]>
2022
15
3
ISSN 2008-1898
80
Controllability of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay via resolvent operator
Controllability of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay via resolvent operator
en
en
This paper is concerned with the controllability results of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay in a real separable Hilbert space. The controllability results are obtained using stochastic analysis, the theory of resolvent operator in the sense of Grimmer and Krasnoselskii fixed point theorem. An example is provided to illustrate the obtained theory.
172
185
D.
Chalishajar
Department of Mathematics and Computer science, Mallory Hall
Virginia Military Institute
United States
chalishajardn@vmi.edu
K.
Ramkumar
Department of Mathematics
PSG College of Arts and Science
India
ramkumarkpsg@gmail.com
A.
Anguraj
Department of Mathematics
PSG College of Arts and Science
India
angurajpsg@yahoo.com
K.
Ravikumar
Department of Mathematics
PSG College of Arts and Science
India
ravikumarkpsg@gmail.com
M. A.
Diop
Departement de Mathematiques
Universite Geston Berger de Sanit-Louis
Senegal
dipu17370@gmail.com
Controllability
impulsive systems
fractional Brownian motion
neutral functional integrodifferential equations
resolvent operator
infinite delay
Article.1.pdf
[
[1]
A. Anguraj, K. Ramkumar, Approximate controllability of semilinear stochastic integrodifferential system with nonlocal conditions, Fractal Fract., 2 (2018), 1-13
##[2]
G. Arthi, J. H. Park, H. Y. Jung, Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Commun. Nonlinear Sci. Numer. Simul., 32 (2016), 145-157
##[3]
K. Balachandran, S. Karthikeyan, J. H. Kim, Controllability of semilinear stochastic integrodifferential equations, Kybernetika, 43 (2007), 31-44
##[4]
B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558
##[5]
B. Boufoussi, S. Hajji, E. Lakhel, Functional differential equations in Hilbert spaces driven by a fractional Brownian motion, Afr. Mat., 23 (2012), 173-194
##[6]
M. Chen, Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motions, Stoch. Dyn., 15 (2015), 1-16
##[7]
M. A. Diop, R. Sakthivel, A. A. Ndiaye, Neutral stochastic integrodifferential equations driven by a fractional Brownian motion with impulsive effects and time varying delays, Mediterr. J. Math., 13 (2016), 2425-2442
##[8]
R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349
##[9]
J. Klamka, Stochastic controllability of linear systems with delay in control, Bull. Pol. Acad. Sci. Tech. Sci., 55 (2007), 23-29
##[10]
J. Klamka, Controllability of dynamical systems--A survey, Bull. Pol. Acad. Sci. Tech. Sci., 61 (2013), 221-229
##[11]
E. Lakhel, Controllability of neutral stochastic functional integrodifferential equations driven by fractional Brownian motion, Stoch. Anal. Appl., 34 (2016), 427-440
##[12]
E. Lakhel, S. Hajji, Existence and Uniqueness of Mild Solutions to NFSDEs Driven by a Fractional Brownian Motion With Non-Lipschitz Coefficients, J. Numer. Math. Stoch., 7 (2015), 14-29
##[13]
Y. Li, B. Liu, Existence of solution of nonlinear neutral functional differential inclusion with infinite delay, Stoch. Anal. Appl., 25 (2007), 397-415
##[14]
J. Liang, J. H. Liu, T.-J. Xiao, Nonlocal problems for integrodifferential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 815-824
##[15]
J. Y. Park, K. Balachandran, G. Arthi, Controllability of impulsive neutral integrodifferential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybrid Syst., 3 (2009), 184-194
##[16]
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York (1983)
##[17]
Y. Ren, X. Cheng, R. Sakthivel, On time dependent stochastic evolution equations driven by fractional Brownian motion in Hilbert space with finite delay, Math. Methods Appl. Sci., 37 (2014), 2177-2184
##[18]
R. Sakthivel, R. Ganesh, Y. Ren, S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3498-3508
##[19]
R. Sakthivel, J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett., 79 (2009), 1219-1223
]
A different approach for behavior of fractional plant virus model
A different approach for behavior of fractional plant virus model
en
en
In the last few decades many authors
pointed out that derivatives and integrals of non-integer order are very
suitable for the description of properties of various real problems. It has
been shown that fractional-order models are more adequate than previously used
integer-order models. In this work, we aim to investigate of different
features of the plant virus model with its fractional order equivalent. We
present an application for reproduction number for these kind of epidemic
models with next generation matrix method. Also, existence and uniqueness of
solutions have been showed for this fractional order system. Finally we
present some figures according to the given numerical scheme.
