pump current induces rare jumps to coexisting subharmonic states with high-amplitude pulses perceived as rogue waves. The probability of these events depends on the noise filtered frequency and grows up when the noise amplitude increases. The probability distribution of spike amplitudes confirms the rogue wave character of the observed phenomenon. The results of numerical simulations are in good agreement with experiments.
We study synchronization phenomena in scale-free networks of asymmetrically coupled dynamical systems featuring degree-degree correlation in the topology of the connection wiring. We show that when the interaction is dominant from the high-degree to the low-degree (from the low-degree to the high degree) nodes, synchronization is enhanced for assortative (disassortative) degree-degree correlations.Keywords: Complex Network, 05.45Xt Recent years have witnessed increasing interest in the study of the structure of complex networks. Networks are present ubiquitously in the real world, including biological systems, genetic chains, social relationships and artificial and engineering architectures [2]. A main quantity that characterizes the structure and dynamics of those networks is the so called degree distribution P (k), the probability that a randomly chosen node within the network has degree k, i.e. there are k connections incident in it. In most cases, this distribution has been found to obey a power law P (k) ∼ k −γ , with γ usually ranging between 2 and 3. The slow decay of the power-law distribution, when the exponent is within the observed values, is responsible for the presence of few high-degree nodes, usually termed as hubs, which play a leading role in characterizing the network dynamical behavior. Thus the network structure (termed in this case as scale free) displays high heterogeneity, with the presence of a few high-degree vertices and many low-degree ones.Beyond the specific interest in the study of the main topological properties of such networks, there is an ongoing research effort to understand how these statistical features affect the dynamical processes taking place over them, as e.g. epidemic spreading [24,3,19], communication models [23,10,11,1] and spin systems [8,9,14]. Among these, a relevant phenomenon is the emergence of collective synchronized motion. This corresponds to the case of several dynamical systems, starting from different initial conditions appropriately coupled with a network topology, stabilizing in a common evolution.In this framework, we consider the case of networks of coupled identical systems, that can be conveniently * Email: f sorrent@unina.it 1 described by the following set of ordinary differential equations:
We demonstrate experimental evidence of noise-induced attractor hopping in a multistable fiber laser. Multistate hopping dynamics displays complex statistical properties characterized by nontrivial scalings. When hopping is encountered between two states, the dynamics of the system is characterized by the -32 power law for the probability distribution of periodic windows versus their length, just as in the case of two-state on-off intermittency. A surprising noise saturation effect is found: average output noise in the hopping regime is almost independent of input noise. Such robustness of the system against external noise may be beneficial for some applications: for example, for communications with multistable systems or for designing noise-insensitive detectors.
We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it. Synchronization is a common phenomenon in a diversity of natural and technological systems [1]. Synchrony, however, is not always achieved spontaneously, and reaching or maintaining a synchronous state often requires an external action. An elegant way to enhance synchronization is the use of relay units between the systems to be synchronized [see Fig. 1(a)]. Relay synchronization (RS) consists of achieving complete synchronization (CS) of two dynamical systems by indirect coupling through a relay unit, whose dynamics does not necessary join the synchronous state. RS is especially useful in bidirectionally coupled systems with a certain delay in the coupling line. In these cases, indeed, the coupling delay may induce instability of the synchronous state [2], which can be restored again thanks to a relay system. Lasers [3,4] and electronics circuits [5] have been the benchmark for experimental demonstration of the feasibility of RS, showing its robustness against noise or parameter mismatch. In semiconductor lasers, for instance, zero-lag synchronization between two delaycoupled oscillators can be achieved by relaying the dynamics via a third mediating element, which surprisingly lags behind the synchronized outer elements. With electronic circuits, RS has been used as a technique for transmitting and recovering encrypted messages, which can be sent bidirectionally and simultaneously [6]. Apart from its technological applications, RS has also been proposed as a possible mechanism at the basis of isochronous synchronization between distant areas of the brain [7]. Despite such evidence of RS, there are still open questions of a fundamental nature. The main issue is to characterize properly the relationship, established in RS, between the dynamics of the relay system and that of the synchronized systems. When a certain delay is introduced in the coupling lines, lag synchronization has been reported [3,4]. Nevertheless, relay units may have certain parameter mismatch [8] or even be completely different systems [5], thus having dynamics with unclear a priori relationship with the systems they are synchronizing.In this paper, we give evidence that RS in fact corresponds to t...
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