Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Chaotic and Coexistence Attractors of Classical Complex Exotic Oscillator with Position-Dependent Mass

J. G. Houeto, B. N. Tokpohozin, C. H. Miwadinou*, A. A. Koukpemedji, A. V. Monwanou

Laboratoire de Mécanique des Fluides, de la Dynamique Nonlinéaire et de la Modélisation des Systèmes Biologiques (LMFDNMSB); Institut de Mathématiques et de Sciences Physiques, Porto-Novo, Bénin

Abstract: This work analyzes the chaotic dynamics and the coexistence of attractors of the complex exotic oscillator. From the Hamiltonian of the basic system, the oscillator is obtained. The complete dynamics of the exotic oscillator is studied and the coexistence of attractors analyzed using fourth order Runge-Kutta algorithm. It is obtained for appropriate conditions the coexistence of chaotic and regular attractors. The study showed that for domains of values of the pair of parameters (a,b) of the exotic oscillator and given initial conditions, the dynamics of the system can be regular, or chaotic or can have very large amplitudes. The basins of attraction and the drawn phase portraits confirmed the PT symmetry and the breaking of this symmetry of the oscillator.

Keywords: Exotic oscillator, Hamiltonian, PT-symmetry, chaos and bifurcation, coexistence of attractors

Received December 01, 2024; Revised January 01, 2025; Accepted January 07, 2025

Citation: J. G. Houeto, B. N. Tokpohozin, C. H. Miwadinou, A. A. Koukpemedji, A. V. Monwanou, Chaotic and Coexistence Attractors of Classical Complex Exotic Oscillator with Position-Dependent Mass. International Journal of Physics, 13 (1). 1-10. January 2025. doi:10.12691/ijp-13-1-1

1. Introduction

In nature almost everything is dynamic since a system, whether natural or artificial, has at least one of its parameters which change over time quickly, slowly or relatively slowly depending on the time scale considered. To understand the evolution of a system or a variable in time and space, we use the notions of dynamic systems [1,2]. To do this, it is necessary to model the system to be studied. The approach used for modeling depends on the nature of the system and the area of use of the latter. The fields often concerned are mechanics, electronics, biochemistry, biophysics, chemistry, economics, finance, engineering, nonlinear optics, mathematics, etc. [1,2,3,4,5,6]. Thus, the modeling can be based on Newton’s second law for mechanical systems, the laws of chemical kinetics for chemical systems, Kirchoff’s laws for electronic systems.... [1,2,3,4,5,6]. With the notable advances in research on dynamic systems in general and nonlinear dynamics in particular, the areas of intervention of the notions of dynamics are no longer limited. Indeed, the modeling of the dynamics of certain classical or quantum systems is well managed using the Lagrange and Hamilton approaches when the system is mechanical [7,8,9,10,11] [12,13,14,15] [16,17,18,19] [20,21,22,23]. In this sense, several researchers have constructed Hamiltonians to study quantum systems in order to determine their quantum states and the eigenvalues of their energies. Among the research carried out by scientists in the field, it is shown that the obligation of hermiticity when one wants to construct Hamiltonians which take into account the unitary temporal evolution of quantum states and eigenvalues of energy can be replace with PT symmetry [17]. Thus, several types of Hamiltonians previously rejected for reasons of non-hermiticity are now considered thanks to this discovery. PT symmetry defined as symmetry in time and space plays several roles in non-Hermitian physical models and models with individually symmetric Hamiltonians P or T PT [17]. This is the case of the harmonic oscillator where the particles in a quartic potential present symmetrical closed trajectories PT [17,18].

According to the literature, closed but not PT-symmetric trajectories are very rare to find. This is the case of the complexified exotic oscillator studied by Gosh and Modak [17]. In their work, they considered the harmonic oscillator complexified by taking into account a specific form of position-dependent mass. Their motivation for studying this particular form of effective mass oscillator is because in the classical sense it is a PT-symmetric model. Indeed, they showed that this oscillator has an interesting form of Hamiltonian dynamics and moreover is exactly solvable as a quantum system. They also proved that it is possible to have closed trajectories that are mostly PT symmetric and closed orbits that are not PT symmetric. They therefore concluded that there is a possible link between this phenomenon and the classic analogue of exceptional.

