### Abstract

Several aspects of global dynamics are studied for the scalar differential-difference equation ε over(x, ̇) (t) + x (t) = f (x ([t])), 0 < ε ≪ 1, where [{dot operator}] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation ε over(x, ̇) (t) + x (t) = f (x (t - 1)). Sufficient conditions for the invariance, global stability of equilibria, existence, stability/instability, and shape of periodic solutions, and the chaotic behavior are derived. The principal analysis is based on the reduction of its dynamics to the one-dimensional map F : x → f (x) + [x - f (x)] e^{- 1 / ε}, many relevant properties of which follow from those of the interval map f.

Original language | English (US) |
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Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 71 |

Issue number | 12 |

DOIs | |

State | Published - Dec 15 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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**Global dynamics of a differential equation with piecewise constant argument.** / Ivanov, Anatoli.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Global dynamics of a differential equation with piecewise constant argument

AU - Ivanov, Anatoli

PY - 2009/12/15

Y1 - 2009/12/15

N2 - Several aspects of global dynamics are studied for the scalar differential-difference equation ε over(x, ̇) (t) + x (t) = f (x ([t])), 0 < ε ≪ 1, where [{dot operator}] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation ε over(x, ̇) (t) + x (t) = f (x (t - 1)). Sufficient conditions for the invariance, global stability of equilibria, existence, stability/instability, and shape of periodic solutions, and the chaotic behavior are derived. The principal analysis is based on the reduction of its dynamics to the one-dimensional map F : x → f (x) + [x - f (x)] e- 1 / ε, many relevant properties of which follow from those of the interval map f.

AB - Several aspects of global dynamics are studied for the scalar differential-difference equation ε over(x, ̇) (t) + x (t) = f (x ([t])), 0 < ε ≪ 1, where [{dot operator}] is the integer part function. The equation is a particular case of the special discretization (discrete version) of the singularly perturbed differential delay equation ε over(x, ̇) (t) + x (t) = f (x (t - 1)). Sufficient conditions for the invariance, global stability of equilibria, existence, stability/instability, and shape of periodic solutions, and the chaotic behavior are derived. The principal analysis is based on the reduction of its dynamics to the one-dimensional map F : x → f (x) + [x - f (x)] e- 1 / ε, many relevant properties of which follow from those of the interval map f.

UR - http://www.scopus.com/inward/record.url?scp=72449211707&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72449211707&partnerID=8YFLogxK

U2 - 10.1016/j.na.2009.05.030

DO - 10.1016/j.na.2009.05.030

M3 - Article

VL - 71

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 12

ER -