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A novel 3D chaotic oscillator that incorporates quadratic and absolute-function nonlinearities is introduced in this paper. The system dynamics are explored using the Lyapunov direct method, phase plane trajectories, time response, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The uniqueness and existence of the system solution have been proven. The analytical investigations show that the system has two stable equilibrium points along with one unstable equilibrium point and a line of equilibria. The positive half of this line represents unstable equilibria, while the negative half is associated with stable equilibria. Additionally, it is found that the oscillator exhibits a chaotic basin of attraction centered along the line of equilibria and surrounded by a fixed-point attractor. An electronic circuit using Multisim software is designed to demonstrate the possibility of physical implementation for the considered mathematical model. Chaos control is addressed using adaptive and sliding mode control strategies. The performance of both control methods is compared, with the sliding mode control demonstrating superior results in both fast response and small transient overshoot. Furthermore, a novel sliding mode controller for master–slave synchronization is introduced, showing high performance compared to other sliding-mode and adaptive control methods applied in the literature. Finally, the proposed oscillator is employed as a Pseudo Random Number Generator (PRNG) for image encryption applications using a new encryption model. The experimental results confirm the high security and robustness of the proposed encryption algorithm against various attack methods.

Field programmable gate array (FPGA) implemented array of Duffing oscillators for weak signal detection in noisy environments is presented in this paper. The array of Duffing oscillators is an efficient method for detecting weak signals with unknown exact frequency and phase delay. Because of the high computation resource requirement, especially for online weak signal detection, hardware implementation is a proper solution. In this FPGA implementation, the array was constructed using multiprocessor Nios II system. In the proposed multiprocessor system, each oscillator was implemented by a separate Nios II processor. Also, a separate Nios II processor was dedicated to control and detect the state of other oscillators. The proposed system was implemented using one of the Altera's Cyclone IV GX FPGAs. In this paper, beside the structure and resource utilization of the array of oscillators, an experimental result was also presented in order to show effectiveness of the implementation.

A novel chaotic oscillator based on "cross-coupled" inverter rings is presented. The oscillator consists of a 3-ring to which higher odd n-rings are progressively coupled via diodes and pass gates; it does not contain reactive or resistive elements, and is thus suitable for area-efficient implementation on a CMOS integrated circuit. Numerical simulation based on piece-wise linear approximation predicted the generation of positive spikes having approximately constant periodicity but highly variable cycle amplitude. Simulation Program with Integrated Circuit Emphasis (SPICE) simulations and experimental data from a prototype realized on 0.7 μm technology confirmed this finding, and demonstrated increasing correlation dimension (D₂) as 5-, 7- and 9-rings were progressively coupled to the 3-ring. Experimental data from a ring of 24 such oscillator cells showed phase synchronization and partial amplitude synchronization (formation of small clusters), emerging depending on DC gate voltage applied at NMOS transistors implementing diffusive coupling between neighboring cells. Thanks to its small area, simple synchronizability and digital controllability, the proposed circuit enables experimental investigation of dynamical complexity in large networks of coupled chaotic oscillators, and may additionally be suitable for applications such as broadband signal and random number generation.

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.

This paper presents the design of novel autonomous and non-autonomous inductorless chaotic circuit using lambda diode. The autonomous chaotic circuit is implemented using Chua’s circuit, where the piece-wise linear element of Chua’s circuit called Chua’s diode is replaced by lambda diode. The lambda diode used as a nonlinear resistor in Chua’s circuit comprises of BJT, FET and resistors. The non-autonomous chaotic circuit is studied by replacing the piece-wise linear element of Murali–Lakshmana–Chua (MLC) circuit by lambda diode. The reason for employing lambda diode is that it has a wide range of negative resistance characteristics, which enable the circuit to operate at higher frequency ranges. The resulting chaotic oscillator can easily be made to operate at both low and high frequencies. The chaotic behavior of the circuit is established through Multisim simulations in the time and frequency domains. Both theoretical analysis and electronic circuit experiments are presented. The circuit’s chaotic characteristics are further confirmed by means of Poincare plot and the Bifurcation diagram. The observed route to chaos is period-adding.

This paper presents six different meminductor emulator circuits based on operational amplifiers. Five circuits of meminductor emulators have been proposed using two operational amplifiers, one memristor, three resistors and one capacitor, whereas the sixth circuit uses two operational amplifiers, two memristors, one resistor and two capacitors. All circuits of the proposed meminductor emulators are very simple over most of the realizations of meminductor emulators in the literature. The behaviors of meminductor emulators are satisfactory over a wide range of frequencies. The proposed configurations of meminductor emulators have been simulated by the LTspice tool. The SPICE models of both operational amplifier (AD711) and memristor have been used for simulation. The workability of the proposed meminductor emulators has also been verified using the basic and well-known structure of operational amplifier. In addition, the pinched hysteresis loop obtained by the simulation results of meminductor emulator has been achieved by the experimental results as well. Chaotic oscillator has been designed using the proposed meminductor emulator to prove the worthiness of the design.

