Chaos Generation Research Articles

Chaotic Dynamics of Colpitts Oscillator Under Control of MEMS Feedback

The nonlinear dynamics of Colpitts oscillator under control of MEMS varactor in feedback connectivity has been analyzed with objectives for generation and control of high frequency chaotic signals. The feedback signal derived from the capacitive divider in the standard Colpitts oscillator is modified by the MEMS varactor response mirrored by a voltage-controlled current multiplier. The latter implements MEMS capacitance multiplication and serves as a control parameter. The effects of voltage nonlinearity of the MEMS capacitance and the capacitance multiplication factor (α) have been analyzed by employing Lyapunov exponent, bifurcation diagram, phase portrait and Fourier transform methods. The modified feedback network facilitates high frequency chaos generation due to frequency doubling and high pass filtering effects of the MEMS capacitance. The latter emphasizes high frequency generation and attenuates lower frequencies. The variation of capacitance multiplication factor allows systematic changes in the qualitative nature of oscillator dynamics from a stable low frequency noisy state to Hopf bifurcation to period doubling/ tripling to chaos generation. The analysis suggests new MEMS based tuning and control of chaotic Colpitts oscillations.

Laser-based optoelectronic generation of narrowband microwave chaos for radars and radio-communication scrambling.

This Letter proposes an optoelectronic oscillator architecture for narrowband microwave chaos generation. In the time domain, the microwave signal features a slowly varying envelope with amplitude and phase chaos while, in the frequency domain, it is quasi-indistinguishable from a band-limited white noise. A full theoretical analysis is performed to investigate the stability properties and route to chaos for the microwave oscillations. We experimentally generate the narrowband microwave chaos with a central frequency of 3GHz and a bandwidth of only 16MHz, and we discuss the applications for radar engineering and radio-communication scrambling.

FPGA based hardware and device-independent implementation of chaotic generators

In this paper we will describe the use of Matlab/Simulink without third party toolboxes as a single integrated environment to study and generate the HDL description of chaotic systems for rapid prototyping purposes. We will provide a review of different approaches to implement chaos generators in both analog and digital forms with a focus on digital one. We will present a rapid prototyping method of digital chaos generators on FPGAs that is of great use for a seamless study, simulation and implementation of those systems in one environment. This will ease the design phase of the chaotic systems and will provide more flexibility for their study. Multiple chaos generators, such as Lorenz, Rössler, Chua, Linz-Sprott, and Sprott types have been designed using this method to showcase its effectiveness. In addition, for a better investigation of the chaotic dynamics, this work makes use of an implementation scheme that permits the visualization of the 3D chaotic attractor on the oscilloscope. The main focus of this paper is to provide a common implementation framework of chaos generators to target various applications, mainly cryptographic one. Newcomers to the field will easily dive in and active researchers will be able to speak the same language and build upon one anothers works more efficiently by adopting the presented guidelines.

White Gaussian Chaos

A discrete-time white Gaussian noise (WGN) is a random process with impulsive autocorrelation function. Also, the random variables obtained by sampling the process at any time instants are jointly Gaussian. WGN is widely used to model noise in engineering and physics. In this letter, we propose a way to generate chaotic signals that behave like WGN, due to the features of its autocorrelation function and its invariant density. From the tent map and by using as conjugacy map transformations commonly employed to random variables, we obtain a white Gaussian chaos (WGC) map. Numerical simulations are shown to illustrate the technique. WGC can be an interesting choice for chaos generator in chaos-based communication systems.

Simulations of chaos generation from Josephson junctions with various junction parameters

It is well known that voltage waveforms between electrodes of Josephson junctions under irradiation of a microwave behave chaos characteristics under appropriate conditions. In order to apply the chaos to a random number generator, we have been studying Josephson chaos by simulations. In the simulation, the Josephson junction is assumed to fabricate with YBCO materials. We used a RCSJ model in order to present an equivalent circuit of the Josephson junction, and derived a derivative equation. Lyapunov exponents, which determined if the state of the Josephson junction was chaotic or not, were calculated from time evolutions of voltages obtained from the equation. In the simulation, junction parameters were assigned feasible values for an actual YBCO Josephson junctions. As a result, we found that chaos can be generated by adjusting element parameters. Moreover, we found that there were lower limits in the resistance values for generation of chaos. In addition, we found that frequency margins, at which the chaos was obtained, were broadened by decrease of the resistance and increase of capacitance.

A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers

A memristor is the fourth circuit element, which has wide applications in chaos generation. In this paper, a four-dimensional hyperchaotic jerk system based on a memristor is proposed, where the scroll number of the memristive jerk system is controllable. The new system is constructed by introducing one extra flux-controlled memristor into three-dimensional multiscroll jerk system. We can get different scroll attractors by varying the strength of memristor in this system without changing the circuit structure. Such a method for controlling the scroll number without changing the circuit structure is very important in designing the modern circuits and systems. The new memristive jerk system can exhibit a hyperchaotic attractor, which has more complex dynamic behavior. Furthermore, coexisting attractors are observed in the system. Phase portraits, dissipativity, equilibria, Lyapunov exponents and bifurcation diagrams are analyzed. Finally, the circuit implementation is carried out to verify the new system.