Generally, the harmonic oscillator is the classic example often used to move from the classical oscillator to the quantum oscillator using the quantization of position and velocity variables. However, natural phenomena are complex and cannot be better explained by a harmonic oscillator. The oscillator whose mass depends on the position is a good example of a classical oscillator used to model macroscopic classical systems also observed on the microscopic scale. It is to understand the classical behaviors possibly observable at the atomic scale that several authors have studied a Classical oscillator with position-dependent mass in a complex domain [17] [21,22,23]. Since the exotic oscillator considered by Ghosh and Modak is an example of oscillators whose mass depends on position, the exotic oscillator cannot escape the existence of very complex physical phenomena.

Chaos and the coexistence of attractors are two other complex physical phenomena studied for complex systems [24,25,26] [27,28,29] [30,31,32] [33,34,35,36]. Chaos is a complex phenomenon that cannot accurately predict the position and speed of the system. Depending on the domain and the objective, chaos may or may not be desired. In case it is not desired, its control or elimination is suggested [29,30,31]. Finally, the presence of the coexistence of attractors in a system makes it difficult to predict the nature of its dynamics. Indeed, a system for which the coexistence of attractors appears for the same value of one of its parameters is in two or more states or oscillates with several amplitudes, which makes it difficult to control [32,33,34,35] [36,37,38,39] [40,41,42] [43,44]. This phenomenon can be bistable or megastable. Indeed, bistability reflects the coexistence of two attractors, while megastability designates the coexistence of an infinite number of attractors for the same system. Just like chaos, the coexistence of attractors has advantages and disadvantages. One of the advantages of the coexistence of attractors in a system allows it to be a good candidate for reinforcing the key to securing communication while one of its disadvantages is in the difficulty of designing control laws. a system presenting the coexistence of chaotic attractors. To help resolve this drawback, several researchers have embarked on work to predict and control the coexistence of attractors. This explains the numerous works published over the last decade on this subject [32,33,34] [35,36,37] [38,39,40] [41,42] [43,44]. In these different works, several techniques are used by the authors to research, analyze and control coexistence of attractors and very conclusive results are obtained. As applications, these authors showed that great flexibility in system performance is made achievable by the coexistence of different stable states without major parameter changes.

The exotic oscillator studied by Ghosh and Modak [17] is a 4-dimensional nonlinear system whose complexity induces gauge invariance. According to these authors, the existence of PT symmetry made it possible to obtain closed trajectories. Sensitivity to initial conditions was also proven by these authors. Finally, the authors also found that the nature of the possible trajectories of the complex exotic oscillator depends on the values of the latter’s parameters.

From these motivations and the important results obtained by Ghosh and Modak, we believe that it is necessary to find the different domains of the system parameters for the different dynamics of the exotic four-dimensional oscillator. It would also be important to study the chaotic dynamics and the coexistence of the attractors of this oscillator. Thus, the present work specifically concerns the determination of the different dynamics of the exotic oscillator, the determination of the different parameters of this oscillator for which there is each type of dynamic, the determination of the different initial conditions for which the oscillator has finite and infinite amplitudes and the determination of the different initial conditions for which the oscillator presents a phenomenon of coexistence of attractors.

The paper is structured as follows: section 2 gives the mathematical modeling of dynamics of an Elastical Exotic Oscillator. In section 3, the numerical study of the oscillator and the analysis of the different results are carried out. Finally, we provide conclusion in the last section.

2. Model of Classical Complex Exotic Oscillator

In this work, we study the classical Complex Exotic Oscillator (CEO) proposed by Gosh and Modak [17]. Indeed, we consider the CEO in a considerably simplified dimension constituting an oscillator with a function depending on the mass position. The Hamiltonian of the system is:

[picture] | (1)

where a is a stabilizing parameter of the exotic oscillator; b a control parameter; x and p designate the position and the associated conjugate moment. The equation of motion highlighting the PT-symmetry of the Hamiltonian is:

[picture] | (2)

By defining complex variables

and

the complex Hamiltonian is written:

[picture] | (3)

By separating the real and imaginary parts of H, we obtain respectively

[picture] | (4)

and

[picture] | (5)

We consider the case where because in one dimension G is the only first-class constraint inducing gauge invariance since . So the equations of motion are:

[picture] | (6)

3. Numerical Study of Complex Dynamics

Complex systems generally have several possible dynamics, the prediction of which is sometimes not easy.