186
202
I.
Koca
Department of Accounting and Financial Management, Seydikemer School of Applied Sciences
Mugla Sitki Kocman University
Turkey
ilknurkoca@mu.edu.tr
H.
Bulut
Department of Mathematics, Science Faculty
Firat University
Turkey
hbulut@firat.edu.tr
E.
Akcetin
Department of Accounting and Financial Management, Seydikemer School of Applied Sciences
Mugla Sitki Kocman University
Turkey
eyup.akcetin@mu.edu.tr
Fractional differential equation
existence and uniqueness
numerical approximation
Article.2.pdf
[
[1]
B. S. T. Alkahtani, I. Koca, A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 429-442
##[2]
B. Chen-Charpentier, Stochastic Modeling of Plant Virus Propagation with Biological Control, Mathematics, 9 (2021), 1-14
##[3]
M. A. Dokuyucu, Analysis of a Fractional Plant-Nectar-Pollinator Model with the Exponential Kernel, Eastern Anatol. J. Sci., 6 (2020), 11-20
##[4]
M. A. Dokuyucu, Caputo and Atangana-Baleanu-Caputo Fractional Derivative Applied to Garden Equation, Turkish J. Sci., 5 (2020), 1-7
##[5]
M. A. Dokuyucu, E. Çelik, Analyzing a novel coronavirus model (Covid-19) in the sense of Caputo Fabrizio fractional operator, Appl. Comput. Math., 20 (2021), 49-69
##[6]
A. Fereres, Insect vectors as drivers of plant virus emergence, Curr. Opin. Virol., 10 (2015), 42-46
##[7]
M. Jeger, F. Van Den Bosch, L. Madden, J. Holt, A model for analysing plant-virus transmission characteristics and epidemic development, Math. Med. Biol. A J. IMA, 15 (1998), 1-8
##[8]
I. Koca, E. Akcetin, P. Yaprakdal, Numerical approximation for the spread of SIQR model with Caputo fractional order derivative, Turkish J. Sci., 5 (2020), 124-139
##[9]
I. Podlubny, Fractional Differential Equations, Acedemic Press, San Diego (1999)
##[10]
K. B. G. Scholthof, S. Adkins, H. Czosnek, P. Palukaitis, E. Jacquot, T. Hohn, B. Hohn, K. Saunders, T. Candresse, P. Ahlquist, C. Hemenwat, G. D. Foster, Top 10 plant viruses in molecular plant pathology, Mol. Plant Pathol., 12 (2011), 938-954
##[11]
P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48
]
Continuation theorems for weakly inward Kakutani and Strongly inward acyclic maps
Continuation theorems for weakly inward Kakutani and Strongly inward acyclic maps
en
en
In this paper we begin by presenting a general Leray-Schauder alternative and a
topological transversality theorem for Kakutani (upper semicontinuous maps with nonempty convex compact values) compact weakly inward maps. Then with some observations and extra assumptions we present a Leray-Schauder alternative and a
topological transversality theorem for acyclic (upper semicontinuous maps with nonempty acyclic compact values) compact strongly inward maps.
203
208
D.
