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One of the main problems in the study of dynamical systems is to explore the asymptotic behavior of the model when a parameter varies continuously. When these variations lead to the appearance of coexisting states, the study of the global properties of the system becomes an even more complex task, since it is almost impossible to predict the stability change. In this paper, we present a simple method for characterizing qualitative changes in the dynamics of a family of Piece-Wise Linear (PWL) chaotic systems, that transit from monostable to multistable behavior by a single bifurcation parameter. By characterizing the magnitude of the stable and unstable manifolds associated with the eigendirections, it is possible to analytically find tipping points in the linear model that are consistent with the occurrence of coexisting states in the dynamics. The results show agreement between the bifurcation diagrams of the linear operator, the bifurcation diagrams of the PWL system, and the multistability phenomenon validation in analog electronics. The presented work makes it possible to know the mechanism by which the system exhibits the break of its stability and the corresponding basin of attraction. This introduces a new methodology for the analysis of dynamical systems in search of dynamical changes such as coexisting attractors.

Extreme multistability is the coexistence of a large number of attractors which can be reached by varying initial conditions. In this paper we show how this fascinating phenomenon can be used for secure communication. The main advantage of the communication system based on extreme multistability over a conventional chaos-based communication system is its exceptionally high security. The proposed system consists of two identical six-order oscillators; one in the transmitter and another one in the receiver, each exhibiting the coexistence of a large number of chaotic attractors. The oscillators are synchronized using a private channel through one of the system variables, while the information is transmitted via a public channel through another variable. The information is encrypted by varying the initial condition of one of the state variables in the transmitter using a chaotic map, adhering message packages in a staggered form to the coexisting attractors within the same time series of another state variable, which leads to switching among the coexisting chaotic attractors. To ensure communication security, the duration of the packages is shorter than synchronization time, so that synchronization attacks are ineffective.

We study the emergence of synchronization in the network motif of three bistable Duffing oscillators coupled in all possible configurations. The equation of motion is derived for every configuration. For each motif, we vary initial conditions of every oscillator and calculate the bifurcation diagram as a function of the coupling strength. We find transitions of the whole system to a monostable regime with either a fixed point or a limit cycle depending on the motif’s configuration, as the coupling strength is increased. The most complex dynamics is observed the nidirectional chain, where a transition to quasiperiodicity occurs.

We study noise-induced intermittency in a synergetic model of two coupled oscillators with asymmetric nonlinear coupling. This model was previously used to simulate visual perception of ambiguous images. We show that additive noise induces preference for one of the coexisting unstable steady states. When the noise intensity exceeds a certain threshold value, the oscillations of one of the coupled subsystems are interrupted during some time intervals, resulting in intermittent oscillation death, while another subsystem exhibits noisy oscillations in the vicinity of an unstable fixed point.

Recently, it was discovered that the use of fractional derivatives induces the occurrence of multistable states in PWL systems with multiple scrolls. In this paper, we show the path of the system from monostable to multistable behavior by reducing the integration order. Using bifurcation diagrams, which describe the evolution of the attractor under the modification of the bifurcation parameter, and basins of attraction, we show that the system has a path to multistability, revealing the mechanism for the emergence of the behavior. The resulting dynamics shows the coexistence of up to n + 2 attractors, where n is the number of scrolls generated by the integer-order system. Moreover, the existence and uniqueness of the dynamics is proved by Poincar´e sections showing that the behaviors exist and coexist and are not a section of the same solution.

In this work, numerical results of a nonlinear switching system that presents bistable attractors subjected to stochastic modulation are shown. The system exhibits a dynamical modification of the bistable attractor, giving rise to an intermit behavior, which depends of modulation strength. The resulting attractor converge to an intermittent double-scroll, for low amplitude modulation, and a 9-scroll attractor for a higher applied noise amplitude. A Detrended Fluctuation Analysis (DFA) applied to the x state variable, shows a perturbations robustness region, since the increase of noise does not present changes. Due to the applied noise, the final obtained system has higher randomness, compared with the original one. The understanding of the dynamical behavior of multiscrolls systems is highly important for advancing technology in communications, as well in memory systems applications.

