ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Remote Synchronization in a Small Star-like Network of Spin-torque


Pavel V. Kuptsov

Chief Researcher
Doctor of Sciences, Doce
Kotelnikov Institute of Radio-Engineering and Electronics of RAS,
Saratov Branch
Zelenaya 38, Saratov, 410019, Russia

Vyacheslav P. Kruglov

Candidate of Sciences
Kotelnikov Institute of Radio-Engineering and Electronics of RAS,
Saratov Branch
Zelenaya 38, Saratov, 410019, Russia


We consider a mathematical model of field coupled spin-torque oscillators network. The model is described by a set of the Landau–Lifshitz–Gilbert–Slonczewski magnetization equations, which are coupled via an additional term to the effective field. For this model a generic form of the Jacobi matrix is explicitly derived. The network of four oscillators coupled as a star is considered: the central oscillator is coupled with three others and they do not have direct couplings with each other. For this network the Lyapunov's exponent chart is computed and the area in the parameter space is found where the network demonstrates remote synchronization. In this regime the peripheral oscillators are synchronized with each other but not synchronized with the central one.



  1. Prokopenko O., Bankowski E., Meitzler T. et al. Spin-Torque Nano-Oscillator as a Microwave Signal Source. IEEE Magnetics Letters. 2011. Vol. 2. P. 3000104
  2. Zeng Z., Finocchio G., Zhang B. и др. Ultralow-current-density and bias-field-free spin-transfer nano-oscillator. Scientific Reports. 2013. Vol. 3. No 1. P. 1426
  3. Slonczewski J. Current-driven excitation of magnetic multilayers. Journal of Magnetism and Magnetic Materials. 1996. Vol. 159. No 1. P. L1-L7
  4. Zaks M., Pikovsky A. Chimeras and complex cluster states in arrays of spin-torque oscillators. Scientific Reports. 2017. Vol. 7. No 1. P. 4648
  5. Zaks M. A., Pikovsky A. Synchrony breakdown and noise-induced oscillation death in ensembles of serially connected spin-torque oscillators. The European Physical Journal B. 2019. Vol. 92. No 7. P. 160
  6. Turkin Y. V., Kuptsov P. V. [Dynamics of two field-coupled spin-transfer oscillators] Izvestiya VUZ. Applied Nonlinear Dynamics. 2014. Vol. 22. No 6. P. 69-78. (In Russ. )
  7. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A universal concept in nonlinear sciences. Cambridge: Cambridge University Press, 2003. P. 433
  8. Kuznetsov N. V., Blagov M. V., Alexandrov K. D. et al. [Lock-in range of classical PLL with piecewise-linear phase detector characteristic] Differencialnie Uravnenia i Protsesy Upravlenia. 2019. Vol. 3. P. 74-89
  9. Raznoglazova J. V., Plotnikov S. A. [Bifurcation and Synchronization Control of Two Coupled Two-dimensional Hindmarsh-Rose Systems] Differencialnie Uravnenia i Protsesy Upravlenia. 2020. Vol. 4. P. 127-140
  10. Bergner A., Frasca M., Sciuto G. et al. Remote synchronization in star networks. Phys. Rev. E. 2012. Vol. 85. P. 026208
  11. Kuptsov P. V., Kuptsova A. V. Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators. Communications in Nonlinear Science and Numerical Simulation. 2017. Vol. 50. P. 115-127
  12. Kuptsov P. V., Kuptsova A. V. Indirect synchronization control in a starlike network of phase oscillators. Proc. of SPIE. 2018. Vol. 10717. P. 107172G-1
  13. Burkin I. M. [The Method of Transfer to the Derivative Space: 40 Years of Evolution] Differencialnie Uravnenia i Protsesy Upravlenia. 2015. Vol. 3. P. 51-93. 14 . Oseledets V. I. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 1968, 197-231. Moscov. Mat. Obsch. 19, 1968, 179-210
  14. Pikovsky A., Politi A. Lyapunov exponents: a tool to explore complex dynamics. Cambridge University Press, 2016. P. 295
  15. Kuznetsov N. V., Leonov G. A., Mokaev T. N. et al. Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dynamics. 2018. Vol. 92. No 2. P. 267-285
  16. Golub G. H., van Loan C. F. Matrix computations. The Johns Hopkins University Press, Baltimore, MD, 1996. P. 694

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