Chaos is one of those behaviors that is determined from Lyapunov's statement defined by [45]:

[picture] | (7)

where dx, dp, dy and dq are the variations of x, p, y and q, respectively.

The system is chaotic when ; periodic if and quasi-periodic when .

The objective of this section is to numerically research and analyze the different dynamics and coexistence of attractors for the classical exotic complex oscillator. Thus, the system of equations (6) is solved using Runge-Kutta algorithms of order 4 in the Fortran language and parameter basins, basins of attraction, bifurcation diagrams and phase space are represented. Since a and b are the two parameters of the system, the parameter pool is plotted in the plane (a,b). Figure 1 (a) and Figure 1 (b) represents the parameter pool respectively for the initial conditions.

(x₀,p₀,y₀,q₀) = (0.2,0.1,0.1,0.1) and (x₀,p₀,y₀,q₀) = (0.8,0.1,0.1,0.1). In this figure the domain of regular behavior (corresponding to ) is represented in blue, that of chaotic behavior (corresponding to ) in red and that of very large amplitude oscillations in cyan. We see through Figure 1 that for −2 ≤ a ≤ 2 and −10 ≤ b ≤ 10 the domain of existence of large amplitude oscillations is larger than that of periodic behaviors and this is larger than the domain of existence of chaotic behavior. Furthermore, we notice that the domain of chaotic behavior is more abundant for (0.8,0.1,0.1,0.1) than for (0.2,0.1,0.1,0.1). To check the different dynamics obtained in Figure 1, the phase space is plotted for different values of a and b chosen in each domain. Thus Figure 2 is obtained for the chosen values of the couple (a,b). This figure confirms the regular and chaotic dynamics in the (x,p) and (y,q) planes. These two parameters pools therefore show that the dynamics of the exotic classical oscillator depends on the values of (a,b) and its sensitivity to initial conditions. To collect the different initial conditions favorable to each type of dynamics, basins of attractions are drawn. The results obtained for (a = 1,b = 1.24), are represented in Figure 3 (a) and Figure 3 (b) respectively for (y0 = 0.1,q0 = 0.1) and (x0 = 0.2,p0 = 0.1). In this Figure 3, the blue color designates the domain of regular behavior, the red color the domain of chaotic behavior and the cyan color the large amplitude oscillations. These figures confirm the sensitivity to initial conditions of the exotic classical oscillator studied. As proof, we note that regular oscillations do not exist for (x0 = 0.2,p0 = 0.1) in the plane (y,q) while they exist for (y0 = 0.1,q0 = 0.1) in the (x,p). In addition to these figures, it is necessary to notice the symmetry of these basins of attraction around the origin (0,0). Through these figures, we also note that the exotic classical oscillator studied presents the coexistence of attractors. All these predictions of Figure 3 are confirmed by the phase portraits of Figure 4. Indeed, on these phase portraits obtained in the (x,p) and (y,q) planes, we note the coexistence of chaotic and periodic attractors corresponding to each initial conditions thus confirming the periodic and chaotic dynamics predicted by Figure 3 (a). We observe once again for the exotic oscillator the PT symmetry and its breaking through these phase portraits. In order to highlight all the coexisting attractors, the bifurcation diagram is drawn by varying the parameter b from −10 to 10. The result obtained is that of Figure 5 where the blue curve corresponds to the increasing variation of b from -6 to 10 while the red curve corresponds to the decreasing variation of b from 10 to -6. It is clearly noted that the evolution in the decreasing direction largely shows chaotic dynamics while the evolution in the increasing direction shows mono-periodic behavior. We can therefore conclude that chaotic and mono-periodic attractors coexist. This result is confirmed by Figure 6 which shows mono-periodic phase portraits (the blue curve) and chaotic phase portraits (red curve). By taking a as a control parameter, we note that the existence of periodic, quasi-periodic and chaotic behavior as well as the existence of several attractors strongly depend on the parameter a (see Figure 7). These different dynamics and this coexistence of multiple attractors are confirmed by the phase portraits of the oscillators (Figure 8). In short, for a pair of fixed values of (a,b), the exotic oscillator studied can have chaotic or periodic dynamics for well-chosen initial conditions. This therefore does not make it possible to say with accuracy the nature and amplitude of the oscillations and then the position of the exotic oscillator studied, thus confirming its complexity.

