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This paper is reporting on electronic implementation of a three-dimensional autonomous system with infinite equilibrium point belonging to a parabola. Performance analysis of an adaptive synchronization via relay coupling and a hybrid steganography chaos encryption application are provided. Besides striking parabolic equilibrium, the proposed three-dimensional autonomous system also exhibits hidden chaotic oscillations as well as hidden chaotic bursting oscillations. Electronic implementation of the hidden chaotic behaviors is done to confirm their physical existence. A good qualitative agreement is shown between numerical simulations and OrCAD-PSpice results. Moreover, adaptive synchronization via relay coupling of three three-dimensional autonomous systems with a parabolic equilibrium is analysed by using time histories. Numerical results demonstrate that global synchronization is achieved between the three units. Finally, chaotic behavior found is exploited to provide a suitable text encryption scheme by hidden secret message inside an image using steganography and chaos encryption.

The present study addresses the fluid transport behaviour of the flow of gold (Au)-copper (Cu)/biomagnetic blood hybrid nanofluid in an inclined irregular stenosis artery as a consequence of varying viscosity and Lorentz force. The nonlinear flow equations are transformed into dimensionless form by using nonsimilar variables. The finite-difference technique (FTCS) is involved in computing the nonlinear transport dimensionless equations. The significant parameters like angle parameter, the Hartmann number, changing viscosity, constant heat source, the Reynolds number, and nanoparticle volume fraction on the flow field are exhibited through figures. Present results disclose that the Lorentz force strongly lessens the hybrid nanofluid velocity. Elevating the Grashof number values enhances the rate of blood flow. Growing values of the angle parameter cause to reduce the resistance impedance on the wall. Hybrid nanoparticles have a superior wall shear stress than copper nanoparticles. The heat transfer rate is amplifying at the axial direction with the growing values of nanoparticles concentration. The applied Lorentz force significantly reduces the hybrid and unitary nanofluid flow rate in the axial direction. The hybrid nanoparticles expose a supreme rate of heat transfer than the copper nanoparticles in a blood base fluid. Compared to hybrid and copper nanofluid, the blood base fluid has a lower temperature.

This work introduces a three-dimensional, highly nonlinear quadratic oscillator with no linear terms in its equations. Most of the quadratic ordinary differential equations (ODEs) such as Chen, Rossler, and Lorenz have at least one linear term in their equations. Very few quadratic systems have been introduced and all of their terms are nonlinear. Considering this point, a new quadratic system with no linear term is introduced. This oscillator is analyzed by mathematical tools such as bifurcation and Lyapunov exponent diagrams. It is revealed that this system can generate different behaviors such as limit cycle, torus, and chaos for its different parameters’ sets. Besides, the basins of attractions for this system are investigated. As a result, it is revealed that this system’s attractor is self-excited. In addition, the analog circuit of this oscillator is designed and analyzed to assess the feasibility of the system’s chaotic solution. The PSpice simulations confirm the theoretical analysis. The oscillator’s time series complexity is also investigated using sample entropy. It is revealed that this system can generate dynamics with different sample entropies by changing parameters. Finally, impulsive control is applied to the system to represent a possible solution for stabilizing the system.

Here, a novel conservative chaotic oscillator is presented. Various dynamics of the oscillator are examined. Studying the dynamical properties of the oscillator reveals its unique behaviors. The oscillator is multistable with symmetric dynamics. Equilibrium points of the oscillator are investigated. Bifurcations, Lyapunov exponents (LEs), and the Poincare section of the oscillator’s dynamics are analyzed. Also, the oscillator is investigated from the viewpoint of initial conditions. The study results show that the oscillator is conservative and has no dissipation. It also has various dynamics, such as equilibrium point and chaos. The stability analysis of equilibrium points shows there are both stable and unstable fixed points.

Designing chaotic systems with different properties helps to increase our knowledge about real-world chaotic systems. In this article, a piecewise linear (PWL) term is employed to modify a simple chaotic system and obtain a new chaotic model. The proposed model does not have any equilibrium for different values of the control parameters. Therefore, its attractor is hidden. It is shown that the PWL term causes an offset boostable variable. This feature provides more flexibility and controllability in the designed system. Numerical analyses show that periodic and chaotic attractors coexist in some fixed values of the parameters, indicating multistability. Also, the feasibility of the system is approved by designing field programmable gate arrays (FPGA).

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

The elliptic problem with a nonlocal boundary condition is widely applied in the field of science and engineering, such as the chaotic system. Firstly, we construct one high-accuracy difference scheme for a kind of elliptic problem by tactfully introducing an equivalent relation for one nonlocal condition. Then, we obtain the local truncation error equation by the Taylor formula and, initially, prove that the new scheme can reach the asymptotic optimal error estimate O h 2 ln h in the maximum norm through ingeniously transforming a two-dimensional problem to a one-dimensional one through bringing in the discrete Fourier transformation. Numerical experiments demonstrate the correctness of theoretical results.

