We study chaotic synchronization in networks with time-delayed coupling. introduce the notion of strong and weak chaos, distinguished by scaling properties maximum Lyapunov exponent within manifold for large delay times, relate this to condition stable or unstable synchronization, respectively. In simulations laser models experiments electronic circuits, we identify transitions from back chaos upon monotonically increasing coupling strength.

While the assumption that dynamical systems are stationary is common for modeling purposes, in reality, this rarely case. Rather, these can change over time, a phenomenon referred to as concept drift community. there exist numerous statistics-based methods detection on stochastic processes, approaches leveraging nonlinear time series analysis (NTSA) rarer but seeing increased focus cases where processes deterministic. In work, we propose novel approach unsupervised utilizing embedding...

Nonlinear networks with time-delayed couplings may show strong and weak chaos, depending on the scaling of their Lyapunov exponent delay time. We study chaos for semiconductor lasers, either self-feedback or small networks. examine dependence pump current consider question whether can be identified from shape intensity trace, autocorrelations, external cavity modes. The concept sub-Lyapunov λ(0) is generalized to case two time-scale-separated delays in system. give experimental evidence a...

We experimentally study consistency properties of a semiconductor laser in response to coherent optical drive originating from delayed feedback. The is connected short and long fiber loop, switched such that only one providing input the at time. This way, repeating exact same twice, we find consistent or inconsistent responses depending on pump parameter relate kind strong weak chaos. Moreover, are able determine sub-Lyapunov exponent, underlying properties.

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across range of dynamical relationships including generalized synchronization. For drive-response setup, temporal representation the synchronized state is discussed as an alternative to known instantaneous form. The has access both representations through its fading memory property, each with advantages different regimes. also extract signatures maximal conditional Lyapunov exponent performance variations...

Consistency is an extension to generalized synchronization which quantifies the degree of functional dependency a driven nonlinear system its input. We apply this concept echo-state networks, are artificial-neural network version reservoir computing. Through replica test we measure consistency levels high-dimensional response, yielding comprehensive portrait property.

We present a detailed experimental characterization of the autocorrelation properties delayed feedback semiconductor laser for different dynamical regimes. show that in many cases function intensity dynamics can be approximated by analytically derived obtained from linear stochastic model with delay. extract set dynamic parameters fit analytic solutions and discuss limits validity our approximation. The captures multiple fundamental delay systems, such as shift asymmetric broadening echoes....

We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in regime high-frequency modulation delay. The average effect term force is equivalent to distributed interval modulation, and obtained distribution depends on type modulation. In our analysis we use simple generic normal form an focus, investigate effects phase-dependent coupling influence loop latency controllability. addition, have explored delays multiple schemes...

We report on several improvements of the classical time-delayed feedback control method for stabilization unstable periodic orbits or steady states. In an electronic circuit experiment, we were able to realize time-varying and distributed delays in force leading successful large parameter sets, including time delays. The presented techniques make advanced use natural torsion orbits, which is also necessary original work.

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and oscillation patterns. For phase we reduce the system a non-delayed Langevin equation, which allows us analytically compute distribution frequencies, their corresponding residence times. The number stable scales roundtrip time coupling strength, but noisy visits only fraction orbits, square root independent strength....

We study chaotic systems with multiple time delays that range over several orders of magnitude. show the spectrum Lyapunov exponents (LEs) in such possesses a hierarchical structure, different parts scaling delays. This leads to types chaos, depending on maximal LE. Our results are relevant, particular, for synchronisation properties networks (networks networks) where nodes subnetworks coupled shorter and couplings between realised longer delay times. Units within subnetwork can synchronise...

We study swarms as dynamical systems for reservoir computing (RC). By example of a modified Reynolds boids model, the specific symmetries and properties swarm are explored with respect to nonlinear time-series prediction task. Specifically, we seek extract meaningful information about predator-like driving signal from swarm’s response that signal. find naïve implementation computation is very inefficient, permutation symmetry individual agents reduces computational capacity. To circumvent...

We present the idea of reservoir time series analysis (RTSA), a method by which state space representation generated computing (RC) model can be used for analysis. discuss motivation this with reference to characteristics RC and three ad hoc methods generating representative features from space. then develop implement hypothesis test assess capacity these distinguish signals systems varying parameters. In comparison number benchmark approaches (statistical, Fourier, phase space, recurrence...

For delay systems the sign of sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes properties strong and weak chaos consistency. Here we present a robust algorithm based on reconstruction local linearized equations motion, which allows for calculating sub-LE from time series. The is inspired by method introduced Pyragas nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In presented extension to systems, delayed feedback takes over role...

Consistency refers to the property of an externally driven dynamical system respond in similar ways inputs. In a delay system, delayed feedback can be considered as external drive undelayed subsystem. We analyze degree consistency generic chaotic with by means auxiliary approach. this scheme identical copy nonlinear node is exactly same signal original, allowing us verify complete via synchronization. past, phenomenon synchronization delay-coupled systems has been widely studied using...

Consistency is a powerful concept to characterize and classify the behavior of driven dynamical systems, therefore playing an essential role for understanding information processing in by complex photonic systems. Here, we discuss two different experimental configurations semiconductor laser systems with delayed optical feedback allowing unifying view via detailed characterization comparison consistency synchronization properties. The first configuration comprises unidirectionally coupled...

Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well oscillations. We have experimentally observed these in delay-coupled electronic circuits analyzed their stability by numerical simulations analytical calculations. found the conditions for longitudinal transversal largely exclude each other prevent, e.g., synchronization Pyragas-controlled orbits....

We study laminar chaos in an electronic experiment. A two-diode nonlinear circuit with delayed feedback shows chaotic dynamics similar to the Mackey-Glass or Ikeda delay systems. Clock modulation of a single line leads conservative variable delay, which second is augmented dissipative delays, leading regimes. discuss properties this particular and demonstrate experimental aspects terms power spectra return maps.

We investigate the scaling behavior of maximal Lyapunov exponent in chaotic systems with time delay. In large-delay limit, it is known that one can distinguish between strong and weak chaos depending on delay scaling, analogously to instabilities for steady states periodic orbits. Here we show shows significant differences its compared constant or dynamics due fluctuations linearized equations motion. reproduce properties a linear system multiplicative noise. further derive analytic limit...

We study the propagation and distribution of information-carrying signals injected in dynamical systems serving as reservoir computers. Through different combinations repeated input signals, a multivariate correlation analysis reveals measures known consistency spectrum capacity. These are high-dimensional portraits nonlinear functional dependence between state. For multiple inputs, hierarchy capacities characterizes interference from each source. an individual input, time-resolved form...

We report on the observation of noise-free stochastic resonance in an externally driven diode resonator close to interior crisis. At sufficiently high excitation amplitudes shows a strange attractor which after collision with unstable period-three orbit exhibits crisis-induced intermittency. In intermittency regime system jumps between previously stable chaotic and phase space region has been made accessible by This random process can be used amplify weak periodic signal through mechanism...

We investigate the capacity of reservoir computers to reconstruct dynamics a network chaotic oscillators via observation its multivariate time series. The is itself structured echo-state which receives current observations as inputs, and trained produce next outputs. study performance this scheme dependence on separation modularity network, observability system. observe optimal with segregated input structure extremely modular network.

To understand the collective motion of many individuals, we often rely on agent-based models with rules that may be computationally complex and involved. For biologically inspired systems in particular, this raises questions about whether imposed are necessarily an accurate reflection what is being followed. The basic premise updating one's state according to some underlying motivation well suited realm reservoir computing; however, entire swarms individuals yet tasked learning movement...