Abstract
In this paper we consider dynamics of three unidirectionally coupled Duffing oscillators with nonlinear coupling function in the form of third degree polynomial. We focus on the influence of the coupling on the occurrence of different bifurcation’s scenarios. The stability of equilibria, using Routh-Hurwitz criterion, is investigated. Moreover, we check how coefficients of the nonlinear coupling influence an appearance of different types of periodic solutions. The stable periodic solutions are computed using path-following. Finally, we show the two parameters’ bifurcation diagrams with marked areas where one can observe the coexistence of solutions.
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References
S. Watanabe, S.H. Strogatz, Phys. Rev. Lett. 70, 2391 (1993)
S.H. Strogatz, Nature 410, 268 (2001)
O.E. Omel’chenko, C. Hauptmann, Yu.L. Maistrenko, P.A. Tass, Physica D 237, 365 (2008)
L. Lücken, S. Yanchuk, Physica D 241, 350 (2012)
V.V. Astakhov, V.S. Anishchenko, A.V. Shabunin, IEEE Trans. Circuits Syst. 42, 352 (1995)
V.N. Belykh, I.V. Belykh, K.V. Nelvidin, Math. Comput. Simulat. 58, 477 (2002)
W. Lu, T. Chen, G. Chen, Physica D 221, 118 (2006)
P. Perlikowski, B. Jagiello, A. Stefanski, T. Kapitaniak, Phys. Rev. E 78, 017203 (2008)
V. Anishchenko, S. Nikolaev, J. Kurths, Chaos 18, 037123 (2008)
V. Anishchenko, S. Astakhov, T. Vadivasova, Europhys. Lett. 86, 30003 (2009)
J.C. Rekling, J.L. Feldman, J. Neurophysiol. 78, 3508 (1997)
L. Pasti, A. Volterra, T. Pozzan, G. Carmignoto, J. Neurosci. 17, 7817 (1997)
F. Vega-Redondo, Complex Social Networks (Cambridge University Press, Cambridge, 2007)
S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Nature 464, 1025 (2010)
C. Sommer, R. German, F. Dressler, IEEE Trans. Mobile Comput. 10, 3 (2011)
A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C.R. Mirasso, L. Pesquera, K.A. Shore, Nature 438, 343 (2005)
Yo. Horikawa, Hiroyuki Kitajima, Physica D 238, 216 (2009)
Yo. Horikawa, J. Theor. Biol. 289, 151 (2011)
E. Padmanaban, R. Banerjee, S.K. Dana, Int. J. Bifurc. Chaos 22, 1250177 (2012)
M.A. Matías, J. Güémez, Phys. Rev. Lett. 81, 4124 (1998)
P. Perlikowski, S. Yanchuk, O.V. Popovych, P.A. Tass, Phys. Rev. E 82, 036208 (2010)
S. Yanchuk, P. Perlikowski, O.V. Popovych, P.A. Tass, Chaos 21, 047511 (2011)
A. Dvorak, P. Kuzma, P. Perlikowski, V. Astakhov, T. Kapitaniak, Eur. Phys. J. Special Topics 222, 2429 (2013)
E.J. Doedel, Congressus Numerantium 30, 265 (1981)
E.J. Doedel, Auto-07P: Continuation, bifurcation software for ordinary differential equations with major contributions from A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B.E. Oldeman, R.C. Paffenroth, B. Sandstede, X.J. Wang, C. Zhang; available from http://cmvl.cs.concordia.ca/auto/
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Jaros, P., Kapitaniak, T. & Perlikowski, P. Multistability in nonlinearly coupled ring of Duffing systems. Eur. Phys. J. Spec. Top. 225, 2623–2634 (2016). https://doi.org/10.1140/epjst/e2016-60015-7
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DOI: https://doi.org/10.1140/epjst/e2016-60015-7