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Limiting stochastic processes of shift-periodic dynamical systems.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Stochastic Processes and Probability Theory

Background:

  • Shift-periodic maps are one-dimensional maps with periodic-like behavior and potential singularities.
  • Iterative sequences generated by these maps exhibit complex dynamical properties.

Purpose of the Study:

  • To investigate the dynamical behavior of iterative sequences generated by shift-periodic maps.
  • To analyze the convergence properties of these sequences to known stochastic processes.

Main Methods:

  • Analysis of iterative sequences x_{n+1} = F(x_n) generated by shift-periodic maps.
  • Examination of the integer parts of the sequences as discrete-time random walks.
  • Asymptotic analysis to determine convergence to Brownian motion and Lévy processes.

Main Results:

  • Iterative sequences display rich and complex dynamical behavior.
  • The integer parts of the sequences form discrete-time random walks for suitable initial distributions.
  • Convergence to Brownian motion and more general Lévy processes is demonstrated in certain limits.

Conclusions:

  • Shift-periodic maps provide a framework for generating complex stochastic processes.
  • The study establishes connections between deterministic map dynamics and stochastic processes like Brownian motion and Lévy processes.
  • Further analysis shows convergence to continuous-time random walks for maps with specific properties.