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Cross section of the basins of atraction of five different coexisting attractors (see Fig. 15) in a symmetrical mode of oscillation a and the five different coexisting attractors (see Fig. 12) in a asymmetrical mode of oscillation b. The blue basin corresponds to the chaotic attractor whose average x is positive, the yellow basin corresponds to the chaotic attractor whose average x is negative, the red basin corresponds to the period-3 limit-cycle whose average x is negative, the magenta basin corresponds to the period-3 limit-cycle whose average x is positive and finally the green basin corresponds to the hidden attractor period-1 limit-cycle. Noting that the blue basin corresponding to period-1 limit cycle does not intersect with neighborhood of any equilibrium points plotted in cyan
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This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equ...
Citations
... The application of Duffing oscillators can be found in problems related to electronic circuits. We refer the reader to latest works involving the application of Duffing type oscillators which are examined by [2,3,4,5,6,7]. ...
... Based on the oscillator, a set of modified versions have been presented to explore the different properties in nonlinear systems, such as the multistability [Qriouet & Mira, 2000], the competition of attracting basins [Salas et al., 2021] and the sequences of period-doubling bifurcations [Demina, 2021]. For example, in the modified 3D van der Pol-Duffing circuit, expressed in nondimensional form as [Kengne et al., 2020] x = −m(x 3 − wx − y + μ), ...
... We refer to the modified van der Pol-Duffing circuit in (1) described as [Kengne et al., 2020] ...
This paper is devoted to exploring the effect of the coexisting attractors in the fast subsystem of a full nonlinear system on the complex dynamics. As an example, a modified 3D van der Pol–Duffing circuit is discussed, in which the coexistence of five attractors including a hidden one was reported in 2020. By introducing a parametric excitation, when the exciting frequency is far less than the natural frequency, a slow–fast model with the two-scale coupling is established in the frequency domain. Regarding the exciting term as a slow-varying parameter, all the attractors as well as the bifurcations of the fast subsystem are derived. The coexistence conditions as well as the coexisting attractors, including the equilibrium points and limit cycles, are presented. With the variation of the exciting amplitude, different types of bursting oscillations are observed, the mechanism of which is revealed using a modified slow–fast analysis method, by combining the transformed phase portraits and the attractors as well as the bifurcations. It is found that the coexisting attractors of the fast subsystem may lead to coexisting bursting solutions in the full system, in which a trajectory may visit one of the attracting basins of the attractors, or result in a merged bursting motion, which may visit the attracting basins in turn. The influence of some attractors and the associated bifurcations may vanish in the case that the trajectory passes directly across the parameter interval corresponding to the attractors because of the inertia of the motion; this may also occur in the case that the trajectory does not visit the associated attracting basins, but it starts initially from the basins. It should be pointed out that, when the exciting frequency is small enough, the disappeared effect of stable attractors may reoccur. Furthermore, if the parameter interval between two types of codimension-1 bifurcation points is short enough, the trajectory may behave in a combination of two bifurcations, which is somewhat similar to that caused by a codimension-2 bifurcation.
... may find importance in secure communication. We explore 'antimonotonicity', 'multistability' type striking features in the modified system which are also important in nonlinear study [34][35][36]. In spite of this, by the introduction of filter the damping term of the system under study has been modified with delay (as a filter within a system makes it a delayed one). ...
Studies on the nonlinear dynamics of a modified Van der Pol oscillator based
electronic circuit has been reported. It has been found that inclusion of an active RC
section (or filter) in the Van der Pol oscillator is able to generate rich nonlinear
behavior. Effect of a low pass filter as well as an all pass filter has been reported. The
behavior of the proposed system has been explained analytically and numerically with
respect to its parameters by exploiting standard nonlinear analysis tools such as
bifurcation diagrams, Lyapunov exponent, phase portraits, Poincare sections, and
basins of attraction. Occurrence of some fascinating features like antimonotonicity,
hysteresis and coexisting attractors have been found in the modified system. The
analytical and numerical studies has been verified experimentally with a prototype
experiment.
... Chaotic dynamical systems are highly coveted for applications in many fields requiring effective security of communication, the generation of sufficiently random bits, the setting up of cryptosystems, etc. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Several researchers have focused on the study, analysis, and construction of such systems intending to obtain even more complex behaviors from those already existing, given their importance [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Some important works based on the construction and study of multistable chaotic systems or chaotic systems with the coexistence of multiple attractors have recently been published by [38][39][40][41][42]. ...
