Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling

Xiaojun Huang; Zigen Song; Jian Xu; Xiaojun Huang; Zigen Song; Jian Xu

doi:10.3934/era.2023353

Electronic Research Archive

2023, Volume 31, Issue 11: 6964-6981. doi: 10.3934/era.2023353

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Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling

1.
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
2.
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China

Received: 14 June 2023 Revised: 11 October 2023 Accepted: 24 October 2023 Published: 30 October 2023

In this paper, we investigated the dynamics of a pair of VDP (Van der Pol) oscillators with direct-indirect coupling, which is described by five first-order differential equations. The system presented three types of equilibria including HSS (homogeneous steady state), IHSS (inhomogeneous steady state) and NPSS (no-pattern steady state). Employing the corresponding characteristic equations of the linearized system, we obtained the necessary conditions for the pitchfork and Hopf bifurcations of the equilibria. Further, we illustrated one-dimensional bifurcation and phase diagrams to verify theoretical results. The results show that the system exhibited two types of oscillation quenching, i.e., amplitude death (AD) for HSS equilibria and oscillation death (OD) for IHSS equilibria. In some special regions of the parameters, the system proposed multiple types of stable coexistence including HSS and IHSS equilibria, periodic orbits or quasi-periodic oscillations.
- VDP oscillator,
- direct-indirect coupling,
- oscillation death,
- bifurcation,
- multistability
Citation: Xiaojun Huang, Zigen Song, Jian Xu. Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling[J]. Electronic Research Archive, 2023, 31(11): 6964-6981. doi: 10.3934/era.2023353

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Abstract

In this paper, we investigated the dynamics of a pair of VDP (Van der Pol) oscillators with direct-indirect coupling, which is described by five first-order differential equations. The system presented three types of equilibria including HSS (homogeneous steady state), IHSS (inhomogeneous steady state) and NPSS (no-pattern steady state). Employing the corresponding characteristic equations of the linearized system, we obtained the necessary conditions for the pitchfork and Hopf bifurcations of the equilibria. Further, we illustrated one-dimensional bifurcation and phase diagrams to verify theoretical results. The results show that the system exhibited two types of oscillation quenching, i.e., amplitude death (AD) for HSS equilibria and oscillation death (OD) for IHSS equilibria. In some special regions of the parameters, the system proposed multiple types of stable coexistence including HSS and IHSS equilibria, periodic orbits or quasi-periodic oscillations.

