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(a) Bloc diagram of the Arduino microcontroller based realization of system (2) followed by a fast data acquisition and representation module; (b) the experimental set-up for practical experiment.
Source publication
We investigate the dynamics of a pair of coupled non oscillatory Rayleigh-Duffing oscillators (RDOs here after). The RDO serves as a model for a class of nonlinear oscillators including microwave Gunn oscillators [Guin et al., Comm. in Nonlinear Sci. Numerical Simulat, 2017]. Here, the coupling between the two oscillators is obtained by superimposi...
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Citations
... Instead of using self-oscillatory systems, in this scheme, double-scroll chaos from one oscillator is forced into another similar oscillator in a resting state. Very recently, a four-scroll chaotic attractor was obtained from a pair of coupled non-oscillatory Rayleigh-Duffing oscillators [4]. ...
It is known that the coupling of nonlinear oscillators can result in extremely rich and complex dynamics not found in an isolated oscillator. This work introduces a novel four-dimensional autonomous system designed by coupling two second-order damped oscillators with hyperbolic sine nonlinearity in such a way that each oscillator is perturbed by a signal proportional to the amplitude of the other one. The model presents both odd and rotational symmetry. The stability and nature of the equilibrium points are studied by resorting to the Routh–Hurtwitz criterion and the Hopf bifurcation theorem. A direct numerical integration of the evolution equation reveals rich and striking features such as multiple Hopf bifurcations, four-scroll chaotic attractors, and the coexistence of multiple modes of oscillations (i.e. homogeneous and heterogeneous multistability) when adjusting the coupling coefficients and the dissipation as well. These striking features are visualized by utilizing phase space trajectory plots, basins of attraction, plots of maximum Lyapunov exponents and bifurcation diagrams. A simple analogue electronic circuit emulating the coupled oscillators system is designed by using only basic electronic components consisting of operational amplifiers, resistors, capacitors and two pairs of diodes (implementing the hyperbolic sine nonlinearities). PSpice simulations and the microcontroller-based implementation are in accordance with the predictions of the theoretical investigations. One of main conclusions of the present study is that the coupling of second-order damped oscillators with hyperbolic sine function (only) can be seen as an alternative method to design multiscroll chaos generators. The proposed approach then differs from previous ones employing piecewise-linear functions, sine functions, and hysteresis functions or coupled third-order oscillators where the electronic design of the nonlinearity is relatively complex.
... In computational neuroscience, therefore, several research studies have been carried out to improve our knowledge to the electrical activity of coupled neurons models [3] [3][4][5]. As a result, several phenomena were observed in those works, including firing activity [6,7], multi-scroll dynamics [8][9][10], and multistability [11][12][13]. The design of simple chaotic models with complex chaotic orbits and multiple coexistence of attractors is still a hot topic in nonlinear dynamics area [14,15]. ...
... Bistability, offset boosting and microcontroller implementation are also reported in the model. Balamurali et al explore the dynamics of a coupled two non-oscillatory Rayleigh-duffing oscillators and shows extremely complex nonlinear behaviors (multistability, multi-scroll generation and coexisting bifurcation) [9]. Since the nervous system is made up of an interconnection of a large number of neurons with different roles and functionalities, this paper introduces a coupled Hopfield inertial neurons with different activation functions. ...
This work deals with the regular and chaotic dynamics of a system made up of two Hopfield-type neurons with two different activation functions: the hyperbolic tangent function and the Crespi function. The mathematical model is in the form of an autonomous differential system of order four with odd symmetry. The analysis highlights nine equilibrium points and four of these points experience a Hopf bifurcation at the same critical value of a control parameter which can be either the diss1ipation parameter or one of the coupling coefficients. This makes plausible the presence of four parallel bifurcation branches as well as the coexistence of multiple attractors in the behavior of the system. One of the highlights revealed in this work is the coexistence of three double-scroll type attractors of particular topology as well as the presence of a four-spiral attractor. Furthermore, the coexistence of both self-excited and hidden dynamics is also reported. All this plethora of dynamics is elucidated by making use of the usual tools for analyzing nonlinear systems such as bifurcation diagrams, the maximum of Lyapunov exponent, basins of attractions as well as phase portraits. A physical implementation of the microcontroller-based system is envisaged in order to confirm the plethora of behaviors observed theoretically.
... In the past twenty years, a variety of design methods for multi-scroll and multi-wing chaotic attractors have been proposed. In the multi-scroll system, the number of scrolls is extended by using some nonlinear functions, such as the PWL function [9,10], step function [11], sine function [12], switching flow pattern [13,14], saturated delay sequence [15], parameterized n-order polynomial transformation [16], TanSig activation function [17], hyperbolic tangent function [18], non-oscillatory rayleigh-duffing oscillators [19], etc. These nonlinear functions can better control the scroll amount of chaos by adjusting the parameters of the system. ...
In comparison to traditional chaotic systems, the multi-scroll and multi-wing chaotic systems are more complicated. The design and execution of sophisticated multi-scroll or multi-wing chaotic attractors attract a lot of attention. However, these constructed nonlinear functions cannot be applied to extended multi-scroll and multi-wing attractors at the same time. To this end, this paper proposes a new function which can be used to generate multi-scroll and multi-wing chaotic attractors in both double-scroll and double-wing chaotic systems. Using this function, multi-scroll and multi-wing chaotic systems can be constructed directly without relying on whether the chaotic system has some symmetry (odd symmetry or even symmetry). The construction method presented is generally applicable to chaotic systems with multi-scroll and multi-wing self-excited attractors.The main point of this method is as follows: firstly, the piecewise linear (PWL) saturation function is nested within the cosine nonlinearity function , and the resulting nested COS-PWL function. Secondly, to enable the expansion of multi-wing and multi-scroll, the nested COS-PWL function is incorporated into the double-wing and double-scroll systems in different manners. The maximum Lyapunov exponent (MLE) and the bifurcation diagram route for increasing the number of wings and scrolls confirm the feasibility and effectiveness of the method. Finally, the three-element method is used to determine a Sinusoidal function, which can generate attractor self-reproduction in the corresponding dimension by replacing the state variables of the multi-scroll and multi-wing systems, so that an infinite number of coexisting attractors can be obtained by simply changing the initial values of the variables, i.e., multiple stability can be generated.
