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The dynamic behavior of many physical, biological, and other systems, are organized according to the synchronization of chaotic oscillators. In this paper, we have proposed a new method with low sensitivity to noise for detecting synchronization by mapping time series to complex networks, called the ordinal partition network, and calculating the permutation entropy of that structure. We show that this method can detect different kinds of synchronization such as complete synchronization, phase synchronization, and generalized synchronization. In all cases, the estimated permutation entropy decreases with increased synchronization. This method is also capable of estimating the topology of the network graph from the time series, without knowledge of the dynamical equations of individual nodes. This approach has been applied for the two identical and nonidentical coupled Rössler systems, two nonidentical coupled Lorenz systems, and a ring of coupled Lorenz96 oscillators.

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden.

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.

Many literatures have discussed fractional order memristor and memcapacitor-based chaotic oscillators but the entire oscillator model is considered to be of fractional order. My interest is to propose a nonlinear oscillator with considering only the memcapacitor element of fractional order. Hence, I propose a fractional order memcapacitor (FMC)-based novel chaotic oscillator. The complete mathematical model for the proposed oscillator is derived and presented in this paper. The dimensionless state equations are then analyzed by using the equilibrium points and their stability, Eigen values, Kaplan–Yorke dimensions and Lyapunov exponents. To understand the complete dynamical behavior, bifurcation graphs are obtained and presented. Finally, the proposed fractional memcapacitor oscillator is implemented by using the shelf components.

We construct local and global conjugate variable coupled chaotic systems with an arbitrary number of identical chaotic oscillators. Amplitude death phenomena are found in these two kinds of systems. General methods are proposed to theoretically analyze conditions for amplitude death under local and global conditions, respectively. Taking the Rössler chaotic system as the oscillator unit, we apply these methods to determine the analytical conditions for amplitude death. And the transition relations between local and global coupling induced amplitude deaths are also given. All theoretical results are well confirmed by numerical simulations.

In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its forced version and choosing a proper set of parameters, we introduce a chaotic system with some very interesting features. In this system, not only can we see the coexistence of different types of attractors, but also a fascinating phenomenon: some initial conditions can escape from the gravity of nearby attractors and travel far away before being trapped in an attractor beyond the usual access.

The problem of synchronization of chaotic oscillators, using state observers, is handled through the use of a simplified perturbed linear integration model of the chaotic system output dynamics. The linear simplified model of the chaotic system does not entitle approximate linearizations, nor state coordinate transformations, but, simply, a pure linear integration model with additive unknown but bounded perturbation inputs lumping all the output dynamics nonlinearities. An extended linear state observer (here addressed as Generalized Proportional Integral (GPI) observer) is proposed for the accurate estimation of the phase variables and the perturbation input of the nonlinear output dynamics. The effectiveness of the approach is tested in the synchronization of two study cases: The Genesio–Tesi chaotic system and the Rossler oscillator. As an application of the estimation process, a coding–decoding process involving encrypted messages, in transmitted phase variables, is implemented using a Rossler chaotic system.

A system for generating a multi-butterfly chaotic attractor using the multi-level-logic pulse-excitation technique is proposed. Two-butterfly, three-butterfly and four-butterfly attractors are demonstrated. Results from an experimental setup are also shown.

In the present paper, we briefly discuss the construction of mapping model of coupled oscillator systems for limit-cycle oscillators and chaotic oscillators. A comparison of the proposed mapping model and other models, i.e., the Ikeda map and the phase map model is made. Furthermore, the critical dynamics associated with the breakdown of chaotic phase synchronization based on the present mapping model is discussed.

The paper considers the evolution of a particular class of networks of identical chaotic oscillators, namely that of ecological networks. In these networks, nodes represent patches where a certain number of plant and animal populations interact on ecological timescale, arcs represent migration flows due to dispersal, and Darwinian evolution is responsible for variations, on a longer evolutionary timescale, of the demographic parameters characterizing the populations. Up to now, this problem has been mainly studied with reference to single-population patches described by one-dimensional discrete-time models and by considering only the dispersal rates of migrating populations as an evolving trait. Here, we propose a method of investigation which allows to study multipopulation patches described by continuous-time models with evolving traits influencing various demographic parameters (including or not dispersal). The method is casted within the frame of the so-called master stability function approach for the analysis of synchronization of coupled systems, and the results obtained in a first and very simple application support the conjecture that evolution drives ecological networks toward weak forms of synchronization.

Coupled chaotic oscillators are found to be able to achieve the partial synchronization state in some networks. The scaling relation between the synchronization time and the coupling at the synchronous threshold is found and explained. In the studies, it was found that the coupling of the synchronization properties of networks is not homogeneous.

We discuss the key features of theoretical models of networks of coupled chaotic oscillators that produce seizure-like events and bear striking similarities to dynamics of epileptic seizures. Our long-term objective is to understand the basic functional mechanisms that can produce seizures and may ultimately lead to strategies for seizure suppression and control. We show that, from a dynamical systems point of view, a plausible cause of seizures is a pathological feedback in the brain circuitry. This suggests new seizure control approaches, as well as systematic methods to tune existing ones. While the suggested models and control approaches are far from being considered optimal for epileptic seizures, they have interesting physical interpretation and implications for treatments of epilepsy. They also have close ties with a variety of recent practical observations in the human and animal epileptic brain, and with theories from adaptive systems, optimization and chaos.

