Synchronization in the van der Pol-Duffing system via elastic and dissipative couplings
Keywords:Nonlinear dynamics, control of chaos, synchronization
The classical master-slave configuration allows synchronizing pairs of unidirectionally coupled systems in a relatively easy manner. However, it has been found that this scheme has a limitation: for certain systems including those with chaotic dynamics this scheme fails to induce synchronization. In this work a modified master-slave scheme, based on combining elastic and dissipative couplings is presented. We focuses on a possible solution for this limitation by illustrating our method employing the va der Pol and Dung oscillators and analyzing three dierent ways of coupling. We obtain, synchronization in both oscillators.
L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821, https://doi.org/10.1103/PhysRevLett.64.821.
I. Pastor-Diaz and A. Lopez-Fraguas, Dynamics of two coupled van der Pol oscillators, Phys. Rev. E 52 (1995) 1480. https://doi.org/10.1103/PhysRevE.52.1480.
C. Reick and E. Mosekilde, Emergence of quasiperiodicity in symmetrically coupled, identical period-doubling systems, Phys. Rev. E 52 (1995) 1418, https://doi.org/10.1103/PhysRevE.52.1418.
M. Z. Ding and W. H. Yang, and H. J. Zhang, Observation of intermingled basins in coupled oscillators exhibiting synchronized chaos Phys. Rev. E 54 (1995) 2489. https://doi.org/10.1103/PhysRevE.54.2489.
H-W. Yin and J-H. Dai, Phase effect of two coupled periodically driven Duffing oscillators, Phys. Rev. E 58 (1998) 5683, https://doi.org/10.1103/PhysRevE.58.5683.
K-J. Lee, Y. Kwak and T-K. Lim, Phase jumps near a phase synchronization transition in systems of two coupled chaotic oscillators, Phys. Rev. Lett. 81 (1998) 321. https://doi.org/10.1103/PhysRevLett.81.321.
L.O. Chua, M. Itoh, L. Kocarev and K. Eckert, Chaos synchronization in Chua’s circuit, J. Circuits Syst. Comput. 3 (1993) 93, https://doi.org/10.1142/S0218126693000071.
A.P. Kuznetsov, and J.P. Roman, Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization, Phys. D 238 (2009) 1499. https://doi.org/10.1016/j.physd.2009.04.016.
M. G. Rosenblum, A. Pikovsky and J. Kurths, From phase to lag synchronization in coupled chaotic oscillators, Phys. Rev. Lett. 78 (1997) 4193. https://doi.org/10.1103/PhysRevLett.78.4193.
K. Murali and M. Lakshmanan, Transmission of signals by synchronization in a chaotic Van der Pol-Duffing oscillator, Phys. Rev. E 48 (1993) R1624, https://doi.org/10.1103/PhysRevE.48.R1624.
A.N. Njah, Synchronization via active control of parametrically and externally excited Φ 6 Van der Pol and Duffing oscillators and application to secure communications, J. Vib. Control 17 (2010) 504, https://doi.org/10.1177/1077546309357024.
L. Lu, F. Zhang and C. Han, Synchronization transmission of the target signal in the circuit network based on coupling
technique, Physica A: Statistical Mechanics and its Applications 535 (2019) 122412, https://doi.org/10.1016/j.physa.2019.122412.
M. S. Dutra, A.C. de Pina Filho and V.F. Romano, Modeling of a bipedal locomotor using coupled nonlinear oscillators of Van der Pol, Biol. Cybern. 88 (2003) 292, https://doi.org/10.1007/s00422-002-0380-8.
F. Jasni and A. A. Shafie, Van Der Pol Central Pattern Generator (VDPCPG) Model for Quadruped Robot, in Intelligent Robotics, Automation, and Manufacturing (Springer, Berlin, 2012), Vol. 330, p. 175, https://doi.org/10.1007/978-3-642-35197-6 18.
S. Mall and S. Chakraverty, Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation, Neural Comput. 28 (2016) 1598, https://doi.org/10.1162/NECO a 00858.
S. S. Chaharborj, S. S. Chaharborj, and P. P. See, Application of Chebyshev neural network to solve Van der Pol equations, Int. J. Basic Appl. Sci. 10 (2021) 7, https://www.sciencepubco.com/index.php/ijbas/article/view/31431.
D. Pazo and E. Montbri ´ o, Low-dimensional dynamics of populations of pulse-coupled, Phys. Rev. X 4 (2014) 011009, https://doi.org/10.1103/PhysRevX.4.011009.
M. Zhang, G.S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, Synchronization of micromechanical oscillators using light, Phys. Rev. 109 (2012) 233906. https://doi.org/10.1103/PhysRevLett.109.233906.