We provide compelling numerical evidence of deterministic coherence and anti-coherence resonance in small networks of unidirectional coupled chaotic Rossler oscillators in star and star-ring configurations in the presence of a small mismatch between natural frequencies of the oscillators. The resonance phenomena are found in both the normalized standard deviations of the peak amplitude and inter-peak time interval with respect to the coupling strength and frequency mismatch. The deterministic coherence/anti-coherence resonance resembles self-stabilization/destabilization of the network when the coupling makes collective dynamics more regular/irregular.

We propose the use of error-feedback control for converting a multistable system into a monostable one. The method is tested with a multistable erbium-doped fiber laser which exhibits the coexistence of four periodic orbits. A desired orbit is selected by an external harmonic modulation which frequency is equal to the frequency of one of the coexisting attractors. Additionally, the error signal, proportional to the difference between the external modulation and the laser output intensity, is applied to the pump current of the diode pumped laser. For certain combinations of the amplitude of the external modulation and the error signal, the laser displays monostability with a single periodic orbit at frequency determined by the external modulation. The efficiency of the control is demonstrated by both numerical simulations and experimentally.

We propose an experimental arrangement to increase pulse energy in the array of erbium-doped fiber lasers (EDFLs). The experimental setup consists of three coupled periodically modulated multistable EDFLs; one master laser and two slave lasers. The master laser is tuned to the coexisting attractor with the largest peak amplitude and synchronizes the slave lasers to maintain them in the same attractor as the master laser. Summing the outputs of the slave lasers we obtain laser pulsations with a double peak intensity. The number of slave lasers can be augmented to further increase the sum peak power.

We study synchronization of two chaotic oscillators coupled with time delay in a master-slave configuration and with delayed positive feedback in the slave oscillator which acts as memory. The dynamics of the slave oscillator is analyzed with bifurcation diagrams of the peak value of the system variable with respect to the coupling and feedback strengths and two delay times. For small coupling, when the oscillators’ phases synchronize, memory can induce bistability and stabilize periodic orbits, whereas for stronger coupling it is not possible. The delayed feedback signal impairs synchronization, simultaneously enhancing coherence of the slave oscillator.

This paper presents a novel electronic locking key based on discrete-time chaos synchronization. Two Chen chaos generators are synchronized using the Model-Matching Approach, from non-linear control theory, in order to perform the encryption/decryption of the signal to be transmitted. A model/transmitter system is designed, generating a key of chaotic pulses in discrete-time. A plant/receiver system uses the above mentioned key to unlock the mechanism. Two alternative schemes to transmit the private chaotic key are proposed. The first one utilizes two transmission channels. One channel is used to encrypt the chaotic key and the other is used to achieve output synchronization. The second alternative uses only one transmission channel for obtaining synchronization and encryption of the chaotic key. In both cases, the private chaotic key is encrypted again with chaos to solve secure communication-related problems. The results obtained via simulations contribute to enhance the electronic locking devices.

In this paper, a physical interpretation of the fractional-order-derivatives effects in a jerk system, based on Unstable Dissipative Systems (UDS), and a Saturated Non-Linear Function (SNLF), is presented. The system is electronically implemented in Multisim development platform for a 9-scrolls attractor generation. The behavior is analyzed trough the Detrended Fluctuation Analysis (DFA), Probability Density Function (PDF), bifurcation diagrams, and the implementation of a geometrical analysis of the phase space. The changes that the system undergoes when a fractional-order are analyzed. The results show that when the fractional integration-orders are considered, the areas of the generated attractor are modified with respect to the integer-order dynamic. The long-range correlations in the system are also modified because of the fractional-orders. Besides, a particular phenomenon in the equilibrium points preference occurs, which is induced when the fractional integration-order is applied in only one of the state variables.