O'Regan
School of Mathematical and Statistical Sciences
National University of Ireland
Ireland
donal.oregan@nuigalway.ie
Essential maps
homotopy
inward maps
acyclic maps
Article.3.pdf
[
[1]
R. P. Agarwal, D. O'Regan, Fixed points for admissible multimaps, Dynam. Systems Appl., 11 (2002), 437-448
##[2]
R. P. Agarwal, D.O'Regan, A note on the topological transversality theorem for acyclic maps, Appl. Math. Lett., 18 (2005), 17-22
##[3]
K. Deimling, Multivalued differential equations, Walter de Gruyter & Co., Berlin (1992)
##[4]
P. M. Fitzpatrick, W. V. Petryshyn, Fixed point theorems for multivalued noncompact acyclic mappings, Pacific J. Math., 54 (1974), 17-23
##[5]
G. Fournier, H.-O. Peitgen, On some fixed point principles for cones in linear normed spaces, Math. Ann., 225 (1977), 205-218
##[6]
A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York (2003)
##[7]
B. Halpern, Fixed point theorems for set--valued maps in infinite dimensional spaces, Math. Ann., 189 (1970), 87-98
##[8]
D. O'Regan, A continuation theory for weakly inward maps, Glasgow Math. J., 40 (1998), 311-321
##[9]
D. O'Regan, Homotopy and Leray--Schauder type results for admissible inward multimaps, J. Concr. Appl. Math., 2 (2004), 67-76
##[10]
D. O'Regan, Continuation theorems for acyclic maps in topological spaces, Commun. Appl. Anal., 13 (2009), 39-46
]
An inertia-based algorithm for pseudomonotone variational inequality and fixed point problems in real Hilbert space
An inertia-based algorithm for pseudomonotone variational inequality and fixed point problems in real Hilbert space
en
en
The aim of this work is to study a pseudomonotone variational
inequality and a fixed point problem involving pseudocontractive
mappings in real Hilbert spaces. We introduce an inertia-based
iterative algorithm for finding a common solution to this problem.
The strong convergence of the proposed algorithm is proved.
Finally, numerical examples are provided and also meaningful
comparisons of these results with those in [Y. Yao, M. Postolache, J. C. Yao, Mathematics, \(\textbf{7}\) (2019), 14 pages],
proving that at our proposed numerical schemes are more efficient.
209
224
J. N.
Ezeora
Department of Mathematics and Statistics
University of Port Harcourt
Nigeria
jeremiah.ezeora@uniport.edu.ng
R. C.
Ogbonna
Department of Computer Science and Mathematics
Evangel University
Aka-eze
richardmathematics@evangeluniversity.edu.ng
F. E.
Bazuaye
Department of Mathematics and Statistics
University of Port Harcourt
Nigeria
febazuaye@yahoo.com
Pseudomonotone variational inequality
pseudocontractive mapping
fixed point problem
Hilbert space
Article.4.pdf
[
[1]
G. L. Acedo, H.-K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67 (2007), 2258-2271
##[2]
Q. H. Ansari, J. C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl., 108 (2001), 527-541
##[3]
J. Y. Bello Cruz, A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert space, Numer. Funct. Anal. Optim., 30 (2009), 23-36
##[4]
R. I. Bot, E. R. Csetnek, P. T. Vuong, The forward-backward-forward method from discrete and continuous perspective for pseudomonotone variational inequalities in Hilbert spaces, arXiv, 2018 (2018), 1-12
##[5]
L. C. Ceng,, A. Petrus¸el, J. C. Yao, Y. Yao, Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, 19 (2018), 487-501
##[6]
Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich´s extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119-1132
##[7]
C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer-Verlag, London (2009)
##[8]
C Chidume, S Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractioons, Proc. Amer. Math. Soc., 129 (2001), 2359-2363
##[9]
P. Cholamjiak, D. V. Thong, Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., 169 (2020), 217-245
##[10]
V. Dadashi, M. Postolache, Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators, Arab. J. Math. (Springer), 9 (2020), 89-99
##[11]
Q. L. Dong, Y. J. Cho, L. L. Zhong, T. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., 70 (2018), 687-704
##[12]
J. N. Ezeora, Approximation of solution of a system of variational inequality problems in real Banach spaces, JP J. Fixed Point Theory Appl., 11 (2017), 207-220
##[13]