... In addition, multi-scroll chaotic attractor can also be generated via smooth state transformation [30], switching control [31][32][33], fractal [34][35][36][37], fractional-order derivatives [38], non-autonomous approach [39][40][41][42], amplitude control method [43], system coupling [44][45][46][47][48] and so on. Some approaches are also applicable for generating multi-scroll hidden attractors [49][50][51] and fractional-order multi-scroll attractors [52,53]. ...
The oscillatory potential well can cause the mass points in it to move chaotically, which can be considered as a physical mechanism of chaos generation. Based on this physical mechanism, the oscillatory multi-well potential with multi-concave is used to generate controllable multi-scroll chaotic attractors with multi-wing. These attractors have two kinds of topological units: scroll and wing. The topological structure of the attractor depends on the shape of the potential well, that is, the wells and the concaves correspond to the scrolls and wings of the attractor. By constructing potential wells with different shapes, we can obtain attractors with different topo-logical structure. The number of scrolls and wings of the chaotic attractor can be controlled by the oscillation amplitude of the potential well and damping coefficient, that is, stronger oscillation or smaller damping allows the mass points to enter the higher potential wells or concaves, resulting in attractors with more scrolls or wings. As the number of scrolls or wings increases, the attractors become more complex, as expressed by Poincaré maps and quantified by the Kaplan-Yorke dimensions. Kaplan-Yorke dimensions of attractors can be adjusted continuously from 2 to 3 by the two control parameters. Finally, the results are confirmed by Multisim simulation circuit and actual analog circuit.
... Mechanical dynamical systems have several applications in chaos theory [1][2][3][4]. For instance, Reis and Savi [1] studied quasi-periodic and spatiotemporal chaotic responses in a conservative Duffing-type mechanical system with cubic nonlinearity. ...
... Madiot et al. [3] demonstrated dissipative chaotic motion in an electromechanical resonator. Balamurali et al. [4] investigated the generation of multi-scroll chaos in coupled Rayleigh-Duffing oscillators. Other related investigations can be found in recent papers, which in addition, include the hardware implementation of the chaotic systems. ...
Mechanical jerk systems have applications in several areas, such as oscillators, microcontrollers, circuits, memristors, encryption, etc. This research manuscript reports a new 3-D chaotic jerk system with two unstable balance points. It is shown that the proposed mechanical jerk system exhibits multistability with coexisting chaotic attractors for the same set of system constants but for different initial states. A bifurcation analysis of the proposed mechanical jerk system is presented to highlight the special properties of the system with respect to the variation of system constants. A field-programmable gate array (FPGA) implementation of the proposed mechanical jerk system is given by synthesizing the discrete equations that are obtained by applying one-step numerical methods. The hardware resources are reduced by performing pipeline operations, and, finally, the paper concludes that the experimental results of the proposed mechanical jerk system using FPGA-based design show good agreement with the MATLAB simulations of the same system.
... Mechanical dynamical systems have several applications in chaos theory [1][2][3][4]. For instance, Reis and Savi [1] studied quasi-periodic and spatiotemporal chaotic responses in a conservative Duffing-type mechanical system with cubic nonlinearity. ...
... Madiot et al. [3] demonstrated dissipative chaotic motion in an electromechanical resonator. Balamurali et al. [4] investigated the generation of multi-scroll chaos in coupled Rayleigh-Duffing oscillators. Other related investigations can be found in recent papers, which in addition, include the hardware implementation of the chaotic systems. ...
Network dynamics is the subject of a great deal of scientific research these days. Its scope is not only restricted to the study of the collective dynamic behavior of the internal subunits of a network or the interaction of connected systems with their neighbors, but it is also applicable to some of today's cutting-edge technologies, and helps us to understand the complex phenomena that define the world around us. In this study, a new simple dynamical network is constructed from a special bidirectional coupling between three jerk oscillators. The jerk oscillator has been widely studied in the literature, but little is known about the collective dynamics described by networks of jerk systems linked by a special bidirectional coupling scheme. The used coupling scheme is special in the sense that it favors the creation of twenty-seven additional equilibrium points in each subunit of the network. This extra point induces new complex behaviors in each subsystem of the network, in particular multiscroll behaviors, which have an extraordinary dynamic richness and are applicable to chaos-based technologies. However, as the methods used in the literature to generate multiscroll dynamics are not very explanatory, the special coupling scheme highlighted in this work constitutes a new method for generating multiscroll behavior in a system. New complex phenomena are understood through theoretical and numerical study, including symmetry, stability of equilibrium points, bifurcation diagrams, phase portraits, multistability, basins of attraction, coexisting bifurcation branches, one- and two-parameter diagrams, maximum Lyapunov exponent diagram and Lyapunov exponent spectrum to show the hyperchaotic aspect. Furthermore, as in most chaos-based applications, digital or analog implementation is an unavoidable exercise, we explored a practical realization of the new dynamical network using two different techniques: an analog circuit-based implementation with PSpice simulations and a microcontroller-based implementation, followed by a data acquisition module. These implementations validate the different results obtained numerically.