In this article, a new model of Hopfield Neural Network (HNN) with two neurons considering a synaptic weight with a hyperbolic-type memristor is studied. Equilibrium points analysis shows that the system has an unstable line of equilibrium in the absence of the external stimuli (i.e. \(I_{1} =0)\) and presents no equilibrium point in the presence of the external stimuli (i.e. \(I_{1} \ne 0)\); hence the model admits hidden attractors. Analyses are carried out for both cases \(I_{1} =0\) and \(I_{1} \ne 0\) using appropriate tools (bifurcation diagrams and the Lyapunov exponents, phase portraits, etc.). For both modes of operations, the system exhibits complex homogeneous and heterogeneous bifurcations, respectively marked by a large number of coexisting attractors. The roads to chaos unfold in the same scenario of period doubling. The Hamiltonian plot for the case \(I_{1} =0\) allows us to observe an increase in the energy of the neuronal structure when it migrates from regular oscillations to irregular ones. Moreover, the existence of infinitely many coexisting homogeneous solutions (chaotic or periodic) is revealed for case \(I_{1} =0\). In contrast, for \(I_{1} \ne 0\) (i.e \(I_{1} =0.1)\) the new model presents infinitely many coexisting hidden heterogeneous attractors (periodic and chaotic). An electronic circuit design of the new hyperbolic memristor enables the analog computer of the whole system to be designed for future engineering applications. Simulation results based on this analog computer in PSpice confirm those of the numerical investigations.
... It is made of five elements connected in parallel namely, two capacitors, an inductor, a resistor and an active nonlinear resistor constructed using a set of diodes and an operational amplifier [3]. The VdPD circuit is governed by a common nonlinear differential equation which can be used to describe several physically interesting systems in engineering, biology, physics, neurology and so on [4][5][6][7][8][9][10][11][12]. In order to obtain interesting dynamics in the VdPD circuit, its implementation has been modified in at least two physically interesting situations: (i) By adding a serial resistor to its induc- To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal business use only by the author(s), editor(s), reviewer(s), Elsevier and typesetter VTeX. ...
An absolute memristor Van der Pol–Duffing circuit is proposed and analyzed in this paper. The proposed circuit is described by a four-dimensional autonomous system with a line equilibrium point. The stability of the line equilibrium point of this system is analyzed. With a suitable choice of the parameters, the proposed circuit can exhibit extreme multistability features, periodic attractors and hidden chaotic attractors. By introducing two additional parameters in the system describing the proposed circuit, it is possible to either partially or totally control the amplitude of its signals. The physical existence of the dynamical behaviors found in the absolute memristor autonomous Van der Pol–Duffing circuit is assessed using PSIM software. Good qualitative agreement is shown between the numerical simulations and the PSIM results.
... In fact, the physical memristor usually possesses an asymmetric hysteresis loop Hua, Yang et al., 2020;Sah et al., 2015). Moreover, it is known that deliberately induced asymmetry in nonlinear dynamical circuits may help to discover new nonlinear phenomena as illustrated by various relevant works (Kahlert, 1993;Kengne, Kengne et al., 2020a;Léandre Kamdjeu Kengne et al., 2020a, 2020bKengne, Kengne et al., 2020a;Lucarini & Fraedrich, 2009;Tong et al., 2017). To the best of the author's knowledge, the literature is relatively poor concerning the dynamics of memristor-based oscillators possessing an asymmetric hysteresis loop (see Table 1). ...
... In fact, the physical memristor usually possesses an asymmetric hysteresis loop Hua, Yang et al., 2020;Sah et al., 2015). Moreover, it is known that deliberately induced asymmetry in nonlinear dynamical circuits may help to discover new nonlinear phenomena as illustrated by various relevant works (Kahlert, 1993;Kengne, Kengne et al., 2020a;Léandre Kamdjeu Kengne et al., 2020a, 2020bKengne, Kengne et al., 2020a;Lucarini & Fraedrich, 2009;Tong et al., 2017). To the best of the author's knowledge, the literature is relatively poor concerning the dynamics of memristor-based oscillators possessing an asymmetric hysteresis loop (see Table 1). ...
The dynamics of memristor–based chaotic oscillators with perfect symmetry is very well documented. However, the literature is relatively poor concerning the behavior of such types of circuits when their symmetry is perturbed or destroyed. In this paper, we consider the dynamics of a memristive twin-T oscillator recently introduced by Ling Zhou et al. (International Journal of Bifurcation and Chaos, 28(04), 1850050, 2018). Here, the symmetry is broken by assuming a memristor with an asymmetric pinched hysteresis loop i−v characteristics. A variable disturbance term is introduced into the current-voltage relationship of the memristor in order to obtain an asymmetric characteristic. Phase portraits, bifurcation plots, basins of attraction, and Lyapunov exponents spectra are used to illustrate various nonlinear patterns experienced by the underlined memristive circuit. It is shown that in the absence of the disturbance term, the i−vcharacteristic of the memristor is perfectly symmetric which induces typical behaviors such as period doubling, coexisting symmetric bifurcation and bubbles, spontaneous symmetry-breaking, symmetry recovering, and coexistence of several (i.e. two, four and up to six) pairs of mutually symmetric attractors. With the perturbation term, the symmetry of the oscillator is destroyed resulting in more complex nonlinear phenomena such as coexisting asymmetric bubbles of bifurcation, critical transitions, and multiple coexisting (i.e. up to five) asymmetric attractors. To the best of the authors’ knowledge, the plethora of behaviors revealed in this works, has not yet been reported in the literature concerning twin-T network circuits and therefore deserves dissemination. Also, PSpice simulation studies confirm well the results of theoretical predictions.