In this study, a second-generation positive current conveyor (CCII+)-based analog circuit is proposed for the electronic implementation of a different dynamical system which is an adaptation of the chaotic Lorenz differential equation set. The proposed circuit is more cost-effective and contains less active and passive elements than the circuit obtained by applying the classical parallel synthesis method with opamps. Mathematical analyses and SPICE simulations are performed for chaotic phase portraits and bifurcation diagrams. The proposed dynamical circuit is implemented on the board by using commercially available active and passive elements on the market and an experimental study is conducted. In order to demonstrate the usability of this proposed circuit in secure communication studies, three different synchronization methods are applied and one of them is implemented. The obtained experimental results are in good agreement with the mathematical analysis and simulation results.

In this paper, we have studied, designed and realized a single-transistor chaotic generator with a smooth power spectrum of about −35 dBm in frequency bandwidth up to 1 GHz. The chaos generator model is described by a continuous-time six-dimensional autonomous system assuming an exponential nonlinearity. Chaotic behavior is characterized by bifurcation diagrams, Lyapunov exponents, phase portraits of the attractors and spectra of the oscillations, using both numerical and circuit simulations. Advanced Design System (ADS)-based simulations were carried out to support the theoretical analysis, and to validate the mathematical model. The simulations are carried out using real transistor parameters and taking into account the properties of the substrate, the influence of the board topology and the characteristics of the layout material. This simulation method known as EM/Circuit Co-Simulation allows the simulation results to approach as closely as possible to those of the experiments. In the light of the positive simulation results, the proposed structure is realized to demonstrate its feasibility, and to confirm the numerical results. The prototype is manufactured and mounted on a breadboard using the surface-mount devices and BFP193 bipolar junction transistor. After realizing the generator, we pushed it further towards perfection, through the proposal and realization of a broadband amplifier, in order to gain more bandwidth to improve the spectral characteristic of the generator, which makes it promising for many communication applications such as spread spectrum communication, direct chaotic communication, etc.

In this paper, a new design of meminductor emulator has been suggested using voltage differencing buffered amplifier (VDBA), current differencing buffered amplifier (CDBA), a resistor and two grounded capacitors. The key advantages of the proposed meminductor emulator circuits are their design simplicity, implementation without memristors and easily adjustable design for grounded/ floating decremental to incremental configurations. The proposed circuits have been designed and simulated using 180-nm process technologies in the Mentor Graphics Eldo simulation tool. The nonvolatility and Monte Carlo analysis of the proposed meminductor emulators have been given. A comparison table has also been presented to discuss the benefits of the suggested meminductor emulator designs with existing designs. The operation of the designed emulator has been corroborated by using it in the chaotic oscillator and neuromorphic circuit applications.

In this work, a new cubic-like smooth nonlinearity is generated by modifying Chua’s piecewise-linear segmental nonlinear function using logarithmic cos-hyperbolic function implementation. A logarithmic cos-hyperbolic function possessing smooth symmetric nonlinear characteristics is implemented through CMOS-based circuit design using the current mode approach. The nonlinear design is then incorporated in a new third-order chaotic oscillator configuration to produce chaotic oscillations. This chaotic circuit is tuned to develop different attractors through the bifurcation parameter. Moreover, the dynamics of chaos such as antimonotonicity and coexistence of attractors are also depicted in circuit simulation by tuning various controlling parameters. Additionally, some numerical analyses are performed on this dynamic system to justify the existence of chaoticity and attractors’ development. This design has been optimized for low-voltage and moderately high dominant frequency of oscillations. Simulations are done using 180-nm CMOS technology in Cadence Virtuoso. Experimental results are presented to verify the workability of the proposed chaotic system.

This paper shows how the qualitative theory of dynamical systems, that is intensively developed during the last forty years, is currently applied in practice to provide effective wireless ultrawideband communications using chaotic information carrier.

We fabricated an analog integrated circuit (IC) that implements the Lotka–Volterra (LV) chaotic oscillator presented by Mimura and Kan-on [1986]. The LV system describes periodic or chaotic behaviors in prey–predator systems in simple mathematical form, and is suitable for analog IC implementation [Asai et al., 2003]. The proposed circuit consists of a small number of metal-oxide-semiconductor field-effect transistors (MOS FETs) operating in their subthreshold region. A new scaling factor of system variables, which was not discussed in [Asai et al., 2003], is also introduced for quantitative studies of designing practical hardware LV systems.