Generation of flat wideband chaos with suppressed time delay signature by using optical time lens.

We propose a flat wideband chaos generation scheme that shows excellent time delay signature suppression effect, by injecting the chaotic output of general external cavity semiconductor laser into an optical time lens module composed of a phase modulator and two dispersive units. The numerical results demonstrate that by properly setting the parameters of the driving signal of phase modulator and the accumulated dispersion of dispersive units, the relaxation oscillation in chaos can be eliminated, wideband chaos generation with an efficient bandwidth up to several tens of GHz can be achieved, and the RF spectrum of generated chaotic signal is nearly as flat as uniform distribution. Moreover, the periodicity of chaos induced by the external cavity modes can be simultaneously destructed by the optical time lens module, based on this the time delay signature can be completely suppressed.

Mesoscopic chaos mediated by Drude electron-hole plasma in silicon optomechanical oscillators

Chaos has revolutionized the field of nonlinear science and stimulated foundational studies from neural networks, extreme event statistics, to physics of electron transport. Recent studies in cavity optomechanics provide a new platform to uncover quintessential architectures of chaos generation and the underlying physics. Here, we report the generation of dynamical chaos in silicon-based monolithic optomechanical oscillators, enabled by the strong and coupled nonlinearities of two-photon absorption induced Drude electron–hole plasma. Deterministic chaotic oscillation is achieved, and statistical and entropic characterization quantifies the chaos complexity at 60 fJ intracavity energies. The correlation dimension D2 is determined at 1.67 for the chaotic attractor, along with a maximal Lyapunov exponent rate of about 2.94 times the fundamental optomechanical oscillation for fast adjacent trajectory divergence. Nonlinear dynamical maps demonstrate the subharmonics, bifurcations and stable regimes, along with distinct transitional routes into chaos. This provides a CMOS-compatible and scalable architecture for understanding complex dynamics on the mesoscopic scale.

A Novel Micro-Architecture Using a Simplified Logistic Map for Embedded Security

A novel scheme for the introduction of chaos-based security into embedded systems at an architectural (hardware) level is presented in this letter. The logistic map is simplified for use in resource-constrained microcontrollers in the encryption of digital data through interface protocols like SPI etc. This letter, as an example, presents an outline of the registers and their associated logic for such an 8-bit chaotic modulator. Correlation analyses along with a Gottwald-Melbourne test proved its robustness for security. This letter develops a micro-architecture that involves the proposed chaos generator and discusses the control methods by which data is to be handled for seamless encryption/decryption without processor involvement. The descriptive work presented is proposed to be a silicon level encryption technique and its implications are generalized to be used with other word sizes and interface protocols, in any microcontroller.

Chaos generation in fractional-order switched systems and its digital implementation

A double-scroll chaotic attractor generated by a fractional-order switched system is presented. Based on unstable dissipative systems (UDS) and a switching law, a fractional-order UDS is proposed. Chaos behavior is found with a fractional order as low as 2.568 by satisfying the stability criterion for fractional order chaotic systems. The fractional-order switched system is transformed to an equivalent switched system with augmented states. Then, the equivalent system is straightforward implemented on an ARM system-on-chip board by using Python high level programming language. Numerical simulations are in good agreement with experimental results, which demonstrates the usefulness of proposed method. Additionally, the test 0–1 and a chaos definition for finite state sets are given to demonstrate chaotic behavior. Finally, a random number generator is also proposed as a possible application.

Formal verification of stability and chaos in periodic optical systems

Optical systems are widely used in a wide range of safety, cost and mission-critical applications including biomedical devices and high-speed communication networks. Therefore, modeling and verification of such high-consequence systems is crucial for both theoretical and application viewpoints. In this paper, we propose a formal methods based approach to model and verify the properties of periodic optical systems which allow a cyclic passage of light through a sequence of optical components. We focus on two important properties namely stability and chaotic map generation which ensure the confinement of light and chaos generation, respectively. We use higher-order logic as a specification and reasoning framework and develop a library of necessary notions of periodic optical systems. Consequently, we demonstrate the utilization and effectiveness of our development by a couple of case studies: a Fabry Pérot resonator with fiber-rod lens and a phase-conjugated ring resonator.

Chaos and hyperchaos in simple gene network with negative feedback and time delays.

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.

Mechanical analysis of Chen chaotic system

The Chen chaotic system is transformed into Kolmogorov type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. By the combinations of different torques, five cases are studied to discover key factors of chaos generation and the physical meaning. The conversion among Hamiltonian energy, kinetic energy and potential energy is investigated in these five cases. The relationship between the energies and the parameters is studied. It concludes that the combination of these four types of torques is necessary conditions to produce chaos, and any combination of three types of torques cannot produce chaos in Chen system.