Figure 1. (Color online) Basin of parameters with (a) (0.2,0.1,0.1,0.1) and (b) (0.8,0.1,0.1,0.1)

Figure 2. (Color online) Phase portraits with parameters of Figure 1(a) in (a)(x,p) and (b)(y,q). Blue curve: a = −0.85, b = −0.7; red curve a = 1,b = 1.3; cyan curve a = 1.3,b = 2.283 and green curve a = 0.7,b = 1.24

Figure 3. (Color online) Basin of attraction with a = 1 ; b = 1.24: (a)(x₀,p₀) and (b)(y₀,q₀)

Figure 4. (Color online) Coexistence of attractors with parameters of Figure 3: (a) (x,p) where blue curve correspond to (0.1,0.1,0.1,0.1), red curve correspond to (0.3,0.34,0.1,0.1), green curve correspond to (0.8,0.1,0.1,0.1), cyan curve correspond to (1,0.9,0.1,0.1); (b) (y,q) red curve correspond to (0.8,0.1,0.001,0.03), cyan curve correspond to (0.8,0.1,0.1,0.3)

Figure 5. (Color online) Bifurcation diagram and its corresponding Lyapunov exponents for a = 1: where blue curve correspond to (0.1,0.1,0.1,0.1) and red curve correspond to (0.3,0.32,0.1,0.1)

Figure 6. (Color online) Coexistence of attractors with parameters of Figure 5 for b = 5 where blue curve correspond to (0.1,0.1,0.1,0.1) and red curve correspond to (0.8,0.1,0.1,0.1)

Figure 7. (Color online) Bifurcation diagram and its corresponding Lyapunov exponents for b = 3.5: where blue curve correspond to (0.1,0.1,0.1,0.1) and red curve correspond to (0.3,0.32,0.1,0.1)

Figure 8. (Color online) Coexistence of attractors with parameters of Fig.7 where blue curve correspond to (0.1,0.1,0.1,0.1) and red curve correspond to (0.3,0.32,0.1,0.1). (a),(b) : a = 0.5; (c),(d) : a = 1.1 and (d),(e) : a = 1.4

4. Conclusion

In this work, the exotic classical oscillator is numerically studied. Using numerical simulations of the parameter diagram, the basins of attraction, the bifurcation diagram and phase spaces made it possible to identify the different dynamics and conditions of coexistence of attractors of the exotic classical oscillator. Indeed, the domains of values of a and b for which the exotic oscillator studied has a regular, chaotic behavior or a large amplitude oscillation are obtained. Sensitivity to the initial conditions of the oscillator was obtained as well as the coexistence of chaotic and mono-periodic attractors. These different results confirm the complexity of the exotic oscillator and complement the results obtained by Ghosh and Modak. This complexity is linked to the two phenomena obtained: chaos and the coexistence of attractors. They do not allow an exact prediction of the position, conjugate moment, amplitude and nature of the oscillations of the exotic oscillator. This could be a disadvantage or an advantage for systems modeled by the exotic oscillator. A disadvantage if this oscillator is used in systems where the vibration must be periodic or where position precision is mandatory. But an advantage if this oscillator could be quantized to be used in quantum systems used for securing data or information because this would strengthen the security key by synchronization. In this sense, it would be necessary to study the dynamics of the quantum equivalent of this exotic oscillator, to achieve its chaotic synchronization in order to use it in the coding of information. Finally, it would also be important to study the conditions in which the exotic oscillator studied will be hyperchaotic.

ACKNOWLEDGMENTS

We thank Professor A. F. Dossa and Professor S. Ghosh for the rich discussions on PT symmetry and the interest of chaos in the exotic oscillator. We would like to thank the anonymous reviewers for corrections, comments and suggestions that would helped us to significantly improve our paper.

Author Contribution Statement

J. G. Houeto and B. N. Tokpohozin contributed to the modeling, to the various calculations, to the drafting of the paper.

C. H. Miwadinou contributed to the modeling, the various calculations and the analyzes of the results of the paper.

A. A. Koukpemedji and A. V. Monwanou contributed to the analysis of the results and supervised this work.

Data Availability Statement

All data are included in text directly.

Declarations

The authors declare that there are no conflicts of interest.

