Oscillator representations and systems of ordinary differential equations

Oscillator representations and systems of ordinary differential equations.

Using representation-theoretic methods, we determine the spectrum of the 2 x 2 system. Q(x, D(x)) = A- partial differential(2)(x)2 + x(2)2 + Bx partial differential(x) + 1/2, x in, with A, B in Mat(2)(R) constant matrices such that A = (t)A > 0 (or <0), B = -(t)B not equal 0, and the Hermitian matrix A + iB positive (or negative) definite. We also give results that generalize (in a possible dir...

Ordinary Differential Equations and Dynamical Systems

Ordinary Differential Equations and Dynamical Systems

This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous...

Inferring reaction systems from ordinary differential equations

In Mathematical Biology, many dynamical models of biochemical reaction systems are presented with Ordinary Differential Equations (ODE). Once kinetic parameter values are fixed, this simple mathematical formalism completely defines the dynamical behavior of a system of biochemical reactions and provides powerful tools for deterministic simulations, parameter sensitivity analysis, bifurcation an...

Summary: Ordinary Differential Equations

1 Initial Value Problem We are given a right hand side function f(t, y) with f : [t0, T ]×Rn → Rn and an initial value y0 ∈ Rn. We want to find a function y(t) with y : [t0, T ] → Rn such that y′(t) exists, is continuous and satisfies the initial value problem y′(t) = f (t, y(t)) , y(t0) = y0. (1) We assume that f(t, y) satisfies a Lipschitz condition with respect to y (at least for y with |y −...