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Oscillatory modes in a nonlinear second-order differential equation with delay

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Solution properties of the nonlinear second-order delay-differential equation x(t)=−ax(t)+f[x(t−Τ)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.

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an der Heiden, U., Longtin, A., Mackey, M.C. et al. Oscillatory modes in a nonlinear second-order differential equation with delay. J Dyn Diff Equat 2, 423–449 (1990). https://doi.org/10.1007/BF01054042

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Key words

  • Nonlinear differential equations of second order with deviating argument
  • oscillations
  • periodic solutions