References

[1]   | Hale J. & Ko¸cak H., “Dynamics and Bifurcations”, Applied Mathematics, Springer-Verlag, Inc, New York, 1996.
[2]   | A. H. Neyfeh, D. T Mook, 1979, Non linear oscillations (Willey, New York).
[3]   | C Zerbe and P Hanggi, Brownian parametric quantum oscillator with dissipation, Physical Review E 52, 1533 − 1543(1995).
[4]   | H. F. Dozounh´ekpon, A. Koukp´em´edji, A.V. Monwanou, C. H. Miwadinou, L. A. Hinvi, J. B. Chabi Orou, 2022, Eur. Phys. J. B , 95: 22.
[5]   | L. A. Miller, 2005, Journal of Theoretical Biology , 234, 511-524.
[6]   | H. F. Dozounh´ekpon, A. Koukp´em´edji, A.V. Monwanou, C. H. Miwadinou, L. A. Hinvi, J. B. Chabi Orou, 2022, Phys. Scr. 97 (2022) 045209.
[7]   | B. Bagchi, S. Das, S. Ghosh and S. Poria, J. Phys. A: Math. Theo. 46, 032001 (2013).
[8]   | B. Bagchi, A. Banerjee, C. Quesne and V. M. Tkachuk, J. Phys. A: Math. Gen. 38, 2929 (2005).
[9]   | V. C. Ruby, M. Senthilvelan and M. Lakshmanan, J. Phys. A: Math. Theo. 45, 382002 (2012).
[10]   | J. F. Carin˜ena, M. F. Ran˜ada and M. Santander, Reports Math. Phys. 54, 285 (2004).
[11]   | J. F. Carin˜ena, M. F. Ran˜ada and M. Santander, Annals of Physics 322, 434 (2007).
[12]   | J. F. Carinèna, M. F. Ran˜ada and M. Santander, Proceding 10th International Conference in Modern Group Analysis (Larnaca, Cyprus, 39 − 46, 2005). (arXiv:math-ph/0505028).
[13]   | S. C. y Cruz, J. Negro and L. M. Nieto, Phys. Lett. A 369, 400 (2007).
[14]   | S. C. y Cruz, J. Negro and L. M. Nieto, J. Phys.: Conference Series 128, 012053 (2008).
[15]   | S. C. y Cruz, and O. Rosas-Ortiz, J. Phys. A: Math. Theo. 42, 185205 (2009).
[16]   | S. C. y Cruz, and O. Rosas-Ortiz, SIGMA 9, 004 (2013).
[17]   | S. Ghosh, and S. K. Modak, Phys. Lett. A 373, 1212 (2009).
[18]   | C. M. Bender, D. D. Holm, D. Hook, J. Phys. A 40 F793 (2007).
[19]   | A. Venkatesan and M. Lakshmanan, Phys. Rev. E 55, 5134 (1997).
[20]   | P. M. Mathews and M. Lakshmanan, Quarterly of Applied Mathematics 32, 215 (1974).
[21]   | T. O. Roy-Layinde, U. E. Vincent, S. A. Abolade, O. O. Popoola, J. A. Laoye and P. V. E. McClintock, Philosophical Transactions A 379, 20200227(2021).
[22]   | Hinvi, L. A., Koukp´em`edji, A. A., Monwanou, V. A., Miwadinou, C. H., Kamdoum Tamba, V. and Chabi Orou, J. B. Resonance, chaos and coexistence of attractors in a position dependent mass-driven Duffing-type oscillator. J. Korean Phys. Soc. 79, 755-771 (2021).
[23]   | D. Ghosh, B. Roy, Nonlinear dynamics of classical counterpart of the generalized quantum nonlinear oscillator driven by position dependent mass, Annals of Physics, 353, 222−237(2015).
[24]   | Siewe M, Moukam Kakmeni F M, Tchawoua C (2004) Resonant oscillation and homoclinic bifurcation in a φ6 -Van der Pol oscillatorv. Chaos Soliton Fract 21:841-853.
[25]   | Siewe M, Moukam Kakmeni F M, Tchawoua C, Woafo P (2006) Nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ6 -Rayleigh oscillator combined to parametric excitations. Journal of Computational and Nonlinear Dynamics 1(3):196-204.
[26]   | Miwadinou C H, Hinvi L A, Monwanouv A V, Chabi Orou J B (2017) Nonlinear Dynamics of a φ⁶- Modified Rayleigh-Duffing Oscillator : Resonant Oscillations and Homoclinic Bifurcation. Nonlinear Dyn 88:97-113.
