Polynomial normal forms of constrained differential equations with three parameters

Polynomial Chaos for Linear Differential Algebraic Equations with Random Parameters

Technical applications are often modeled by systems of differential algebraic equations. The systems may include parameters that involve some uncertainties. We arrange a stochastic model for uncertainty quantification in the case of linear systems of differential algebraic equations. The generalized polynomial chaos yields a larger linear system of differential algebraic equations, whose soluti...

Computability with Polynomial Differential Equations

In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

Polynomial and Linearized Normal Forms for Almost Periodic Differential Systems

For almost periodic differential systems ẋ = εf(x, t, ε) with x ∈ Cn, t ∈ R and ε > 0 small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system ẋ = ε limT→∞ 1 T ∫ T 0 f(x, t, 0) dt, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non–resonant.

Computer algebra derives normal forms of stochastic differential equations

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term dynamics from undesirably detailed microscale dynamics. I aim to explore normal forms of stochastic differential equations when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic...

Tau Numerical Solution of Volterra Integro-Differential Equations With Arbitrary Polynomial Bases