In this Letter, we propose a class of nonlinear dynamic translinear circuits as a viable tool for the rapid and systematic implementation of chaotic oscillators in analog integrated circuits. In this regard, our primary focus is on multiple-input translinear element (MITE) networks and their particular merits. We also describe, as an illustrative example, our monolithic implementation of the Lorenz equations, and report on results from a test chip fabricated in a 0.5 μm CMOS process through MOSIS. We also show that Chua's circuit can be implemented easily in our framework.

Phase synchronization of a chaotic oscillator that is driven by two chaotic signals is investigated. The anti-biased PS in the presence of biased coupling is found, i.e., the response oscillator can be phase synchronized by the drive with a weaker coupling rather than the stronger driver. The mechanism for this behavior is explored. In the non-PS region, alternating phase-locking is observed.

This paper presents a chaotic oscillator based on the nonlinearity of the typical transconductance (Gm). This chaotic circuit only consists of 13 MOS transistors and three grounded capacitors, which is one of the smallest chaotic oscillators. This circuit operates on low voltage supply (± 1.5 V). The dimensionless form of the circuit is also introduced to confirm the circuit simulation.

A novel system of nonlinear differential equations is proposed. This system is capable of generating a complex four-wing butterfly chaotic attractor by relying on two embedded state-controlled binary switches. Hence, the system is fully autonomous and does not require external forcing to create this attractor. Furthermore, digital logic operations (e.g. AND/OR) performed on the outputs of the two switches are permitted and effectively alter the dynamics of the system. Our findings are validated via experimental results.

A chaotic system can be used to transform a periodic input sequence of events, e.g. impulses, into a sequence of output events. Even after transient, the output sequence is often aperiodic. In particular, it might happen that the output events are basically synchronized with the input events, but that some of them are randomly skipped from the output time series (imperfect phase synchronization [Zacks et al., 1999, 2000]). This phenomenon has been neatly observed by Schimz and Hildebrand [1992] in a rich series of experiments on the locomotor behavior of Halobacterium salinarium, where the input event was a light impulse and the output event was the reversal in the swimming direction of the bacterium. In this paper, we show that the same phenomenon occurs when classical, low dimensional chaotic oscillators are forced by a periodic sequence of impulses. This proves, on one side, that imperfect phase synchronization is a common phenomenon in chaotic oscillators subjected to periodic sequences of impulses, and, on the other side, that there are high chances that the locomotor behavior of Halobacterium salinarium can be modeled by a low dimensional deterministic model.

A simple chaotic system with a single nonquadratic term is developed to be a switchable chaotic signal generator in this paper. Additional nonlinearity of the absolute value function is introduced for reforming the structure without damaging the basic dynamics but yielding a new independent amplitude–frequency controller. A switchable chaotic experimental oscillator is designed afterwards, where two coefficients corresponding to two independent rheostats rescale the amplitude and frequency of the chaotic signals smoothly. To our knowledge, this has never been found in other chaotic oscillators.

The intermittent lag synchronization of two nonidentical symmetrically coupled Rössler systems is investigated numerically. This phenomenon orginates by intermittent bursts away from a principal lag synchronization regime, that correspond to jumps of the system toward other lag configurations. The global scenario of transitions toward a perfect lag synchronization regime may be tracked by the identification of the different lag configurations and the measure of the fraction of time spent by the system in each one of them. The statistical properties of this intermittent behavior are described and compared with the ones of on–off intermittency.

Rings of chaotic oscillators coupled unidirectionally through driving are studied. While synchronization is observed for small sizes of the ring, beyond a certain critical size a desynchronizing transition occurs. In the two examples studied here the system exhibits a transition to periodic rotating waves for rings of Lorenz systems, while one finds a sort of chaotic rotating waves when Chua's circuit is used.

We consider star networks of chaotic oscillators, with all end-nodes connected only to the central hub node, under diffusive coupling, conjugate coupling and mean-field diffusive coupling. We observe the existence of chimeras in the end-nodes, which are identical in terms of the coupling environment and dynamical equations. Namely, the symmetry of the end-nodes is broken and coexisting groups with different synchronization features and attractor geometries emerge. Surprisingly, such chimera states are very wide-spread in this network topology, and large parameter regimes of moderate coupling strengths evolve to chimera states from generic random initial conditions. Further, we verify the robustness of these chimera states in analog circuit experiments. Thus it is evident that star networks provide a promising class of coupled systems, in natural or engineered contexts, where chimeras are prevalent.

Experimental evidence is provided for the synchronization of chaotic self-excited oscillators and chaotic synchronous response. With unidirectional coupling between two chaotic nonlinear circuits, synchronous chaotic oscillations can result from either forced synchronization or the response phenomenon. With mutual coupling, two originally chaotic oscillators can become time periodic, a result that we have shown to persist for values of the coupling down to noise level.

Using the damped harmonic oscillator equations as a mathematical template, several novel chaotic oscillators are developed with an emphasis on mathematical simplicity and ease of electronic circuit implementation. These chaotic systems offer an intuitive introduction to chaos theory, enabling comparison of mathematical and computational analyses with experimental results.