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Journal of Applied Nonlinear Dynamics 13(2) (2024) 247--267 | DOI:10.5890/JAND.2024.06.006

Satadal Datta$^{1,2}$, Jayanta Kumar Bhattacharjee$^{3}$, Dibya Kanti Mukherjee$^{4,5,6}$

$^1$ Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea

$^2$ Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

$^3$ Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India

$^4$ Department of Physics, Indiana University, Bloomington, IN 47405

$^5$ Quantum Science and Engineering Center, Indiana University, Bloomington, IN 47408

$^6$ Laboratoire de Physique des Solides, CNRS UMR 8502, Universit'e Paris-Saclay, 91405 Orsay Cedex, France

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Abstract

Invariant tori (or ``fixed surfaces": attractors for neighboring periodic orbits in a dissipative dynamical system) have been widely studied but the corresponding generalization for quasi periodic orbits have rarely been discussed. Here we investigate ``higher dimensional fixed surfaces" by analyzing a pair of coupled non identical Van der Pol oscillators and also the Kuznetsov oscillator. We find that the renormalization group based approach introduced by Chen, Goldenfeld and Oono is ideally suited for analyzing the quasi periodic analogues of limit cycles. We also address entrainment issues in a pair of forced and coupled Van der Pol oscillators. Our principle finding there is that if two such independent oscillators one with frequency entrainment and the other without are coupled linearly, then it is possible to produce an entrained state via the coupling.

References

1. [1]	Jordan, D. and Smith, P. (2014), Nonlinear Ordinary Differential Equations An Introduction for Scientists and Engineers, Oxford University Press, 4th ed.

2. [2]	Strogatz, S.H. (2018), Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second Edition, CRC Press.

3. [3]	Drazin, P.G. (1992), Nonlinear Systems, Cambridge Texts in Applied Mathematics, Cambridge University Press.

4. [4]	Lorenz, E.N. (1963) Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130-141.

5. [5]	Rossler, O. (1976), An equation for continuous chaos, Physics Letters A, 57, 397-398.

6. [6]	Shahruz, S.M. (2001), Bounded-input bounded-output stability of a class of lasers, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228),1, 901-902.

7. [7]	Fortin, J.-F., Grinstein, B., and Stergiou, A. (2012), Limit cycles in four dimensions, Journal of High Energy Physics, 2012, 112.

8. [8]	Das, A., Bhattacharjee, J.K., and Kirkpatrick, T.R. (2020), Transition to turbulence in driven active matter, Physical Review E, 101, 023103.

9. [9]	Sprott, J.C. (2014), A dynamical system with a strange attractor and invariant tori, Physics Letters A, 378, 1361-1363.

10. [10]	Chen, L.-Y., Goldenfeld, N., and Oono, Y. (1996), Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Physical Review E, 54, 376-394.

11. [11]	Paquette, G.C. (2000), Renormalization group analysis of differential equations subject to slowly modulated perturbations, Physica A: Statistical Mechanics and its Applications, 276, 122-163.

12. [12]	Kovalev, V.F. and Shirkov, D.V. (1999), Functional self-similarity and renormalization group symmetry in mathematical physics, Theoretical and Mathematical Physics, 121, 1315-1332.

13. [13]	Landau, L. and Lifshitz, E. (1976), Mechanics, Butterworth-Heinemann, third edition edn.

14. [14]	Nayfeh, A. and Mook, D. (1995), Nonlinear Oscillations, John Wiley \& Sons, Ltd.

15. [15]	Krylov, N. and Bogoliubov, N. (1950), Introduction to Non-Linear Mechanics. (AM-II), vol.~II. Princeton University Press.

16. [16]	Chiba, H. (2009), Extension and unification of singular perturbation methods for odes based on the renormalization group method, SIAM Journal on Applied Dynamical Systems, 8, 1066-1115.

17. [17]	Kuznetsov, A., Kuznetsov, S., and Stankevich, N. (2010), A simple autonomous quasiperiodic self-oscillator, Communications in Nonlinear Science and Numerical Simulation, 15, 1676-1681.

18. [18]	Pastor, I., Perez-Garc{\i}a, V.M., Encinas, F., and Guerra, J.M. (1993), Ordered and chaotic behavior of two coupled van der pol oscillators, Physical Review E, 48, 171-182.

19. [19]	Ohsuga, M., Yamaguchi, Y., and Shimizu, H. (1985) Entrainment of two coupled van der pol oscillators by an external oscillation, Biological Cybernetics, 51, 325-333.

20. [20]	Choubey, B. (2010), An experimental investigation of coupled van der Pol oscillators, Journal of Vibration and Acoustics, 132, 031013.

21. [21]	Chiba, H. (2009), Simplified renormalization group method for ordinary differential equations, Journal of Differential Equations, 246, 1991--2019.

22. [22]	Luo, A.C.J. (2014), Toward Analytical Chaos in Nonlinear Systems, John Wiley \& Sons.