### Abstract

We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system,when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.

Original language | English (US) |
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Title of host publication | Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012 |

Publisher | Springer New York LLC |

Pages | 267-296 |

Number of pages | 30 |

Volume | 180 |

ISBN (Print) | 9783662529263 |

DOIs | |

State | Published - 2016 |

Event | 18th International Conference on Difference Equations and Applications, ICDEA 2012 - Barcelona, Spain Duration: Jul 23 2012 → Jul 27 2012 |

### Other

Other | 18th International Conference on Difference Equations and Applications, ICDEA 2012 |
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Country | Spain |

City | Barcelona |

Period | 7/23/12 → 7/27/12 |

### Fingerprint

### Keywords

- Differential inclusions
- Switched systems
- Symbolic dynamics

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012*(Vol. 180, pp. 267-296). Springer New York LLC. https://doi.org/10.1007/978-3-662-52927-0_19

**Continuous limit in dynamics with choice.** / Kapitanski, Lev; Živanović Gonzalez, Sanja.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012.*vol. 180, Springer New York LLC, pp. 267-296, 18th International Conference on Difference Equations and Applications, ICDEA 2012, Barcelona, Spain, 7/23/12. https://doi.org/10.1007/978-3-662-52927-0_19

}

TY - GEN

T1 - Continuous limit in dynamics with choice

AU - Kapitanski, Lev

AU - Živanović Gonzalez, Sanja

PY - 2016

Y1 - 2016

N2 - We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system,when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.

AB - We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system,when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.

KW - Differential inclusions

KW - Switched systems

KW - Symbolic dynamics

UR - http://www.scopus.com/inward/record.url?scp=84994841002&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994841002&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-52927-0_19

DO - 10.1007/978-3-662-52927-0_19

M3 - Conference contribution

AN - SCOPUS:84994841002

SN - 9783662529263

VL - 180

SP - 267

EP - 296

BT - Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012

PB - Springer New York LLC

ER -