On the fundamental solution of linear delay differential equations with multiple delays

For a class of linear autonomous delay differential equations with parameter $\alpha$ we give upper bounds for the integral \int_{0}^{\infty}\left|X\left(t,\alpha\right)\right|\mbox{d}t of the fundamental solution $X\left(\cdot,\alpha\right)$. The asymptotic estimations are sharp at a critical value $\alpha_{0}$ where $x=0$ loses stability. We use these results to study the stability properties of perturbed equations

A Harrod modell strukturális stabilitása (Structural stability of the Harrod model)

In this study it is shown that the nontrivial hyperbolic fixed point of a nonlinear dynamical system, which is formulated by means of the adaptive expectations, corresponds to the unstable equilibrium of Harrod. We prove that this nonlinear dynamical (in the sense of Harrod) model is structurally stable under suitable economic conditions. In the case of structural stability, small changes of the functions (C1-perturbations of the vector field) describing the expected and the true time variation of the capital coefficients do not influence the qualitative properties of the endogenous variables, that is, although the trajectories may slightly change, their structure is the same as that of the unperturbed one, and therefore these models are suitable for long-time predictions. In this situation the critique of Lucas or Engel is not valid. There is no topological conjugacy between the perturbed and unperturbed models; the change of the growth rate between two levels may require different times for the perturbed and unperturbed models

Periodic orbits and the global attractor for delayed monotone negative feedback

We study the delay differential equation $\dot x(t)=-\mu x(t)+f(x(t-1))$ with $\mu\ge 0$ and $C^1$-smooth real functions $f$ satisfying $f(0)=0$ and $f'<0$. For a set of $\mu$ and $f$, we determine the number of periodic orbits, and describe the structure of the global attractor as the union of the strong unstable sets of the periodic orbits and of the stationary point $0$

Local stability implies global stability for the 2-dimensional Ricker map

Consider the difference equation $x_{k+1}=x_k e^{\alpha-x_{n-d}}$ where $\alpha$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.Comment: for associated C++ program, mathematica worksheet and output, see http://www.math.u-szeged.hu/~krisztin/ricke