Henrik Farkas

Explodator: A new skeleton mechanism for the halate driven chemical oscillators

In the first part of this work, some shortcomings in the present theories of the Belousov–Zhabotinskii oscillating reaction are discussed. In the second part, a new oscillatory scheme, the limited Explodator, is proposed as an alternative skeleton mechanism. This model contains an always unstable three‐variable Lotka–Volterra core (the ‘‘Explodator’’) and a stabilizing limiting reaction. The new scheme exhibits Hopf bifurcation and limit cycle oscillations. Finally, some possibilities and problems of a generalization are mentioned.

Involutes: the geometry of chemical waves rotating in annular membranes

According to earlier theories certain parts of a chemical wave front propagating in a 2-D excitable medium with a convex obstacle should be involutes of that obstacle. The present paper discusses a special case where self-sustained chemical waves are rotating around a central obstacle in an annular 2-D excitable region. A simple geometrical model of wave propagation based on the Fermat principle (minimum propagation time) is suggested. Applying this model it is shown that the wave fronts in the case of an annular excitable region should be purely involutes of the central obstacle in the asymptotic state. This theory is supported by experiments in a novel membrane reactor where a catalyst of the Belousov-Zhabotinsky reaction is fixed on a porous membrane combined with a gel medium. Involutes of circular and triangular obstacles are observed experimentally. Deviations from the ideal involute geometry are explained by inhomogeneities in the membrane. (c) 1995 American Institute of Physics.

On the phenomenological theory of heat conduction

Abstract In this paper the foundations of the phenomenological theory of heat conduction is generalized, then the principle of least dissipation of energy and the Gyarmati variational principle are treated. The second-order differential of Lagrangian of the Gyarmati principle, the general Γ picture and nonlinear cases are investigated. In the last part nonlinear constitutive theories are involved: the method of power series and the method of mollification. Generalizing the mollification method we demonstrate that results obtained by mollifications of different kinds are not equivalent, that is, this method is not unique.

Geometric theory of trigger waves — A dynamical system approach

We propose a geometric model for wave propagation in excitable media. Our model is based on the Fermat principle and it resembles that of Wiener and Rosenblueth. The model applies to the propagation of excitations, such as chemical and biological wave fronts, grass fire, etc. Starting from the Fermat principle, some consequences of the assumptions are derived analytically. It is proved that the model describes a dynamical system, and that the wave propagates along “ignition lines” (extremals). The theory is applied to the special cases of tube reactor and annular reactor. The asymptotic shape of the wave fronts is derived for these cases: they are straight lines perpendicular to the tube, and involutes of the central obstacle, respectively.

Generalized Lotka–Volterra schemes and the construction of two-dimensional explodator cores and their liapunov functions via‘critical’ Hopf bifurcations

A new class of generalized Lotka–Volterra schemes has been investigated. It is shown that such systems are conservative. Their first integrals can be used as Liapunov functions to prove the globally stable or explosive behaviour of a wide class of modifications. On modifying the value of an exponent in one rate law, a ‘critical’ Hopf bifurcation occurs and the stable and the explosive regions are separated by a critical value at which conservative oscillations take place. The explosive schemes can be regarded as two-dimensional explodator cores. Limit-cycle oscillators can be constructed using these cores and one or more limiting reactions.

Rotating chemical waves: theory and experiments

After a brief introduction in the first theoretic part of this work the geometrical wave theory and its application for rotating waves are discussed. Here the waves are rotating around a circular obstacle which is surrounded by two homogeneous wave conducting regions with different wave velocities. The interface of the inner slow and the outer fast region is also a circle but the two circles (the obstacle and interface) are not concentric. The various asymmetric cases are classified and described theoretically. In the second experimental part chemical waves rotating in a so-called moderately asymmetric reactor are studied. A piecewise homogeneous wave conducting medium is created applying a novel reactor design. All the three theoretical cases of the moderately asymmetric arrangement are realized experimentally and qualitative and quantitative comparison of these results with the theoretical predictions show a good agreement.

Constructing global bifurcation diagrams by the parametric representation method

The parameter dependence of the solution x of equation f0(x)+u1f1(x)+u2f2(x) = 0 is considered. Our aim is to divide the parameter plane (u1;u2) according to the number of the solutions, that is to construct a bifurcation curve. This curve is given by the singularity set, but in practice it is dicult to depict it, because it is often derived in implicit form. Here we apply the parametric representation method which has the following advantages: (1) the singularity set can be easily constructed as a curve parametrized by x, called D-curve; (2) the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the tangential property; (3) the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve. c 1999 Elsevier Science B.V. All rights reserved.

Chemical waves in confined regions by Hamilton-Jacobi-Bellman theory

Abstract Propagation of concentration disturbances in the form of (bio)chemical waves satisfying Fermats principle of minimum time is analysed by methods of the classical variational calculus, Finsler geometry and optimal control theory. The optimal control methods are most general and effective as they can assure the minimum time and can be applied to constrained states. In particular, the dynamic programming approach leads to the Hamilton-Jacobi-Bellman equation for chemical waves, which describes (along with its characteristic set) the link between the constrained motions of wave fronts and associated ‘rays’ or extremal trajectories. The geodesic constraints due to an obstacle influence the state changes and especially the entering (leaving) conditions of a ray as the tangentiality condition for rays that begin to slide over the boundary of an obstacle. The analysis determines also the deviations of rays from straightlinearity in inhomogeneous media. It may handle complex transversality conditions. In a complex case an optimal arc of a chemical wave is composed of an internal and boundary parts.

Waves of excitation on nonuniform membrane rings, caustics, and reverse involutes

Chemical wave experiments on concentric nonuniform membrane rings are presented together with their theoretical description. A new technique is applied to create a slow inner and a fast outer zone in an annular membrane. An abrupt qualitative change of the wave profile was observed while decreasing the wave velocity in the inner zone. This phenomenon and all the experimental wave profiles can be adequately described by assuming that waves are involutes of a relevant caustic. A possible connection with recent models of atrial flutter is also set forth. (c) 1997 American Institute of Physics.

Use of the parametric representation method in revealing the root structure and hopf bifurcation

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =uo +u1x +g(x), whereuo andu1 are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the “stable region” on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.