#### For citation:

Kuznetsov S. P., Sedova Y. V. Hyperbolic chaos in the Bonhoeffer–van der Pol oscillator with additional delayed feedback and periodically modulated excitation parameter. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 1, pp. 77-95. DOI: 10.18500/0869-6632-2019-27-1-77-95

# Hyperbolic chaos in the Bonhoeffer–van der Pol oscillator with additional delayed feedback and periodically modulated excitation parameter

Topic and aim. The aim of the work is to consider an easy-to-implement system demonstrating the Smale–Williams hyperbolic attractor based on the Bonhoeffer–van der Pol oscillator, alternately manifesting a state of activity or suppression due to periodic modulation of the parameter by an external control signal, and supplemented with a delayed feedback circuit. Investigated models. A mathematical model is formulated as a non-autonomous second-order equation with delay. The scheme of the electronic device that implements this type of chaotic behavior is proposed. Results. The results of numerical simulating of the system dynamics, including waveforms, oscillation spectra, plots of Lyapunov exponents, a chart of regimes on the parameters plane are presented. The circuit simulation of the electronic device using the software Multisim is carried out. Discussion. The Smale–Williams attractor in the system appears due to the fact that the transformation of the phases of the carrier for the sequence of radio-pulses generated by the system corresponds to a circle map expanding by an integer factor. The important feature of the system is that the transfer of excitation from one to the next stage of activity with doubling (or tripling) of the phase occurs due to the resonance mechanism involving a harmonic of the developed oscillations that have twice (or triple) longer period than that of small oscillations. Due to the hyperbolic nature of the attractor, the generated chaos is rough, that is, it is characterized by low sensitivity to variations in the parameters of the device and its components. Our scheme corresponds to a low-frequency device, but it can be adapted for chaos generators also at high and ultrahigh frequencies.

- FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1961, vol. 1, no. 6, pp. 445–466.
- Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 1962, vol. 50, no. 10, pp. 2061–2070.
- Izhikevich E.M., FitzHugh R. FitzHugh–Nagumo model. Scholarpedia, 2006, vol. 1, no. 9, p. 1349.
- Izhikevich E.M. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, Cambridge, MA. 2010.
- Dmitrichev А.S., Klinshov V.V., Kirillov S.Y., Maslennikov O.V., Shapin D.S., Nekorkin V.I. Nonlinear dynamical models of neurons: Review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 4, pp. 5–58 (in Russian).
- Smale S. Differentiable dynamical systems. Bulletin of the American mathematical Society, 1967, vol. 73, no. 6, pp. 747–817.
- Dynamical Systems with Hyperbolic Behaviour, D.V.Anosov (Ed.). Encyclopaedia Math. Sci., Dynamical Systems, vol. 9, Berlin, Springer, 1995.
- Shilnikov L. Mathematical problems of nonlinear dynamics: A tutorial. International Journal of Bifurcation and Chaos, 1997, vol. 7, no. 09, pp. 1953–2001.
- Sinai Ya.G. The Stochasticity of Dynamical Systems. Selected Translations. Selecta Math. Soviet., 1981, vol. 1, no. 1, pp. 100–119.
- Katok A. and Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia Math. Appl., vol. 54, Cambridge, Cambridge Univ. Press, 1995.
- Afraimovich V. and Hsu S.-B. Lectures on chaotic dynamical systems. AMS/IP Stud. Adv. Math., 2003, vol. 28.
- Bonatti C., Daz L.J., Viana M. Dynamics beyond uniform hyperbolicity: A global geometric and probobalistic perspective. Encyclopaedia of Mathematical Sciences, 2005, vol. 102, Mathematical Physics, III, Springer-Verlag, Berlin, p. 2.
- Ruelle D. Strange attractors. The Mathematical Intelligencer, 1980, vol. 2, no. 3, p. 126–137.
- Andronov A.A., Vitt A.A., Khaikin S.E. Theory of oscillators. Pergamon Press, Oxford, 1966.
- Pugh C., Peixoto M.M. Structural stability. Scholarpedia, 2008, vol. 3, no. 9. 4008.
- Rabinovich M.I., Trubetskov D.I. Oscillations and Waves in Linear and Nonlinear Systems. Kluwer Academic Publisher, 1989.
- Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: From mathematics to physics. Physics-Uspekhi, 2011, vol. 54, no. 2, pp. 119–144.
- Kuznetsov S.P., Ponomarenko V.I. Realization of a strange attractor of the Smale–Williams type in a radiotechnical delay-fedback oscillator. Technical Physics Letters, 2008, vol. 34, no. 9, pp. 771–773.
- Arzhanukhina D.S., Kuznetsov S.P. Robust chaos in autonomous time-delay system. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, no. 2, pp. 36–49 (in Russian).
- Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems. Phys. Rev. E, 2016, vol. 94, no. 1, 010201.
- Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, vol. 56, pp. 227–239.
- Kuznetsov S.P., Sedova Yu.V. Hyperbolic chaos in systems based on FitzHugh–Nagumo model neurons. Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 329–341.
- Doroshenko V.M., Kruglov V.P., Kuznetsov S.P. Smale–Williams solenoids in a system of coupled Bonhoeffer–van der Pol oscillators. Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, p. 435–451.
- Kruglov V.P., Kuznetsov S.P. Hyperbolic chaos in a system of two Froude pendulums with alternating periodic braking. Communications in Nonlinear Science and Numerical Simulation, 2019, vol. 67, pp. 152–161.
- Bellman R.E., Cooke K.L. Differential-difference equations. Academic Press, 2012.
- El’sgol’ts L.E., Norkin S.B. Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, 1973.
- Farmer J. D. Chaotic attractors of an infinite-dimensional dynamical system. Physica D: Nonlinear Phenomena, 1982, vol. 4, no. 3, pp. 366–393.
- Yanchuk S., Giacomelli G. Spatio-temporal phenomena in complex systems with time delays. Journal of Physics A: Mathematical and Theoretical, 2017, vol. 50, no. 10, p. 103001.
- Balyakin А.А., Ryskin N.M. Peculiarities of calculation of the Lyapunov exponents set in distributed self-oscillated systems with delayed feedback. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, no. 6, pp. 3–21 (in Russian).
- Koloskova A.D., Moskalenko O.I., Koronovskii A.A. A method for calculating the spectrum of Lyapunov exponents for delay systems. Technical Physics Letters, 2018, vol. 44, no. 5, pp. 374–377.
- Sveshnikov A.A. Applied methods of the theory of random functions. Elsevier, 2014.
- Jenkins G.M., Watts D.G. Spectral analysis and its applications. Holden-Day, 1969.

- 2350 reads