Finding safety in partially controllable chaotic systems

Juan Sabuco, Samuel Zambrano, Miguel A F Sanjuán, James A. Yorke

Research output: Contribution to journalArticlepeer-review


Many discrete-time dynamical systems have a region Q from which all or almost all trajectories leave, or at least they leave in the presence of perturbations that we call disturbances. We partially control systems so that despite disturbances the trajectories of a dynamical system stay in the region Q at least for some initial points in Q. The disturbances can be thought of as either noise or as purposeful, hostile efforts of an enemy to drive the trajectory out of the region. Our goal is to keep trajectories inside Q despite the disturbances and our partial control of chaos method succeeds.Surprisingly this goal can be achieved with a control whose maximum allowable size is smaller than the maximum allowed disturbance. A fundamental step towards this goal is to compute a set called the safe set that had, until now, been found only in certain very special situations.This paper provides a general algorithm for computing safe sets. The algorithm is able to compute the safe sets for a specified region in phase space, the maximum disturbance value, and the maximum allowed control. We call it the Sculpting Algorithm. Its operation is analogous to removing material while sculpting a statue. The algorithm sculpts the safe sets. Our Sculpting Algorithm is independent of the dimension and is fast for one- and two-dimensional dynamical systems. As examples, we apply the algorithm to two paradigmatic nonlinear dynamical systems, namely, the Hénon map and the Duffing oscillator.

Original languageEnglish
Pages (from-to)4274-4280
Number of pages7
JournalCommunications in Nonlinear Science and Numerical Simulation
Issue number11
Publication statusPublished - Nov 2012


  • Escaping dynamics
  • Partial control of chaos
  • Transient chaos

ASJC Scopus subject areas

  • Modelling and Simulation
  • Numerical Analysis
  • Applied Mathematics


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