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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.

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