B.-S. He, Z.-H. Yang, X.-M. Yuan, An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., 300 (2004), 362-374
##[14]
D. A. Lorenz, T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325
##[15]
P.-E. Mainge, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236
##[16]
B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Comput. Math. Phys., 4 (1964), 1-17
##[17]
Y. Shehu, O. S. Iyiola, Projection methods with alternating inertial steps for variational inequalities: weak and linear convergence, Appl. Numer. Math., 157 (2020), 315-337
##[18]
W. Sintunavarat, A. Pitea, On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis, J. Nonlinear Sci. Appl., 9 (2016), 2553-2562
##[19]
M. Tian, B.-N. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Inequal. Appl., 2017 (2017), 1-17
##[20]
D. Van Hieu, P. K. Anh, L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932
##[21]
D. Van Hieu, L. D. Muu, P. K. Quy, L. Van Vy, Explicit extragradient-like method with regularization for variational inequalities, Results Math., 74 (2019), 1-20
##[22]
J. Yang, H. W. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer. Algorithms, 80 (2019), 741-752
##[23]
Y. Yao, Y. C. Liou, J. C. Yao, Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction, Fixed Point Theory Appl., 2015 (2015), 1-19
##[24]
Y. Yao, M. Postolache, J. C. Yao, Iterative algorithms for pseudomonotone variational inequalities and fixed point problems of pseudocontractive operators, Mathematics, 7 (2019), 1-14
##[25]
H. Y. Zhou, Strong convergence of an explicit iterative algorithm for continuous pseudocontractions in Banach, Nonlinear Anal., 70 (2009), 4039-4046
]
The Dirichlet-type Laplace transforms
The Dirichlet-type Laplace transforms
en
en
We show that it is possible to define extensions of the Laplace transform
that use a general Dirichlet series as a kernel. These transforms, denoted by DLTs, further generalize
those, considered in previous papers, in which the kernels were related to Laguerre-type exponentials or
Bell polynomials. Computational techniques, exploiting expansions in Laguerre polynomials, and using Tricomi's
method, have been considered. Since it turns out that the transforms considered are obtained as linear
combinations of ordinary Laplace transforms, it is also possible to define an approximation of the relevant
inverse transforms. Numerical experiments, performed with the algebra program Mathematica,
show that the introduced technique is fast and efficient.
225
239
D.
Caratelli
Electromagnetics Group, Department of Electrical Engineering
Eindhoven University of Technology
The Netherlands
d.caratelli@tue.nl
S.
Pinelas
Academia Militar-Departamento de Ciências Exatas e Engenharias
Portugal
sandra.pinelas@gmail.com
P. E.
Ricci
Sezione di Matematica
International Telematic University UniNettuno
Italy
paoloemilioricci@gmail.com
General Dirichlet series
Laguerre-type exponentials
Bell polynomials
Laplace transform
Article.5.pdf
[
[1]
R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie, Fourier and Laplace Transforms, Cambridge University Press, Cambridge (2003)
##[2]
E. T. Bell, Exponential polynomials, Ann. of Math. (2), 35 (1934), 258-277
##[3]
J. Blissard, Theory of generic functions, Quarterly J. Pure Appl. Math., 4 (1861), 279-305
##[4]
D. Caratelli, C. Cesarano, P. E. Ricci, Computation of the Bell-Laplace transforms, Dolomites Res. Notes Approx., 14 (2021), 74-91
##[5]
D. Caratelli, P. E. Ricci, Approximating generalized Laplace transforms, J. Comput. Math., 6 (2021), 15-40
##[6]
L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)
##[7]
G. Dattoli, P. E. Ricci, Laguerre-type exponentials, and the relevant L-circular and L-hyperbolic functions, Georgian Math. J., 10 (2003), 481-494
##[8]
A. Ghizzetti, A. Ossicini, Trasformate di Laplace e calcolo simbolico, Unione Tipografico–Editrice Torinese (UTET), Turin (1971)
##[9]
V. Kiryakova, From the hyper-Bessel opertors of Dimovski to the generalized fractional calculus, Fract. Calc. Appl. Anal., 17 (2014), 977-1000
##[10]
] V. S. Kiryakova, A. C. McBride, Explicit solution of the nonhomogeneous hyper-Bessel differential equation, C. R. Acad. Bulgare Sci., 46 (1993), 23-26
##[11]