A sliding mode control technique is introduced for generalized fractional chaotic systems. These systems are governed by a set of fractional differential equations of incommensurate orders. The proposed design method relies on the fact that the stability region of a fractional system contains the stability region of its underlying integer-order model. A sliding mode controller designed for an equivalent integer-order chaotic system is used to stabilize all its corresponding fractional chaotic systems. The design technique is demonstrated using two generalized fractional chaotic models; a chaotic oscillator and the Chen system. The effect of the total fractional order is investigated with respect to the controller effort and the convergence rate of the system response to the origin. Numerical simulations validate the main results of this work.

Chaotic circuits have been widely used in the teaching of nonlinear dynamics disciplines, where a common practice is pedagogically based on the circuit point of view. Chua's circuit is the most popular platform for the demonstration of its components with rich dynamical patterns. However, engineering students majoring in control systems are more familiar with feedback systems rather than physical electronics with nonlinear components. This may lead to some difficulty in understanding the nonlinear properties of Chua's circuit, at least on first sight. This paper provides an alternative approach to teaching and learning chaotic oscillators by using the inherent understanding of feedback systems with simple modules at the undergraduate level. Utilizing the idea of chaotification, which means to make a nonchaotic module chaotic, the modules consist of only four blocks yet can generate more than thirty types of chaotic patterns via their various combinations. Conceptually, the blocks can be assembled as various mixed-mode chaotic circuits. Functionally, the blocks are easy-to-use in a Simulink-like fashion. Structurally, they are a LEGO-like educational kit. With low-cost implementation employing a few op-amps for each block, the newly proposed modules are simple, self-instructional and suitable for teaching and training students in school laboratories and in experimental environments.

A method of targeting synchronization and its control is reported in chaotic oscillators. A design of appropriate coupling is proposed using an open-plus-closed-loop (OPCL) scheme based on Hurwitz stability to realize a desired state of synchrony between the oscillators. A general theory of the coupling definition is described for unidirectional as well as bidirectional mode. In a synchronization state, a chaotic attractor can be scaled up or down in size relative to another attractor. Additionally, a technique of controlling synchronization is introduced that allows a smooth transition from complete synchronization to antisynchronization or vice versa, by varying a parameter inserted in the coupling definition without loss of stability during the transition. A smooth scaling of the size of the attractor is also implemented. Numerical examples are given using a Sprott system. Physical realization of the OPCL coupling under bidirectional mode and control of synchronization under unidirectional mode is demonstrated in an electronic circuit.

This paper presents a simple autonomous chaotic oscillator. The design method is primarily based on a linear oscillator constructed by a closed loop connection of two building blocks, i.e. an inverting active integrator and a passive second-order LC integrator. A diode is inserted in parallel to the two building blocks for inducing chaos. The mathematical model reveals a set of three-dimensional ordinary differential equations, containing seven terms with four constants and an exponential nonlinearity. The dynamics properties are investigated in terms of an equilibrium point, Jacobian matrix, chaotic attractors, bifurcation, Lyapunov exponents, and chaotic waveforms in time domain. The proposed chaotic oscillator potentially exhibits complex dynamical behaviors through the utilization of only six minimal electronic components.

The chaotic van der Pol oscillator is a powerful tool for detecting defects in electric systems by using online partial discharge (PD) monitoring. This paper focuses on realizing weak PD signal detection in the strong periodic narrowband interference by using high sensitivity to the periodic narrowband interference signals and immunity to white noise and PD signals of chaotic systems. A new approach to removing the periodic narrowband interference by using a van der Pol chaotic oscillator is described by analyzing the motion characteristic of the chaotic oscillator on the basis of the van der Pol equation. Furthermore, the Floquet index for measuring the amplitude of periodic narrowband signals is redefined. The denoising signal processed by the chaotic van der Pol oscillators is further processed by wavelet analysis. Finally, the denoising results verify that the periodic narrowband and white noise interference can be removed efficiently by combining the theory of the chaotic van der Pol oscillator and wavelet analysis.

We study directionally coupled phase-coherent chaotic oscillators in complex networks. We introduce an adjusted Lyapunov function that incorporates the frequencies of the oscillators and the interaction structure. Using the well-known Rössler system as an example, we address two optimization problems: frequency allocation and network design. Through numerical experiments, we demonstrate that the systematic synchrony can be effectively enhanced or inhibited by minimizing or maximizing the objective function, respectively. We then delve into the relationship between the structural and dynamical properties that lead to optimal synchronization. Interestingly, we observe a positive correlation between nodal in-degrees and frequency magnitudes, indicating that nodes with higher in-degrees tend to exhibit larger frequency magnitudes. On the other hand, we also find a negative correlation between nodal frequency and adjacent in-frequencies, suggesting that nodes with higher frequencies tend to be surrounded by nodes with lower frequency values. Finally, we explore the connections between degree correlations and optimal synchronization. We find that when minimizing the objective function, the presence of degree correlations always inhibits the systematic synchrony for frequency allocation, while the act of network design causes the correlations to become negative.