J.C. Chedjou, K. Kyamakya, I. Moussa, H.P. Kuchenbecker, and W. Mathis, Behavior of a self sustained electromechanical transducer and routes to chaos, J. Vib. Acoust. 128 (2006) 282. https://doi.org/10.1115/1.2172255.
J.C. Chedjou, H.B. Fotsin, P. Woafo, and S. Domngang, Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48 (2001) 748. https://doi.org/10.1109/81.928157.
A.P. Kuznetsov, N.V. Stankevich, and L.V. Turukina, Coupled van der Pol-Duffing oscillators: phase dynamics and structure of synchronization tongues, Physica D 238 (2009) 1203. https://doi.org/10.1016/j.physd.2009.04.001.
M.S. Siewe, S.B. Yamgoue, E.M. Moukam Kakmeni and C. Tchawoua, Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory, Nonlinear Dyn. 62 (2010) 379. https://doi.org/10.1007/s11071-010-9725-3.
U.E. Vincent, A. Kenfack, Synchronization and bifurcation structures in coupled periodically forced non-identical Duffing oscillator, Phys. Scr. 77 (2008) 045005. https://doi.org/10.1088/0031-8949/77/04/045005.
U. Uriostegui, E.S. Tututi and G. Arroyo, A new scheme of coupling and synchronizing low-dimensional dynamical systems, Rev. Mex. Fis. 67 (2021) 334. https::https://doi.org/10.31349/RevMexFis.67.334.
J. Kengne, J.C. Chedjou, G. Kenne, K. Kyamakya, and G.H. Kom, Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator, Nonlinear Dyn, 70 (2012) 2163. https://doi.org/10.1007/s11071-012-0607-8.
J. Kengne, F. Kenmogne, and V. Kamdoum Tamba, Experiment on bifurcation and chaos in coupled anisochronous selfexcited systems: Case of two coupled van der Pol-Duffing oscillators, Journal of Nonlinear Dynamics 2014 (2014) 815783. https://doi.org/10.1155/2014/815783.
H. Zhang, D. Liu and Z. Wang, Controlling Chaos: Suppression, Synchronization and Chaotification, Springer, London, (2009). https://doi.org/10.1007/978-1-84882-523-9.
S. Boccaletti, J. Kurths, G. Osipov, DL. Valladares and CS. Zhou, The synchronization of chaotic systems, Physics Reports 366 (2002) 101. https://doi.org/10.1016/S0370-1573(02)00137-0.
L. M. Pecora and T. L. Carroll Synchronization of chaotic systems, Chaos, 25 (2015) 097611. https://doi.org/10.1063/1.4917383.
L. Thomas and L. M. Pecora, Synchronizing nonautonomous chaotic circuits, IEEE Trans. on Circuits and Systems-II, 40 (1993) 646, https://doi.org/10.1109/82.246166.
W-C C. Chan and Y-D Chao, Synchronization of coupled forced oscillators, J. Math. An. and Applications 218 (1998) 97, https://doi.org/10.1006/jmaa.1997.5749.
T-P Chang, Chaotic motion in forced duffing system subject to linear and nonlinear damping, Mathematical Problems in Engineering, 2017 (2017) 3769870. https://doi.org/10.1155/2017/3769870.
M. S. Siewe, C. Tchawoua, and P. Woafo, Melnikov chaos in a periodically driven Rayleigh-Duffing oscillator, Mechanics
Research Communications, 37 (2010) 363, https://doi.org/10.1016/j.mechrescom.2010.04.001.
Y-Z. Wang, and F-M. Li, Dynamical properties of Duffing-van der Pol oscillator subject to both external and parametric excitations with time delayed feedback control, J. Vibration and Control 21 (2015) 371, https://doi.org/10.1177/1077546313483160.
K. Ding, Master-Slave Synchronization of Chaotic Φ 6 Duffing Oscillators by Linear State Error Feedback Control, Complexity 2019 (2019) 3637902, https://doi.org/10.1155/2019/3637902.
A. Buscarino, L. Fortuna, and L. Patane, Master-slave synchronization of hyperchaotic systems through a linear dynamic coupling, Phys. Rev. E 100 (2019) 032215, https://doi.org/10.1103/PhysRevE.100.032215.
F. Aydogmus and E. Tosyali, Master-slave synchronization in a 4D dissipative nonlinear fermionic system, International Journal of Control 2020 (2020) 1, https://doi.org/10.1080/00207179.2020.1808244.
J.P. Ramirez, E. Garcia and J. Alvarez, Master-slave synchronization via dynamic control, Commun Nonlinear Sci Numer Simulat, 80 (2020) 104977. https://doi.org/10.1016/j.cnsns.2019.104977.
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