This paper presents a novel electronic implementation based on a simple methodology to control the number of scrolls generated, by means of operational amplifiers and a single control parameter. Through the use of a Saturated Non Linear Function (SNLF) for the generation of a 9-scroll attractor, the number of scrolls is modified through a single parameter. In addition, the implemented system is also capable of generating a family of bistable behaviors. The phenomenon for the scroll generation, induced by a control parameter, is explained through the system equilibrium points. This methodology can be extended to system’s with higher number of scrolls, or systems with larger basins of attraction. The results shown have high potential application in secure communication systems, mobile surveillance devices, pseudo-random number generators and complex networks, among others.

In this paper, a novel method for increasing the number of scrolls in a hybrid nonlinear switching system is presented. Using the definition of the “Round to the Nearest Integer Function”, as a generalization of a Piece-Wise Linear (PWL) function, which is capable of generating up to a thousand of scrolls. An equation that characterizes the grown in the number of scrolls is calculated, which fits to the behavior of the system measured by means of the coefficient of determination, denoted R 2 . The proposed equation is based on obtaining as many scrolls as desired, based on the control parameters of the linear operator of the system. The work here presented provides a new approach for the generation and control of a high number of scrolls in a hybrid system. The results are verified for all the scenarios that the equations covers.

We experimentally studied dynamics of a dual-cavity Nd:YAG laser with second harmonic generation in one of the cavities and loss modulation in another cavity. Dependent on the frequency of harmonic modulation of the infrared light, we observed the coexistence of different correlated and uncorrelated regimes of the infrared and green outputs from two laser cavities. We present the experimental evidence of oscillation death coexistent with oscillatory states for certain modulation frequencies.

We implement a dynamic logic gate which consists of three elements: a chaotic fiber laser, a threshold controller and the output of the logic gate. The output signal of the fiber laser is sent to the logic gate input as to the threshold controller; threshold controller output signal is sent at the entrance of the logic gate and also feedback to the fiber laser which changes their dynamic behavior. The output of the logic gate consists of a difference amplifier; this compares the signals sent by the threshold controller and the fiber laser, resulting logic output that depend on an accessible parameter in threshold controller. We present the numerical results and experimental of the implementation of a dynamic logic gate using fiber laser, which demonstrates the ability to change the type of logic gate by modifying a parameter of threshold control.

We propose an approach for generation of deterministic Brownian motion. By adding an additional degree of freedom to the Langevin equation and transforming it into a system of three linear differential equations, we determine the position of switching surfaces, which act as a multi-well potential with a short fluctuation escape time. Although the model is based on the Langevin equation, the final system does not contain a stochastic term, and therefore the obtained motion is deterministic. Nevertheless, the system behavior exhibits important characteristic properties of Brownian motion, namely, a linear growth in time of the mean square displacement, a Gaussian distribution, and a −2 power law of the frequency spectrum. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

The proposed Hill-Climbing algorithm incorporates an acceleration operator.The acceleration operator improves the exploration capability of the algorithm.Local search is more adequate than global search for generating typical testors.The proposed algorithm has better performance than other heuristics reported. This paper is focused on introducing a Hill-Climbing algorithm as a way to solve the problem of generating typical testors - or non-reducible descriptors - from a training matrix. All the algorithms reported in the state-of-the-art have exponential complexity. However, there are problems for which there is no need to generate the whole set of typical testors, but it suffices to find only a subset of them. For this reason, we introduce a Hill-Climbing algorithm that incorporates an acceleration operation at the mutation step, providing a more efficient exploration of the search space. The experiments have shown that, under the same circumstances, the proposed algorithm performs better than other related algorithms reported so far.

Synchronization of chaotic oscillators has an important application in cryptography. When two identical oscillators are coupled, they can be completely synchronized and the chaotic output of the transmitter oscillator can be used to mask a message. Although the oscillator parameters are usually used as secret keys, the sensitivity of such cryptosystems to parameter changes has never been systematically analyzed. To cryptanalyze a communication system based on synchro- nization of chaotic oscillators, we use a synchronization attack that allows estimating all unknown parameters by minimizing the synchronization error. Using this attack we cryptanalyze popular communication systems based on the Rossler and Chua chaotic electronic circuits. We suggest to include this attack as a standard security test for crypt-analysis of chaotic communication systems.