References

[1]	A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511755743
[2]	K. Kaneko, Theory and applications of coupled map lattices, in Nonlinear Science: Theory and Applications, Wiley–Blackwell, 1993. Available from: https://cir.nii.ac.jp/crid/1573105973923422464.
[3]	E. Ott, K. Wiesenfeld, Chaos in Dynamical Systems, Phys. Today, 47 (1994). https://doi.org/10.1063/1.2808369 doi: 10.1063/1.2808369
[4]	G. Saxena, A. Prasad, R. Ramaswamy, Amplitude death: the emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521 (2012), 205–228. https://doi.org/10.1016/j.physrep.2012.09.003 doi: 10.1016/j.physrep.2012.09.003
[5]	P. Kumar, A. Prasad, R. Ghosh, Stable phase-locking of an external-cavity diode laser subjected to external optical injection, J. Phys. B: At. Mol. Opt. Phys., 41 (2008), 135402. https://doi.org/10.1088/0953-4075/41/13/135402 doi: 10.1088/0953-4075/41/13/135402
[6]	B. Gallego, P. Cessi, Decadal variability of two oceans and an atmosphere, J. Clim., 14 (2001), 2815–2832. https://doi.org/10.1175/1520-0442(2001)014<2815:DVOTOA>2.0.CO;2 doi: 10.1175/1520-0442(2001)014<2815:DVOTOA>2.0.CO;2
[7]	H. Zhang, D. Xu, C. Lu, E. Qi, J. Hu, Y. Wu, Amplitude death of a multi-module floating airport, Nonlinear Dyn., 79 (2015), 2385–2394. https://doi.org/10.1007/s11071-014-1819-x doi: 10.1007/s11071-014-1819-x
[8]	T. Banerjee, D. Biswas, Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion, Chaos, 23 (2013), 043101. https://doi.org/10.1063/1.4823599 doi: 10.1063/1.4823599
[9]	D. Ghosh, T. Banerjee, Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 90 (2014), 062908. https://doi.org/10.1103/PhysRevE.90.062908 doi: 10.1103/PhysRevE.90.062908
[10]	G. B. Ermentrout, N. Kopell, Oscillator death in systems of coupled neural oscillators, SIAM J. Appl. Math., 50 (1990), 125–146. https://doi.org/10.1137/0150009 doi: 10.1137/0150009
[11]	A. Koseska, E. Volkov, J. Kurths, Oscillation quenching mechanisms: amplitude vs. oscillation death, Phys. Rep., 531 (2013), 173–199. https://doi.org/10.1016/j.physrep.2013.06.001 doi: 10.1016/j.physrep.2013.06.001
[12]	R. Curtu, Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network, Physica D, 239 (2010), 504–514. https://doi.org/10.1016/j.physd.2009.12.010 doi: 10.1016/j.physd.2009.12.010
[13]	A. Koseska, E. Volkov, J. Kurths, Parameter mismatches and oscillation death in coupled oscillators, Chaos, 20 (2010), 023132. https://doi.org/10.1063/1.3456937 doi: 10.1063/1.3456937
[14]	N. Suzuki, C. Furusawa, K. Kaneko, Oscillatory protein expression dynamics endows stem cells with robust differentiation potential, PLoS One, 6 (2011), e27232. https://doi.org/10.1371/journal.pone.0027232 doi: 10.1371/journal.pone.0027232
[15]	D. Biswas, N. Hui, T. Banerjee, Amplitude death in intrinsic time-delayed chaotic oscillators with direct–indirect coupling: the existence of death islands, Nonlinear Dyn., 88 (2017), 2783–2795. https://doi.org/10.1007/s11071-017-3411-7 doi: 10.1007/s11071-017-3411-7
[16]	A. H. Nayfeh, D. T. Mook, Nonlinear Oscillations, John Wiley & Sons, 2008.
[17]	A. Anees, Z. Ahmed, A technique for designing substitution box based on van der pol oscillator, Wireless Pers. Commun., 82 (2015), 1497–1503. https://doi.org/10.1007/s11277-015-2295-4 doi: 10.1007/s11277-015-2295-4
[18]	G. Juárez, M. Ramírez-Trocherie, Á. Báez, A. Lobato, E. Iglesias-Rodríguez, P. Padilla, et al., Hopf bifurcation for a fractional van der Pol oscillator and applications to aerodynamics: implications in flutter, J. Eng. Math., 139 (2023), 1–15. https://doi.org/10.1007/s10665-023-10258-7 doi: 10.1007/s10665-023-10258-7
[19]	S. Dutta, N. R. Cooper, Critical response of a quantum van der Pol oscillator, Phys. Rev. Lett., 123 (2019), 250401. https://doi.org/10.1103/PhysRevLett.123.250401 doi: 10.1103/PhysRevLett.123.250401
[20]	S. Wirkus, R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dyn., 30 (2002), 205–221. https://doi.org/10.1023/A:1020536525009 doi: 10.1023/A:1020536525009
[21]	E. Camacho, R. Rand, H. Howland, Dynamics of two van der Pol oscillators coupled via a bath, Int. J. Solids Struct., 41 (2004), 2133–2143. https://doi.org/10.1016/j.ijsolstr.2003.11.035 doi: 10.1016/j.ijsolstr.2003.11.035
[22]	K. Konishi, Experimental evidence for amplitude death induced by dynamic coupling: van der Pol oscillators, in 2004 IEEE International Symposium on Circuits and Systems (ISCAS), 4 (2004), 792–795. https://doi.org/10.1109/ISCAS.2004.1329123
[23]	T. Endo, S. Mori, Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators, IEEE Trans. Circuits Syst., 25 (1978), 7–18. https://doi.org/10.1109/TCS.1978.1084380 doi: 10.1109/TCS.1978.1084380
[24]	V. Resmi, G. Ambika, R. E. Amritkar, General mechanism for amplitude death in coupled systems, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 84 (2011), 046212. https://doi.org/10.1103/PhysRevE.84.046212 doi: 10.1103/PhysRevE.84.046212
[25]	D. Ghosh, T. Banerjee Mixed-mode oscillation suppression states in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 92 (2015), 052913. https://doi.org/10.1103/PhysRevE.92.052913 doi: 10.1103/PhysRevE.92.052913
[26]	C. O. Weiss, R. Vilaseca, Dynamics of lasers, NASA STI/Recon Tech. Rep. A, 92 (1991), 39875. Available from: https://ui.adsabs.harvard.edu/abs/1991STIA...9239875W/abstract.