Cardiac alternans reduction by chaotic attractors in T ± ϵ feedback control

To achieve control in dynamical systems, one usually avoids the generation of chaos. However, an effective control of an unstable dynamical system can also be practically achieved by deliberately inducing chaos in the system if the induced chaotic attractors are confined in a very small region in phase space. Using nonlinear dynamic analysis of the cardiac restitution model, we reveal that the mechanism of the recently proposed T ± ϵ control for cardiac alternans suppression is due to the confined chaotic attractors so generated. These results are confirmed experimentally by detailed phase-space portrait and transition dynamics measurements in whole heart experiments.

Numerical proof for chemostat chaos of Shilnikov's type

A classical chemostat model is considered that models the cycling of one essential abiotic element or nutrient through a food chain of three trophic levels. The long-time behavior of the model was known to exhibit complex dynamics more than 20 years ago. It is still an open problem to prove the existence of chaos analytically. In this paper, we aim to solve the problem numerically. In our approach, we introduce an artificial singular parameter to the model and construct singular homoclinic orbits of the saddle-focus type which is known for chaos generation. From the configuration of the nullclines of the equations that generates the singular homoclinic orbits, a shooting algorithm is devised to find such Shilnikov saddle-focus homoclinic orbits numerically which in turn imply the existence of chaotic dynamics for the original chemostat model.

On-chip optoelectronic feedback in a micropillar laser-detector assembly

Implementing time-delayed feedback in optoelectronic circuits allows one to uncover the rich physics and application potential of nonlinear dynamics. Important feedback effects are, for instance, the generation of broadband chaos, or laser self-pulsing. We explore the effect of optoelectronic feedback in an ultracompact microlaser–microdetector assembly operating in the regime of cavity quantum electrodynamics (cQED). This system is used to generate self-pulsing at MHz frequencies in the emission of a microlaser, which is qualitatively explained by a rate equation model taking cQED effects into account. The results show promise for exploring chaos in ultracompact nanophotonic systems and for technological approaches toward chaos-based secure communication, random number generation, and self-pulsed single photon sources on a highly integrated semiconductor platform.

Генератор хаоса с аттрактором Смейла–Вильямса на основе эффекта гибели колебаний

Генератор хаоса с аттрактором Смейла–Вильямса на основе эффекта гибели колебаний

Secure Duobinary Signal Transmission in Optical Communication Networks for High Performance & Reliability

A numerical study is presented in this paper based on semiconductor laser chaos generation which is being used to hide multi-level data signal, i.e., duo-binary message to take the advantage of secure environment & higher data rates at the same time. Duobinary message is generated by using the combination of duobinary precoder, duobinary generator & RZ/NRZ pulse generator. Laser rate equations are used to model chaos generation through semiconductor laser whereas message is made secured by hiding it through chaos masking scheme. Propagation of chaos hiding the multi-format message is studied for long distance communication model. Synchronization between transmitter & receiver is achieved to obtain the acceptable eye-diagrams & quality factor (Q-factor). Q-factor is function of optical signal to noise ratio (OSNR) that gives a qualitative performance of receiver. It provides the minimum SNR to obtain specific bit error rate of signal. A comparison is made with & without deployment of chaos masking scheme on RZ & NRZ duobinary optical system to observe the penalty in terms of Q-factor. In addition, response of amplifier on chaotic signal due to its nonlinearities is investigated by varying the gain of amplifier.

VHDL Descriptions for the FPGA Implementation of PWL-Function-Based Multi-Scroll Chaotic Oscillators.

Nowadays, chaos generators are an attractive field for research and the challenge is their realization for the development of engineering applications. From more than three decades ago, chaotic oscillators have been designed using discrete electronic devices, very few with integrated circuit technology, and in this work we propose the use of field-programmable gate arrays (FPGAs) for fast prototyping. FPGA-based applications require that one be expert on programming with very-high-speed integrated circuits hardware description language (VHDL). In this manner, we detail the VHDL descriptions of chaos generators for fast prototyping from high-level programming using Python. The cases of study are three kinds of chaos generators based on piecewise-linear (PWL) functions that can be systematically augmented to generate even and odd number of scrolls. We introduce new algorithms for the VHDL description of PWL functions like saturated functions series, negative slopes and sawtooth. The generated VHDL-code is portable, reusable and open source to be synthesized in an FPGA. Finally, we show experimental results for observing 2, 10 and 30-scroll attractors.

Classical chaos and harmonic generation in laser driven nanorings

A quantum ring driven by an intense laser field emits light in the form of high-harmonic radiation resulting from the strong acceleration experienced by the active electrons forced to move on a curved trajectory. The spectrum of the emitted light is rich and strongly dependent on the parameters of the problem. In order to investigate the physical origin of such variability, we focus on the seemingly simple problem of a laser-driven charge constrained to a ring from a classical standpoint. As it turns out, the dynamics of such a classical electron is governed by a nonlinear equation which results into a chaotic motion—by nature depending on the initial conditions in an unpredictable way. Our results indicate that the quantum harmonic spectra are reminiscent of the classical counterpart and suggest the existence of a line connecting the quantum and classical realms.