István Balázs

University of Szeged · Department of Applied and Numerical Mathematics

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lips...

We prove that all Hopf bifurcations in the Nicholson’s blowfly equation are supercritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is...

First we present the simplest criterion to decide that the Hopf bifurcations of the delay differential equation x′(t)=−μf(x(t−1)) are subcritical or supercritical, as the parameter μ passes through the critical values μk. Generally, the first Lyapunov coefficient, that determines the direction of the Hopf bifurcation, is given by a complicated form...

Consider the delay differential equation \(\dot{x}(t)=a\int _0^r x(t-s)\,d\eta (s)-g(x(t))\) and the neutral type differential equation \({\dot{y}}(t)=a\int _0^r \dot{y}(t-s)\,d\mu (s)-g(y(t))\) where \(a>0, g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is smooth, \(ug(u)>0\) for \(u\ne 0, \int _0^s g(u)\,du\rightarrow \infty \) as \(|s|\rightarrow \i...

We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with state-dependent delay, implicitly defined as the time when the solution of a nonlinear ODE, that depends on the stat...

We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-\mu f(x(t-1))$, as the parameter $\mu$ passes through the critical values $\mu_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain...

We introduce a rigorous computer-assisted method to obtain constructive proofs of existence of solutions to nonlinear differential equations. We introduce all main ideas through examples, accessible to undergraduate students, where we consider radially symmetric solutions of partial differential equations. The proofs are obtained by solving for the...