[27]	K. A. Robbins, A new approach to subcritical instability and turbulent transitions in a simple dynamo, in Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 82 (1997), 309–325. https://doi/org/10.1017/S0305004100053950
[28]	B. Ermentrout, XPPAUT 5.0-the Differential Equations Tool, University of Pittsburgh, Pittsburgh, 2001.
[29]	E. X. DeJesus, C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A: At. Mol. Opt. Phys., 35 (1987), 5288. https://doi.org/10.1103/PhysRevA.35.5288 doi: 10.1103/PhysRevA.35.5288
[30]	G. Saxena, A. Prasad, R. Ramaswamy, Amplitude death: the emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521 (2012), 205–228. https://doi.org/10.1016/j.physrep.2012.09.003 doi: 10.1016/j.physrep.2012.09.003
[31]	A. Sharma, M. D. Shrimali Amplitude death with mean-field diffusion, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 85 (2012), 057204. https://doi.org/10.1103/PhysRevE.85.057204 doi: 10.1103/PhysRevE.85.057204
[32]	A. Sharma, K. Suresh, K. Thamilmaran, A. Prasad, M. D. Shrimali, Effect of parameter mismatch and time delay interaction on density-induced amplitude death in coupled nonlinear oscillators, Nonlinear Dyn., 76 (2014), 1797–1806. https://doi.org/10.1007/s11071-014-1247-y doi: 10.1007/s11071-014-1247-y
[33]	T. Banerjee, D. Ghosh, Experimental observation of a transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 89 (2014), 062902. https://doi.org/10.1103/PhysRevE.89.062902 doi: 10.1103/PhysRevE.89.062902
[34]	N. K. Kamal, P. R. Sharma, M. D. Shrimali, Suppression of oscillations in mean-field diffusion, Pramana, 84 (2015), 237–247. https://doi.org/10.1007/s12043-015-0929-4 doi: 10.1007/s12043-015-0929-4
[35]	A. Zakharova, I. Schneider, Y. N. Kyrychko, K. B. Blyuss, A. Koseska, B. Fiedler, et al., Time delay control of symmetry-breaking primary and secondary oscillation death, Europhys. Lett., 104 (2013), 50004. https://doi.org/10.1209/0295-5075/104/50004 doi: 10.1209/0295-5075/104/50004
[36]	D. V. R. Reddy, A. Sen, G. L. Johnston, Time delay induced death in coupled limit cycle oscillators, Phys. Rev. Lett., 80 (1998), 5019. https://doi.org/10.1103/PhysRevLett.80.5109 doi: 10.1103/PhysRevLett.80.5109
[37]	D. V. R. Reddy, A. Sen, G. L. Johnston, Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators, Phys. Rev. Lett., 85 (2000), 3381. https://doi.org/10.1103/PhysRevLett.85.3381 doi: 10.1103/PhysRevLett.85.3381
[38]	F. M. Atay, Distributed delays facilitate amplitude death of coupled oscillators, Phys. Rev. Lett., 91 (2003), 094101. https://doi.org/10.1103/PhysRevLett.91.094101 doi: 10.1103/PhysRevLett.91.094101
[39]	W. Zou, D. V. Senthilkumar, A. Koseska, J. Kurths, Generalizing the transition from amplitude to oscillation death in coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 88 (2013), 050901. https://doi.org/10.1103/PhysRevE.88.050901 doi: 10.1103/PhysRevE.88.050901
[40]	R. Karnatak, R. Ramaswamy, A. Prasad, Amplitude death in the absence of time delays in identical coupled oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 76 (2007), 035201. https://doi.org/10.1103/PhysRevE.76.035201 doi: 10.1103/PhysRevE.76.035201
[41]	A. Sharma, P. R. Sharma, M. D. Shrimali, Amplitude death in nonlinear oscillators with indirect coupling, Phys. Lett. A, 376 (2012), 1562–1566. https://doi.org/10.1016/j.physleta.2012.03.033 doi: 10.1016/j.physleta.2012.03.033
[42]	C. R. Hens, O. I. Olusola, P. Pal, S. K. Dana, Oscillation death in diffusively coupled oscillators by local repulsive link, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 88 (2013), 034902. https://doi.org/10.1103/PhysRevE.88.034902 doi: 10.1103/PhysRevE.88.034902
[43]	B. K. Bera, C. Hens, D. Ghosh, Emergence of amplitude death scenario in a network of oscillators under repulsive delay interaction, Phys. Lett. A, 380 (2016), 2366–2373. https://doi.org/10.1016/j.physleta.2016.05.028 doi: 10.1016/j.physleta.2016.05.028
[44]	N. K. Kamal, P. R. Sharma, M. D. Shrimali, Oscillation suppression in indirectly coupled limit cycle oscillators, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 92 (2015), 022928. https://doi.org/10.1103/PhysRevE.92.022928 doi: 10.1103/PhysRevE.92.022928
[45]	P. R. Sharma, N. K. Kamal, U. K. Verma, K. Suresh, K. Thamilmaran, M. D. Shrimali, Suppression and revival of oscillation in indirectly coupled limit cycle oscillators, Phys. Lett. A, 380 (2016), 3178–3184. https://doi.org/10.1016/j.physleta.2016.07.041 doi: 10.1016/j.physleta.2016.07.041
[46]	A. Sharma, U. K. Verma, M. D. Shrimali, Phase-flip and oscillation-quenching-state transitions through environmental diffusive coupling, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 94 (2016), 062218. https://doi.org/10.1103/PhysRevE.94.062218 doi: 10.1103/PhysRevE.94.062218
[47]	J. Choi, P. Kim, Reservoir computing based on quenched chaos, Chaos, Solitons Fractals, 140 (2020), 110131. https://doi.org/10.1016/j.chaos.2020.110131 doi: 10.1016/j.chaos.2020.110131
[48]	E. Ullner, A. Zaikin, E. I. Volkov, J. García-Ojalvo, Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication, Phys. Rev. Lett., 99 (2007), 148103. https://doi.org/10.1103/PhysRevLett.99.148103 doi: 10.1103/PhysRevLett.99.148103
[49]	A. Takamatsu, Spontaneous switching among multiple spatio-temporal patterns in three-oscillator systems constructed with oscillatory cells of true slime mold, Physica D, 223 (2006), 180–188. https://doi.org/10.1016/j.physd.2006.09.001 doi: 10.1016/j.physd.2006.09.001

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