Nikola Buric

Time delay in a basic model of the immune response

The eAects of time delay on the two-dimensional system of Mayer et al., which represents the basic model of the immune response, are analysed (cf. Mayer H, Zaenker KS, an der Heiden U. A basic mathematical model of the immune response. Chaos, Solitons and Fractals 1995;5:155‐61). We studied variations of the stability of the fixed points due to the time delay and the possibility for the occurrence of the chaotic solutions. ” 2000 Elsevier Science Ltd. All rights reserved.

Dynamics of delay-differential equations modelling immunology of tumor growth

Abstract A phenomenological model of a tumor interacting with the relevant cells of the immune system is proposed and analysed. The model has a simple formulation in terms of delay-differential equations (DDEs). The critical time-delay, for which a destabilising Hopf bifurcation of the relevant fixed point occurs, and the conditions on the parameters for such bifurcation are found. The bifurcation occurs for the values of the parameters estimated from real data. Local linear analyses of the stability is sufficient to qualitatively analyse the dynamics for small time-delays. Qualitative analyses justify the assumptions of the model. Typical dynamics for larger time-delay is studied numerically.

STABILITY OF A GENERAL DELAY DIFFERENTIAL MODEL OF THE HYPOTHALAMO-PITUITARY-ADRENOCORTICAL SYSTEM

Most of the systems in an organism (human included) function in a regular daily rhythm. Hypothalamo-pituitary-adrenocortical (HPA) axis, although mostly known for its role in stress response, probably has a role in conveying rhythmic signals from the major pacemaker, suprachiazmatic nucleus (SCN), to the periphery. A general qualitative nonphenomenological mathematical model of the HPA axis is constructed and its dynamics is examined using linear stability analysis and Roushes theorem. The results show that this system is asymptotically stable, i.e. it does not generate circadian oscillations, but only responds to the external pacemaker.

BIFURCATIONS DUE TO SMALL TIME-LAG IN COUPLED EXCITABLE SYSTEMS

A system of ODEs is used to attempt an approximation of the dynamics of two delayed coupled FitzHugh–Nagumo excitable units, described by delay-differential equations. It is shown that the codimension 2 generalized Hopf bifurcation acts as the organizing center for the dynamics of ODEs for small time-lags. Furthermore, this is used to explain important qualitative properties, like the phenomenon of oscillator death, of the exact dynamics for small time-delays.

Weak chaos and Poincaré recurrences for area preserving maps

The spectrum F(t) of Poincare recurrence times for the standard map exhibits two distinct limits: an integrable weak-coupling limit with an inverse power law and a chaotic strong-coupling limit with exponential decay. In the domain where chaotic regions coexist with integrable structures, the spectrum F(t) exhibits a superposition of exponential and power law decay. Such a law can be proved to occur in a model of area-preserving map at the boundary of the mixing and integrable components.

Universal scaling of critical quasiperiodic orbits in a class of twist maps

Recently we have shown that the fractal properties of the critical invariant circles of the standard map, as summarized by the spectrum and the generalized dimensions D(q), depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in (Buric N, Mudrinic M and Todorovic K 1997 30 L161). In the present paper this result is extended on the whole class of sufficiently smooth area-preserving twist maps of cylinders. We present numerical evidence that the and D(q) are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail.

Chaos in a Hamiltonian description of interacting quantum spins

Abstract Regular and irregular dynamics of interacting spins are investigated using the Hamiltonian dynamical system, which completely captures the quantum dynamics. The phase-space and the Hamiltonian dynamics of the quantum system are obtained using the continuous representations and the generalised coherent states of the relevant dynamical group SU(2)×SU(2). The considered model is linear in the generators of the dynamical group, so that the Hamiltonian description is exact. The relation between symmetry of the quantum model and stability of the classical counterpart is studied.

Equivalent classes of critical circles

We present numerical evidence that the fractal properties of the critical invariant circles of a typical area-preserving twist map, as summarized by the spectrum and the generalized dimensions D(q), depend only on the tails in the continued fraction expansion of the corresponding rotation numbers. and D(q) are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail.

Singularity spectrum of a fractal diagram for a Hamiltonian system

We determine numerically the relevant spectrum of scaling indices for the fractal diagram of the standard map. Infinite partitions, related by the Gauss transformation and an appropriate measure had to be used in order to obtain convergent results. This choice of the measure and the partitions is motivated by the method of modular smoothing.

Synchronization patterns in neural chains based on hyper-chaotic cells

Synchronization patterns in chains of N bi-directionally delayed coupled systems with delayed feedback are studied in this paper. Each system is hyper-chaotic when decoupled from the chain. It is shown that chains with odd or even number of cites N display different spatial patterns of stable exact synchronization. When N is odd the only stable pattern of exact synchronization is among all of the units. When N is even, next to the nearest neighbors could become exactly synchronized, with the dynamics of the nearest neighbors related in a more complicated way. Sufficiently strong coupling leads to the nearest neighbor synchronization also for even N. No other patterns have been observed.