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Relaxation Oscillations in the Logistic Equation with Delay and Modified Nonlinearity

Author

Listed:
  • Alexandra Kashchenko

    (Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia
    These authors contributed equally to this work.)

  • Sergey Kashchenko

    (Regional Scientific and Educational Mathematical Center «Centre of Integrable Systems», P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia
    These authors contributed equally to this work.)

Abstract

We consider the dynamics of a logistic equation with delays and modified nonlinearity, the role of which is to bound the values of solutions from above. First, the local dynamics in the neighborhood of the equilibrium state are studied using standard bifurcation methods. Most of the paper is devoted to the study of nonlocal dynamics for sufficiently large values of the ‘Malthusian’ coefficient. In this case, the initial equation is singularly perturbed. The research technique is based on the selection of special sets in the phase space and further study of the asymptotics of all solutions from these sets. We demonstrate that, for sufficiently large values of the Malthusian coefficient, a ‘stepping’ of periodic solutions is observed, and their asymptotics are constructed. In the case of two delays, it is established that there is attractor in the phase space of the initial equation, whose dynamics are described by special nonlinear finite-dimensional mapping.

Suggested Citation

  • Alexandra Kashchenko & Sergey Kashchenko, 2023. "Relaxation Oscillations in the Logistic Equation with Delay and Modified Nonlinearity," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1699-:d:1114096
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    References listed on IDEAS

    as
    1. Alexandra Kashchenko, 2022. "Asymptotics of Solutions to a Differential Equation with Delay and Nonlinearity Having Simple Behaviour at Infinity," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    2. Sergey Kashchenko, 2022. "Infinite–Dimensional Bifurcations in Spatially Distributed Delay Logistic Equation," Mathematics, MDPI, vol. 10(5), pages 1-32, February.
    3. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    Full references (including those not matched with items on IDEAS)

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