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Dynamical systems can synchronize to periodic inputs. This study finds that constant inputs are often optimal for maximizing system output, suggesting periodic signals may indicate time-varying objectives in biological systems.

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Area of Science:

  • Dynamical systems theory
  • Optimal control
  • Systems biology

Background:

  • Dynamical systems entrain to periodic inputs, converging to attractors with the same period.
  • Entrainment implies that for constant inputs, systems reach a unique equilibrium.
  • The gain of entrainment quantifies benefits of periodic over constant inputs.

Purpose of the Study:

  • To maximize the weighted average output of a dynamical system along its periodic attractor.
  • To analyze resource allocation problems using periodic optimal control.
  • To investigate optimality of constant versus periodic inputs in entraining systems.

Main Methods:

  • Formulation as a periodic optimal control problem.
  • Analysis using the Pontryagin maximum principle.
  • Numerical solutions via specialized software packages.
  • Application to nonlinear occupancy models in biological synthesis.

Main Results:

  • Constant inputs were found to be optimal for various nonlinear occupancy model architectures.
  • The gain of entrainment was not always positive, challenging intuitive expectations.
  • Optimality of constant inputs suggests underlying time-varying objectives when periodic signals are observed.

Conclusions:

  • Constant inputs can be optimal for maximizing system output in entraining dynamical systems.
  • The prevalence of non-constant periodic signals in biological systems may signal optimization of time-varying objectives.
  • This framework provides insights into resource allocation and signal processing in biological and engineered systems.