Dynamical properties of a two-dimensional airfoil model with higher-order strong nonlinearities are investigated. Firstly, a state-space model is derived considering the plunge and pitch stiffnesses as generalized functions. Then, a stiffness function having square, cubic, and fifth-power nonlinearities is considered for both plunging and pitching stiffnesses, and the dimensionless state equations are derived. Various dynamical properties of the proposed model are investigated using equilibrium points, eigenvalues, and Lyapunov exponents. To further analyze the dynamical behavior of the system, bifurcation plots are derived. It is interesting to note that the new airfoil model with higher-order nonlinearities shows multistability with changing airspeed, and there are infinitely countable number of coexisting attractors generally called as megastability. Both multistability and megastability features of the airfoil model were not captured earlier in the literatures. To be clear, it is the first time a megastable feature is exposed in a physical system. Finally, to analyze the multifrequency effects of the airfoil model, we have presented the bicoherence plots.

Josephson junction devices play a significant role in various physical nonlinear systems because of their complex characteristics. Chaotic phenomenon in various types of Josephson junction devices has been widely reported, but many of those literature studies exempted the analysis into multistability and megastability features of the device. In this work, we investigate the network behaviour using a type of Josephson junction-memristor (JJM) device considering the feedback flux effects while modelling. We have considered both AC- and DC-type external excitation currents, and while considering the AC excitation, the system shows megastability (Ramakrishnan et al. 2020). When analysing the lattice layer network constructed with JJM excited by DC bias current, the network shows a turbulent behaviour thus forming spiral waves. This was not the case when we applied AC bias current for which the network showed a much pattern-like formation confirming localised areas of energy distribution. This energy distribution is due to the homogeneous states of the local nodes which are correlated by the respective periodicity plots. When we apply AC bias current with very low frequency, the network shows small areas of local spirals which are soon dissipated by the inhomogeneous nodes nearby. Thus, we could show that the external bias current plays an important role in the collective performance of the Josephson junction devices. © 2021 Balamurali Ramakrishnan et al.

An electronic implementation of a novel Wien bridge oscillation with antiparallel diodes is proposed in this paper. As a result, we show by using classical nonlinear dynamic tools like bifurcation diagrams, Lyapunov exponent plots, phase portraits, power density spectra graphs, time series, and basin of attraction that the oscillator transition to chaos is operated by intermittency and interior crisis. Some interesting behaviors are found, namely, multistability, hyperchaos, transient chaos, and bursting oscillations. In comparison with some memristor-based oscillators, the plethora of dynamics found in this circuit with current-voltage (i–v) characteristic of diodes mounted in the antiparallel direction represents a major advance in the knowledge of the behavior of this circuit. A suitable microcontroller based design is built to support the numerical findings as these experimental results are in good agreement.

A two-degree-of-freedom shape memory oscillator derived using polynomial constitutive model is investigated. Periodic, quasiperiodic, chaotic, and hyperchaotic oscillations are shown by the shape memory alloy based oscillator for selected values of the operating temperatures and excitation parameters. Bifurcation plots are derived to investigate the system behavior with change in parameters. A fractional order model of the shape memory oscillator is presented and dynamical behavior of the system with fractional orders and parameters are investigated.

We investigate a specific smart grid system and its nonlinear properties. Lyapunov exponents are derived to prove the existence of chaos and bifurcation and bicoherence contours are investigated to show the parameter dependence and existence of quadratic nonlinearities, respectively. A fractional order model of the smart grid system (FOSG) is then derived and bifurcation of the FOSG system with variation in the commensurate fractional order of the system is investigated to show that largest Lyapunov exponent of the system exists in fractional order. Hence we proposed two different control methods to suppress the chaotic oscillations. In the first method we derive fractional order adaptive sliding mode control (FOASMC) algorithm to control chaotic oscillations and in the second method we used genetically optimized fractional order PID controllers (GAFOPID) for chaos control. Numerical simulations are conducted to show the effectiveness of the controllers and also to prove that GAFOPID controllers are more effective than FOASMC controllers for fractional order systems. The GAFOPID controllers are then realized in FPGA to show that the proposed methodology is hardware realizable.

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

A system with an absolute nonlinearity is studied in this work. It is noted that the system is chaotic and has an adjustable amplitude variable, which is suitable for practical uses. Circuit design of such a system has been realized without any multiplier and experimental measurements have been reported. In addition, an adaptive control has been applied to get the synchronization of the system.