Maximally synchronizable networks (MSNs) are acyclic directed networks that maximize synchronizability. In this paper, we investigate the feasibility of transforming networks of coupled oscillators into their corresponding MSNs. By tuning the weights of any given network so as to reach the lowest possible eigenratio lambdaN/lambda2, the synchronized state is guaranteed to be maintained across the longest possible range of coupling strengths. We check the robustness of the resulting MSNs with an experimental implementation of a network of nonlinear electronic oscillators and study the propagation of the synchronization errors through the network. Importantly, a method to study the effects of topological uncertainties on the synchronizability is proposed and explored both theoretically and experimentally. The authors acknowledge J.L. Echenausía-Monroy, V.P. Vera-Ávila, J. Moreno de León, C. Hapo and P.L. del Barrio for assistance in the laboratory, and the support of MINECO (FIS2012-38949-C03-01 and FIS2013-41057-P). One anonymous referee is acknowledged for having provided valuable advice that has influenced our understanding of the origin of the propagation of the synchronization error, and helped us improve the manuscript in several ways. The authors also acknowledge the computational resources, facilities and assistance provided by the Centro computazionale di RicErca sui Sistemi COmplessi (CRESCO) of the Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA). R.S.E. acknowledges Universidad de Guadalajara, CULagos (Mexico) for financial support (PRO-SNI-2015/228069, PROINPEP/005/2014, UDG-CONACyT/I010/163/2014) and CONACyT (Becas Mixtas MZO2015/290842). D.-U. Hwang acknowledges National Institute for Mathematical Sciences (NIMS) funded by the Ministry of Science, ICT & Future Planning (A21501-3).

A diode-pumped erbium-doped fiber laser with pump modulation shows a large dynamic behavior including the coexistence of multiple attractors. In this work it is demonstrated numerically and experimentally that a low-pass noise filtering can control the probability for the appearance of a particular state. The results of numerical simulations with the use of a three-level laser model display good agreement with experimental results. We construct three-dimensional bifurcation diagrams of the probability using the noise amplitude and cutting frequency as control parameters. Finally, it is found the existence of one noise amplitude that cause one periodic orbit to appears more frequently compared to different noise amplitudes.

We study a physiologically realistic implementation of internal stochasticity in a four dimensional Hodgkin-Huxley (HH) type model of cold receptors. We show that in a deterministically tonic-firing regime, changing the parameters can drive the neuron into a state of complex bursting behavior. An explanation of the mechanism behind this effect is given in terms of a bifurcation diagram of inter-spike interval (ISI) as the temperature and external current are altered.

We implement an algorithm to reproduce the behavior of a dynamic logic gate which consists of three elements: a fiber laser in chaotic regime, a threshold controller and the output of the logic gate. The output signal of the fiber laser is sent to the logic gate input as to the threshold controller; threshold controller output signal is sent at the entrance of the logic gate and also fed back to the fiber laser which changes their dynamic behavior. The output of the logic gate consists of a difference amplifier; this compares the signals sent by the threshold controller and the fiber laser, resulting logic output that depend on an accessible parameter in threshold controller. We present the numerical results of the implementation of a dynamic logic gate using fiber laser, which demonstrates the ability to change the type of logic gate by modifying a parameter of threshold control.

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rössler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system. Support from Spanish Ministry of Economy and Competitiveness through Projects No. FIS2011-25167, No. FIS2012-38266, and No. FIS2013-41057-P is also acknowledged. A.A. and J.G.G. acknowledge supportfrom the EC FET-Proactive Projects PLEXMATH (GrantNo. 317614) and MULTIPLEX (Grant No. 317532). J.G.G. acknowledges support from MINECO through the Ramón y Cajal program, the Comunidad de Aragón (Grupo FENOL), and the Brazilian CNPq through the PVE project of the Ciencia Sem Fronteiras program. A.A. acknowledges ICREA Academia and the James S. McDonnell Foundation. R.S.E. ac-knowledges Universidad de Guadalajara, CULagos (Mexico) for financial support (OP/PIFI-2013-14MSU0010Z-17-04,PROINPEP-RG/005/2014, UDG-CONACyT/I010/163/2014) and CONACyT (Becas Mixtas MZO2015/290842).

