Session S12 - Delay and functional differential equations and applications

Monday, July 12, 12:50 ~ 13:25 UTC-3

On stability and asymptotics of Mackey-Glass equations with two delays and neutral systems

Elena Braverman

University of Calgary, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In the first part of our talk, we observe that introduction of two variable delays can change properties of the Mackey-Glass equation $$\dot{x}(t) = r(t) \left[ \frac{a x(h(t))}{1+x^{\nu}(g(t))} - x(t) \right], \quad a>1, ~\nu >0.$$ There may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain $a$ and $\nu$ disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.

In the second part of the talk, we analyze exponential stability and solution estimates for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$ are delayed arguments. Stability tests are applicable to a wide class of linear neutral systems with time-varying coefficients and delays. In addition, explicit exponential estimates for solutions of both homogeneous and non-homogeneous neutral systems are obtained for the first time. These inequalities are not just asymptotic estimates, they are valid on every finite segment and evaluate both short- and long-term behaviour of solutions.

Joint work with Leonid Berezansky (Ben Gurion University of the Negev, Israel).

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