E. Le Roy, Valeurs asymptotiques de certaines séries procédant suivant les puissances entières et positives d'une variable réelle, Darboux Bull. (2), 4 (1899), 245-268
##[12]
P. E. Ricci, Bell polynomials and generalized Laplace transforms, arXiv, 2021 (2021), 1-17
##[13]
P. E. Ricci, D. Caratelli, F. Mainardi, Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials, Symmetry, 13 (2021), 1-13
##[14]
J. Riordan, Introduction to Combinatorial Analysis, John Wiley & Sons, New York (1958)
##[15]
S. Roman, The umbral calculus, Academic Press, New York (1984)
##[16]
S. M. Roman, G.-C. Rota, The umbral calculus, Advances in Math., 27 (1978), 95-188
##[17]
F. G. Tricomi, Ancora sull’inversione della trasformazione di Laplace, (Italian), Rend. Lincei. (6), 21 (1935), 420-426
##[18]
F. G. Tricomi, Trasformazione di Laplace e polinomi di Laguerre, I. Inversione della trasformazione; II. Alcune nuove formule sui polinomi di Laguerre, (Italian), Rend. Accad. Lincei (vi), 21 (1935), 235-242
##[19]
D. V. Widder, The Laplace Transform, Princeton University Press, Princeton (1946)
]
Qualitative analysis, chaos and coexisting attractors in an asymmetric four-well \(\phi^8\)-generalized Liénard oscillator driven by parametric and external excitations
Qualitative analysis, chaos and coexisting attractors in an asymmetric four-well \(\phi^8\)-generalized Liénard oscillator driven by parametric and external excitations
en
en
In this paper, we study the qualitative dynamical analysis, routes to chaos and the coexistence of attractors in a four-well \(\phi^8\)-generalized Liénard oscillator under external and parametric excitations. The local analysis of the autonomous system reveals saddles, nodes, spirals or centers for appropriate choice of stiffness and damping coefficients. The existence of a Hopf bifurcation is proved during the stability analysis of the equilibrium points. The routes to chaos and the prediction of coexisting attractors have been investigated numerically by using the fourth order Runge-Kutta algorithm. The bifurcation structures obtained show that the system displays a rich variety of bifurcation phenomena, such as symmetry breaking, symmetry restoring, period-doubling, period windows, period-m bubbles, reverse period windows, antimonotonicity, intermittency, quasiperiodic, and chaos. In addition, remerging chaotic band attractors and remarkable routes to chaos occur in the system. Further, it is found that the system presents various coexistence of two attractors as well as the monostability and bistability phenomena. On the other hand, for large amplitude of the parametric excitation and with \(\omega = 1\), the coexistence of asymmetric periodic bursting oscillations of different topologies takes place in the system. It has also been shown numerically that for appropriate values of system parameters and initial conditions, the presented system can exhibit up to five types of coexisting multiple attractors.
240
252
Y. J. F.
Kpomahou
Department of Industrial and Technical Sciences, ENSET-Lokossa
UNSTIM-Abomey
Benin
fkpomahou@gmail.com
J. A.
Adechinan
Department of Physics, FAST-Natitingou
UNSTIM-Abomey
Benin
J.
Edou
Department of Physics
University of Abomey-Calavi
Benin
L. A.
Hinvi
Departement de Genie Mecanique et Productique (GMP)
Institut National Superieur de Technologie Industrielle (INSTI)
Benin
Generalized Liénard oscillator
four well potential
parametric excitation
local stability
coexisting attractors
Article.6.pdf
[
[1]
B. Bao, H. Qian, Q. Xu, M. Chen, J. Wang, Y. Yu, Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based hopfield neural network, Front. Comput. Neurosci., 11 (2017), 1-14
##[2]
M. Dutta, H. E. Nusse, E. Ott, J. A. Yorke, G. Yuan, Multiple attractor bifurcations: a source of unpredictability in piecewise smooth systems, Phys. Rev. Lett., 83 (1999), 4281-4284
##[3]
L. Forest, N. Glade, J. Demongeot, Lienard systems and potential-Hamiltonian decomposition-applications in biology, C. R. Biol., 330 (2007), 97-106
##[4]
R. Grimshaw, Nonlinear ordinary differential equations, Blackwell, Oxford, England (1990)
##[5]
M. Han, V. G. Romanovski, On the number of limit cycles of polynomial Lienard systems, Nonlinear Anal. Real World Appl., 14 (2013), 1655-1668
##[6]