We report on the experimental observation of coherence enhancement of noise-induced intermittency in a semiconductor laser subject to optical injection from another laser at the boundary of the frequency-locking regime. The intermittent switches between locked and unlocked states occur more regularly at a certain value of the injecting laser pump current. A shape of probability distribution of the experimental inter-spike-interval fluctuations is used to quantitatively characterize the intermittent behavior.

We fabricate a biometric laser fiber synaptic sensor to transmit information from one neuron cell to the other by an optical way. The optical synapse is constructed on the base of an erbium-doped fiber laser, whose pumped diode current is driven by a pre-synaptic FitzHugh–Nagumo electronic neuron, and the laser output controls a post-synaptic FitzHugh–Nagumo electronic neuron. The implemented laser synapse displays very rich dynamics, including fixed points, periodic orbits with different frequency-locking ratios and chaos. These regimes can be beneficial for efficient biorobotics, where behavioral flexibility subserved by synaptic connectivity is a challenge.

Clear evidence of rogue waves in a multistable system is revealed with an erbium-doped fiber laser driven by harmonic pump modulation (Pisarchik, Jaimes-Reátegui, Sevilla-Escoboza, Huerta-Cuellar & Taki, 2011). We demonstrate numerically and experimentally that a low-pass noise filtering can control the probability for the appearance of a particular state. The results of numerical simulations with the use of a three-level laser model display good agreement with experimental results. 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.

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., Submitted for publication. 5 pages and 4 figures

In our recent Letter [Phys. Rev. Lett. 107, 274101 (2011)], we demonstrated that slow random perturbations of a system parameter were responsible for the emergence of rogue waves in a fiber laser with coexisting attractors. In this paper we investigate how the probability of a particular state to appear in multistate intermittency can be controlled by low-pass noise filtering. We show that the probability of some states depends nonmonotonously on the noise amplitude and cutoff frequency. The conditions for the emergence of extreme pulses in a erbium-doped fiber laser are analyzed numerically and experimentally.

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 report on the experimental evidence of noise amplification in an erbium-doped fiber laser in the vicinity of saddle-node, period-doubling, and crisis bifurcations. We demonstrate this interesting phenomenon by analyzing the laser bifurcation diagrams and power spectra. Numerical simulations on the base of an advanced laser model display good agreement with the experimental results.

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 $\ensuremath{-}3∕2$ 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 propose a robust method that allows a periodic or a chaotic multi-stable system to be transformed to a monostable system at an orbit with dominant frequency of any of the coexisting attractors. Our approach implies the selection of a particular attractor by periodic external modulation with frequency close to the dominant frequency in the power spectrum of a desired orbit and simultaneous annihilation of all other coexisting states by positive feedback, both applied to one of the system parameters. The method does not require any preliminary knowledge of the system dynamics and the phase space structure. The efficiency of the method is demonstrated in both a non-autonomous multi-stable laser with coexisting periodic orbits and an autonomous Rössler-like oscillator with coexisting chaotic attractors. The experiments with an erbium-doped fibre laser provide evidence for the robustness of the proposed method in making the system monostable at an orbit with dominant frequency of any preselected attractor.

Secure optical communication has been realized with two semiconductor lasers driven by two chaotic Rossler oscillators. The communication system contains two channels: optical and electronic; the information is transmitted through an optical fiber, while the Rossler oscillators are synchronized via electronic channel. One of the outputs of the Rossler oscillators serves for modulating the laser pump current, and another for coupling the oscillators. The results of numerical simulations are in good agreement with experiments which demonstrate that the system is viable.

We implement a dynamically flexible logic-gate using a chaotic erbium doped fiber lasers. Experimental results are presented, which demonstrate the ability to change the type of logic-gate by modifying a threshold control parameter.