Global dynamics of a differential equation with piecewise constant argument

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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 languageEnglish (US)
JournalNonlinear Analysis, Theory, Methods and Applications
Volume71
Issue number12
DOIs
StatePublished - Dec 15 2009

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Piecewise Constant Argument
Differential Delay Equations
Interval Maps
One-dimensional Maps
Differential-difference Equations
Stability of Equilibria
Global Dynamics
Chaotic Behavior
Singularly Perturbed
Global Stability
Invariance
Periodic Solution
Differential equations
Discretization
Scalar
Differential equation
Integer
Sufficient Conditions
Operator
Difference equations

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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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.",
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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.

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