T. Harko, S. D. Liang, Exact solutions of the Lienard-and generalized Lienard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator, J. Eng. Math., 98 (2016), 93-111
##[7]
L. A. Hinvi, A. A. Koukpemedji, V. A. Monwanou, C. H. Miwadinou, V. K. Tamba, Resonance, chaos and coexistence of attractors in a position dependent mass driven Duffing type oscillator, J. Korean Phys. Soc., 79 (2021), 755-771
##[8]
D. W. Jordan, P. Smith, Nonlinear ordinary differential equations, Oxford University Press, England (1987)
##[9]
H. N. Kondo, D. Farkas, S. L. Denham, T. Asai, I. Winkler, Auditory multistability and neurotransmitter concentrations in the human brain, Philos. Trans. Royal Soc. B: Biol. Sci., 372 (2017), 1-8
##[10]
H. L. Koudahoun, Y. J. F. Kpomahou, J. Akande, D. K. K. Adjaı, Chaotic dynamics of an extended duffing oscillator under Periodic excitation, World J. Appl. Phys., 3 (2018), 34-50
##[11]
] Y. J. F. Kpomahou, J. A. Adechinan, L. A. Hinvi, Effects of quartic nonlinearities and constant excitation force on nonlinear dynamics of plasma oscillations modeled by a Lienard-type oscillator with asymmetric double well potential, Indian J. Phys., 2022 (2022), 1-20
##[12]
Y. J. F. Kpomahou, R. G. Agbokpanzo, L. A. Hinvi, Regular and chaotic oscillations in a modified Rayleigh-Lienard system under parametric excitation, Int. J. Adv. Math. Mech., 7 (2019), 29-44
##[13]
Y. J. F. Kpomahou, L. A. Hinvi, J. A. Adechinan, C. H. Miwadinou, Chaotic dynamics of a mixed Rayleigh-Lienard oscillator driven by parametric periodic damping and external excitations, Complexity, 2021 (2021), 1-18
##[14]
Q. Lai, A unified chaotic system with various coexisting attractors, Int. J. Bifurcat. Chaos, 31 (2021), 1-11
##[15]
D. Liu, H. Yamaura, Chaos control of a Van der Pol oscillator driven by external excitation, Nonlinear Dyn., 68 (2012), 95-105
##[16]
S. Lynch, C. J. Christopher, Limit cycles in highly nonlinear differential equations, J. Sound Vibr., 243 (1999), 505-517
##[17]
S. Lynch, A. L. Steele, J. E. Hoad, Stability analysis of nonlinear optical resonators, Chaos Solitons Fract., 9 (1998), 936-946
##[18]
A. Maccari, Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation, Nonlinear Dyn., 15 (1998), 329-343
##[19]
A. Maccari, Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Lienard systems, Int. J. Nonlinear Mech., 35 (2000), 239-262
##[20]
A. Maccari, Bifurcation analysis of parametrically excited Rayleigh-Lienard oscillators, Nonlinear Dyn., 25 (2001), 293-316
##[21]
C. H. Miwadinou, A. V. Monwanou, L. A. Hinv, J. B. Chabi Orou, Stability and chaotic dynamics of forced generalized Lienard systems, Pramana-J. Phys., 93 (2019), 1-13
##[22]
C. H. Miwadinou, A. V. Monwanou, A. A. Koukpemedji, Y. J. F. Kpomahou, J. B. Chabi Orou, Chaotic motions in forced mixed Rayleigh-Lienard oscillator with external and parametric periodic excitations, Int. J. Bifurcat Chaos., 28 (2018), 1-16
##[23]
L. Perko, Differential Equations and Dynamical Systems,, Springer, New York (1991)
##[24]
A. Ray, D. Ghosh, A. R. Chowdhury, Topological study of Multiple coexisting attractors in a nonlinear system, J. Phys. A: Math.Theor., 42 (2009), 1-16
##[25]
S. K. Scott, Oscillations, waves and chaos in chemical kinetics, Oxford Science Publication, England (1994)
##[26]
X. Sun, Multiple limit cycles of some strongly nonlinear Lienard-Van der Pol oscillator, Appl. Math. Comput., 270 (2015), 620-630
##[27]
K. Sun, A. Di-li D. Li-kun, Y. Dong, H. Wang, Ke Zhong, Multiple coexisting attractors and Hysteresis in the generalized Ueda oscillator, Math. Prob. Eng., 2013 (2013), 1-7
##[28]
J. Szezyglowski, Influence of eddy currents on magnetic hysteresis loops in soft magnetic materials, J. Magnetic Matter., 223 (2001), 97-102
##[29]
J. Warminski, Regular, chaotic and hyperchaotic vibrations of nonlinear systems with self, parametric and external excitations, Mech. Automat. Control Robot., 3 (2003), 891-905
##[30]
Y. Wu, L. Guo, Y. Chen, Hopf Bifurcation of Z2-equivariant generalized Lienard systems, Int. J. Bifurcat. Chaos, 28 (2018), 1-12
##[31]
J. Yang, W. Ding, Limit cycles of a class of Lienard systems with restoring forces of seventh degree, Appl. Math. Comput., 316 (2018), 422-437
]