[27]   | Miwadinou C H, Monwanou A. V, Hinvi L A, , Tamba V K, Koukp´em`edji A A, Chabi Orou J B (2020) Nonlinear oscillations of nonlinear damping gyros: resonances, hysteresis and multistability. International Journal of Bifurcation and Chaos 30(14): Article ID 2050203.
[28]   | Nwagoum tuwa P R, Miwadinou C H, Monwanou A V, Chabi orou J B, Woafo P (2019) Chaotic vibrations of nonlinear viscoelastic plate with fractional derivative model and subjected to parametric and external excitations. Mechanics Research Communications.
[29]   | Yu H, Cai G, Li Y (2012) Dynamic analysis and control of a new hyperchaotic finance system. Nonlin Dyn 67(3): 2171-2182.
[30]   | D. L Olabodé, C. H Miwadinou, V. A. Monwanou and J B Chabi Orou. Physica D 386 49 (2019).
[31]   | Dousseh PY, Ainamon C, Miwadinou CH, Monwanou AV, Chabi Orou JB (2021) Adaptive control of a new chaotic financial system with integer order and fractional order and its identical adaptive synchronization. Math Probl Eng 2021:15 pages.
[32]   | Li C., Tang S. & Cheke R.A., “Complex dynimics and coexistence of period-doubling and periodhalving bifurcations in an integreted pest management model with nonlinear impulsive control”, Adv. Difference Equ. (514), 2020, 1-23.
[33]   | Lui Q. , “A unified chaotic system with various coexisting attractors”, Int. J. Bifurc. Chaos 31, 2021, 1-11.
[34]   | Lui Q., “A chaotic system with a nonlinear term and multiple coexisting attractors”, Eur. Phys. J. Plus 135, 2020, 1-9.
[35]   | Ngonghala C.N., Feudel U. & Showalter K., “Extreme multistability in a chemical model system”, Phys. Rev. E 83, 2011, 1-10.
[36]   | Mukhopadhyay A. & Chakraborty S., “Chaos and coexisting attractors in replicator-mutator”, J. Phys. Complex. 2, 2021, 1-12.
[37]   | Bao H., Liu W.& Chen M., “Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh-Nagumo circuit”, Nonlinear Dyn. 2, 2019, 1-12.
[38]   | Bao B., Qian H., Xu Q., Chen M., Wang J. & Yu Y., “Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network”, Front. Computat. Neurosci. 11 (81), 2017, 1-14.
[39]   | Lai Q., Akgul A., Varan A., Kengne J. & Erguzel A. T., “Dynamic analysis and synchronization control of an unusual chaotic system with exponential tern and coexisting attractocs”, Chin. J. Phys. 56, 2018, 2837-2851.
[40]   | Xiong L., Zhang S., Zeng Y. & Liu B., “Dynamics of a new composite four-scroll chaotic system”, Chin. J. Phys. 56 (5), 2018, 2381-2394.
[41]   | Xu Q., Cheng S., Ju Z., Chen M. & Wu H., “Asymmetric coexisting bifurcations and multistability in an asymmetric memristive diode-bridge-based jerk circuit”, Chin. J. Phys. 70, 2021, 69-81.
[42]   | Qiusen J., Fuhong M.& Chunbiao L., “Infinitely many coexisting attractors of a dual memristive shinriki attractor and its FPGA digital implementation”, Chin.J. Phys. 62, 2019, 342-357.
[43]   | Pisarchik A.N. & Feudel U., “Control of multistability”, Phys. Rep. 540, 2014, 167-218.
[44]   | Kondo H.N., Farkas D., Denham T.A. S. L. & Winkler I., “Auditory multistability and neurotransmitter concentrations in the human brain”, Philos. Trans. R. Soc. B 372, 2017, 1-8.
[45]   | Enjieu Kadji H.G., Chabi Orou J.B. & Woafo P., “Regular and chaotic behaviors of plasma oscillations modeled by a modified Duffing equation”, Phys. Scr., 77(2), 025503, 2008.