Hopf bifurcation and hidden attractor of a modified Chua’s equation

Nonlinear Dynamics - We made an attempt to provide a realistic picture of the localization of energy in microtubules (MTs), and we intend to model the nonlinear dynamics of MTs using the...     

Memristive oscillator based on Chua’s circuit: stability analysis and hidden dynamics

This paper presents a theoretical stability analysis of a memristive oscillator derived from Chua’s circuit in order to identify its different dynamics, which are mapped in parameter spaces. Since this oscillator can be represented as a nonlinear feedback system, its stability is analyzed using the method based on describing functions, which allows to predict fixed points, periodic orbits, hidden dynamics, routes to chaos, and unstable states. Bifurcation diagrams and attractors obtained from numerical simulations corroborate theoretical predictions, confirming the coexistence of multiple dynamics in the operation of this oscillator.     

Hyperchaos in SC-CNN based modified canonical Chua’s circuit

In this paper, a state-controlled cellular neural network (SC-CNN)-based hyperchaotic circuit is implemented for classical modified canonical Chua’s circuit. The proposed system is modeled by using a suitable connection of four state-controlled generalized CNN cells, while the stability of the circuit is studied by determining the eigenvalues of the stability matrices, the system parameter is varied, and the dynamics as well as the onset of chaos and hyperchaos followed by a period-three doubling bifurcation has been studied through numerical analysis of the generalized SC-CNN equations and real-time experiments. We further validate our findings, the chaotic and hyperchaotic dynamics, characterized by two positive Lyapunov exponents and Lyapunov dimension, is described by a set of four coupled first-order generalized SC-CNN equations. This has been investigated extensively not only analyzing by computer simulation but also demonstrating by laboratory experiments. The experimental results such as phase portraits, Poincaré surface sections and power spectra are in good agreement with those of numerical computations.     

Chaos synchronization and transmission of information in coupled SC-CNN-based canonical Chua’s circuit

In this paper, an experimental implementation of State Controlled Cellular Neural Network (SC-CNN)-based canonical Chua’s circuit is presented. Two such electronic circuits are realized. One of the circuits is configured as driver or transmitter, and other circuit is configured as response or receiver system. Synchronization of chaos between the driver and the driven system is demonstrated. The conditions under which complete synchronization is ensured are derived by applying a strategy based on the Master Stability Function and Lyapunov function. We show that transmitter and receiver systems consisting of cascaded SC-CNN-based canonical Chua’s oscillator can synchronize when they are coupled using direct information from the transmitter into the receiver. The communication performances of different information signals such as sinusoidal, square, and speech signals are demonstrated by real-time hardware experiments. The experimentally observed results are in agreement with the numerically obtained results, and they closely match with each other.     

Single amplifier biquad based inductor-free Chua’s circuit

The present paper reports an inductor-free realization of Chua??s circuit, which is designed by suitably cascading a single amplifier biquad based active band pass filter with a Chua??s diode. The system has been mathematically modeled with three-coupled first-order autonomous nonlinear differential equations. It has been shown through numerical simulations of the mathematical model and hardware experiments that the circuit emulates the behaviors of a classical Chua??s circuit, e.g., fixed point behavior, limit cycle oscillation, period doubling cascade, chaotic spiral attractors, chaotic double scrolls and boundary crisis. The occurrence of chaotic oscillation has been established through experimental power spectrum, and quantified with the dynamical measure like Lyapunov exponents.     

On the Dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors

Nonlinear Dynamics - Chua’s circuit is one of the well-known nonlinear circuits which have been used to study a rich variety of nonlinear dynamic behaviors such as bifurcation, chaos, and...     