... Spontaneous symmetry breaking refers to a situation where given a symmetry of evolution equations, solutions exist which do not remain invariant following the action of the underlined symmetry without the introduction of any term explicitly breaking the symmetry [Lucarini & Fraedrich, 2009;Sprott, 2014;Tong et al., 2017]. This is the case for example for solutions that appear in pairs of asymmetric attractors in some models [Bao et al., 2018;Negou & Tchiotsop, 2018;Pone et al., 2019a;Pone et al., 2019b;Kengne et al., 2020a;Kengne et al., 2020b]. ...
... The dynamical system given by (1) can be easily controlled for the three state variables x, y, and z respectively replaced by x + d, y + d, and z + d, where d represents an offset parameter [Li et al., 2017a;Li et al., 2017b;Kengne et al., 2020a;Kengne et al., 2020b]. Thus by this replacement, we obtain three systems having a particular dynamic behavior after offset boosting control operation. ...
In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.
Today, telediagnosis and telesurgery in the e-healthcare domain use medical images that are sensitive to external disturbances and manipulations leading to huge differences in the final result. To keep safe such images, cryptography is among the best methods and especially when it integrates chaotic systems due to their sensitivity to initial seeds. So, a chaos-based cryptosystem using DNA confusion and diffusion is proposed and applied to biomedical images in this work. It also includes hash functions that compress an indeterminate size of data into fixed size of data. The security and the reliability of the information system are ensured with the combination of two hash functions. The execution time is then considerable, and the integrity of the encrypted image is guaranteed. Encoding/decoding rules and operations are selected using the result of the numerical integration of the logistic map. At the level of diffusion, seven functions are employed reinforcing the security level of our cryptosystem. The construction of the DNA (deoxyribonucleic acid) key is done through the iteration of the new cyclic chaotic system having initial states derived using the keys obtained from the combination of hash functions and external key, thus building a PRNS (pseudorandom number sequence). The entire dynamics of the new system in both symmetric and asymmetric cases is then performed, exhibiting relevant behaviors such as the coexistence of up to eight attractors, intermittency, parallel branches of bifurcations, and metastable chaos very rare in literature, to name a few. PSpice is used to verify the numerical results. Based on confusion and diffusion, the new encryption/decryption algorithm is effective in both processes. The experimental results show that the cryptosystem is able to withstand brute force, exhaustive, statistical, differential, and robustness attacks. Also, the comparison of the algorithm with good ones from the literature shows that it is among the best proposed up to date.
This paper investigates the complex bursting behaviors of the cubic-quintic Duffing-van der Pol system with two slow-varying forcing periodic excitations. Four novel bursting patterns, namely “turnover-of-fold-hysteresis-induced” bursting, “cascade-turnover-of-fold-hysteresis” bursting, “supHopf/supHopf” bursting via turnover-of-fold-hysteresis loops and cascade “supHopf/supHopf” bursting via cascade-turnover-of-fold hysteresis loops, are studied based on the time domain waveform analysis, two-parameter bifurcation diagrams and overlay maps. We find that the “fold/fold” hysteresis loop plays an important role in the generation of the complex bursting characteristics when the second external periodic excitation is introduced. In particular, there are some ups and downs on the stable equilibrium branches, resulting in the emergence of the specific oscillations. These oscillations can be regarded as the spiking states in the “turnover-of-fold-hysteresis-induced” bursting and “supHopf/supHopf” bursting via turnover-of-fold-hysteresis loops. In addition, it is noticeable that the stable equilibrium branches can pass through the fold and supercritical Hopf bifurcation points many times in each bursting period, leading to the “cascade-turnover-of-fold-hysteresis” bursting and cascade “supHopf/supHopf” bursting via cascade-turnover-of-fold hysteresis loops. We also present a fact that the amplitude of the excitations play an important role in the dynamical evolutions among the four bursting oscillations. Finally, the correctness of our study is tested and verified by the numerical simulations.
This chapter proposes a memristor Helmholtz (MH) oscillator which is designed by adding an ideal and active flux-controlled memristor in the circuit of an autonomous Helmholtz jerk oscillator. The dynamical behavior of the MH oscillator is investigated in terms of equilibria states and their stabilities, bifurcation diagrams, Lyapunov exponents, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, reverse period-doubling bifurcation, two different shapes of one-scroll chaotic attractors and coexistence between chaotic and periodic attractors. An electronic implementation of the proposed oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show good agreement with the results of numerical simulations. Moreover, controllers are designed to achieve synchronization of unidirectional coupled identical proposed oscillators. Numerical simulations are also used to verify the effectiveness of the synchronization process. Finally, a single simple controller is designed and added to the MH oscillator in order to suppress the chaotic behavior. The performance of the two proposed simple controllers is illustrated by numerical simulations.