Periodicity detection on the parameter-space of a forced Chua’s circuit

We report a Periodicity-Detection algorithm, implemented in a LabVIEW routine for real-time data analysis on experimental chaos, to evaluate the periodicity P of experimental time series. The Periodicity-Detector (PD) algorithm was applied to the forced Chua’s circuit with the aim to build the Periodicity-parameter-space (P-parameter-space). As results of the P-parameter-space, we could observe very complex dynamical behaviors, as regions of periodic structures, a new sequence of accumulation boundary, and the periodic structures organizing themselves in a period-adding bifurcation cascade. Those results agree with the maximal Lyapunov exponent and the bifurcation diagram analysis, presented in a previous work.     

Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit

This paper proposes a modified canonical Chua’s circuit using an one-stage op-amp-based negative impedance converter and an anti-parallel diode pair. Unlike the conventional Chua’s circuit, this modified canonical Chua’s circuit has one unstable zero node-focus and two stable nonzero node-foci, but complex dynamical behaviors including period, chaos, stable point, and coexisting bifurcation modes are numerically revealed and experimentally verified. Up to six kinds of coexisting multiple attractors, i.e., left-right limit cycles, left-right chaotic spiral attractors and left-right point attractors, are numerically depicted and physically captured. Furthermore, with dimensionless Chua’s equations, dynamical properties of the Chua’s system are investigated, and two symmetric stable nonzero node-foci are validated to exist in the selected parameter regions thus resulting in the emergence of multistability. Specially, multistability with six different steady states is revealed in a narrow parameter range. Within this parameter region, three bifurcation routes are displayed under different initial conditions, and three sets of topologically different and disconnected attractors are observed.     

An experimental study on SC-CNN based canonical Chua’s circuit

In this paper, a State Controlled Cellular Neural Network (SC-CNN) based autonomous canonical Chua’s circuit is presented. The proposed system is modeled by using suitable connection of three simple state controlled generalized CNN cells. The stability of the circuit is studied by determining the eigenvalues of the stability matrices, while the dynamics as well as onset of chaos have been studied through real time experiments and numerical analysis of the generalized SC-CNN equations. The experimental results such as phase portraits and power spectra are in good agreement with those of numerical computations.     

An inductor-free realization of the Chua’s circuit based on electronic analogy

Although literature presents several alternatives, an approach based on the electronic analogy was still not considered for the implementation of an inductor-free realization of the double scroll Chua’s circuit. This paper presents a new inductor-free configuration of the Chua’s circuit based on the electronic analogy. This proposal results in a versatile and functional inductorless implementation of the Chua’s circuit that offers new and interesting features for several applications. The analogous circuit is implemented and used to perform an experimental mapping of a large variety of attractors.     

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua’s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua’s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincaré sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.     

On Saint-Venant’s principle in the dynamics of elastic beams

In dynamics, Saint-Venant’s principle of exponential decay of stress resulting from a self-equilibrating load is not valid. For a beam type structure, a self-equilibrated load may penetrate well inside the beam. Although this effect has been known for a long time, at least since Lamb’s paper [Proc. Roy. Soc. Lon. Ser. A 93 (1916) 114], it was not clear how to characterize it quantitatively. In this paper we propose a “probabilistic approach” to evaluate the magnitude of the penetrating stress state. The key point is that, in engineering problems, the distribution of the self-equilibrated load is usually not known. By assigning to the self-equilibrated load some probabilistic measure one can find probabilistic characteristics of the penetrating stress state. We develop this reasoning for the simplest case: longitudinal vibrations of a two-dimensional semi-infinite, elastic isotropic homogeneous strip, excited by a periodic load at the end. We show the frequency range where Saint-Venant’s principle can be used with good accuracy, and thus, one-dimensional classical beam theory still can be applied. We characterize also the increase in this range which is achieved in the refined plate theory proposed by Berdichevsky and Le [J. Appl. Math. Mech. (PMM) 42 (1) (1978) 140].     

Optical solitons and bifurcation analysis in fiber Bragg gratings with Lie symmetry and Kudryashov’s approach

A combination of Lie symmetry analysis and Kudryashov’s approach secures optical soliton solutions with fiber Bragg gratings. The bifurcation analysis is carried out, and the phase portrait is presented.