Abstract and results
     
      Whenever the heat conductivity l is a given function of temperature (l(T)), we can rescale the temperature by the definition G(T)=l(T)dT (Kelvin's transformation). Gyarmati's variational principle is formulated in terms of this temperature scale for heat conduction. This transformation leads to linear heat equation in the more general case: l=f(r)g(T), too, if g(T) is used in the construction of G.

      3. FARKAS, H. & Z. NOSZTICZIUS 1971. On the non-linear generalization of the Gyarmati Principle and Theorem.  Annalen der Physik 27: 341-348.

      The variational potentials and the Onsagerian reciprocity relations are extended to nonlinear case, that is when the relation between thermodynamic forces and fluxes is not
      linear. An unconditional extremum problem is derived for the case of 1D stationary heat conduction when the conductivity is a function of space and temperature: l(x,T).

      4. FARKAS, H. 1975. Error estimation for approximations of the solution of  the one-dimensional stationary heat conduction equation on the basis of the Governing Principle of dissipative Processes. Int.J.Engng.Sci. 13:1029-1033.

      A formula is derived from an unconditional variational problem (see: 3.) which gives an upper bound for the deviation from the exact unknown solution for 1D stationary heat equation with conductivity l(x,T). Error estimation for a nonlinear second-order ordinary differential equation.

      5. FARKAS, H. 1975. On the phenomenological theory of heat conduction. Int.J.Engng.Sci. 13: 1034-1053.

      Onsager's and Gyarmati's variational principles are discussed for heat equation. Variational potentials for nonlinear cases and temperature dependence, using the G temperature (see: [1]). Coleman and Mizel developed a technique (mollification) aiming at getting a hierarchy of approximations of a general constitutive relation. Here it it is shown that the mollification technique leads to different results if the smoothing process is different. That means that the hierarchy of approximations is not unique: it essentially depends on the selected smoothing process.

      10. FARKAS, H. & Z. NOSZTICZIUS. 1977. A variational method for solving heat conductional problems. Periodica Polytechnica EE.21: 239-242.

      A variational technique is shown for two-point boundary value problem. The problem: to find the minimum of a given functional of a one-variable function provided that at the two
      endpoints the value of the function is prescribed. Starting from a trial function, a perturbation is taken at an inner point. Using piecewise linear functions, the computation is simple. Computing the value of the functional, we reach a better approximation, and so on. The advantages of this method: small memory required, easy to make programming, the errors do not accumulate. As an illustrative example, this technique is applied to the nontypical variational problem in [3].

      11. FARKAS, H. 1978. Thermodynamic concepts for a class of one- ports.  Acta Phys.Hung. 45: 317-326.

      The aim: to establish thermodynamic description to a general electric one-port; the basic quantities are charge and voltage. There is a 'system equation' giving a relation between these quantities and their time derivatives up to a given order. Work is defined for an admissible process. The dissipative character of this system is defined globally (for a finite process) by the requirement of the boundedness of work for all admissible processes. Equivalence of this global dissipativity and the local dissipativity concept used in network thermodynamics is shown. Energy and dissipation is defined as derived concepts. Some special cases are detailed. For a surprising example the minimal dissipation belongs to the infinitely quick process. This formulation does not use any thermodynamic concepts, such as temperature, entropy. Nevertheless, formulation of postulates resemble to the first and the second law of thermodynamics in the approach of Kelvin and Planck.

      12. FARKAS, H. 1980. Generalization of the Fourier law. Periodica Polytechnica ME    24: 291-308.

      A disadvantage of the 'hyperbolic' heat equation is illustrated by an example: it leads to absolute negative temperature even if at the initial time the temperature is positive. On the other hand, the infinite propagation velocity (for the parabolic equation) causes no trouble from a practical point of view. This example shows the importance of clear basic postulates. An "axiomatic treatment for the stationary 1D heat conduction is presented. The postulates means requirement for the temperature spatial dependencies alone. Heat current, conductivity, balance equation and differential equations are not required a priori, they comes out as consequences of the postulates. In essence: a two-point boundary value problem determines a second order differential equation. The postulates refer to monotonity, and an analogon of the maximum principle property /the temperature takes its maximum at the endpoints. /An experiment is proposed to distinguish between the quasilinear and nonlinear cases.

      14. FARKAS, H. 1982. A new proof and generalization of the Maximum Principle of heat conduction. J.Non-Equilibrium Thermodyn. 7: 355-362.

      According to the "maximum principle" the temperature takes its maximum value at the initial moment or at the boundaries. This statement is proved for a very general nonlinear class of nonlinear equations. The indirect proof uses Gyarmati's variational principle.

      15. FARKAS, H. & I. MUDRI 1984. Shape-preserving time- dependencies in heat  conduction. Acta Phys. Hung. 55: 267-273.

      The problem: if the boundary conditions follow a given time dependence, is there a solution of Fourier equation which follows the same time-dependence. In other words: do travelling wave solutions exist? (Change in amplitude and phase is allowed (spatial functions)). Any such solution should satisfy a second-order ordinary linear differential equation with constant coefficients, therefore, the time- functions which can propagate without distortion are:  at+b, exp(at), combination of two exponentials, texp(at),
      exp(at) sin(bt+c). Remark that the linear and sinusoidal waves has special importance, as it had already been known.

      16. NOSZTICZIUS, Z., H. FARKAS, & Z. A. SCHELLY 1984. Explodator: a new  skeleton mechanism for the halate driven chemical oscillators.  J.Chem.Phys.          80: 6062-6070.

      Critical discussion of the Oregonator model of the Belousov- Zhabotinskii oscillating reaction. Adding a third intermediate and a third equation to the Lotka-Volterra model, a new 3D quadratic ODE yields. This Explodator core has a unique positive stationary point, which is always unstable: one eigenvalue is always negative, but the real part of the two others (or the two other eigenvalues itself) is always positive . As Kertész proved, apart from two trajectories belonging to the negative eigenvalue (and hence tends to the stationary point; stable manifold), all the other trajectories in the positive orthant tends to infinity /spirallizing or monotonically/. Adding some "limitary" reactions to the Explodator core, the model undergoes Hopf bifurcation and a stable limit cycle arises. (As Boar Tang proved later, the limit cycle exists for parameters far away from the bifurcation value (global behaviour).

      20. FARKAS H. 1985. A problem in heat conduction: the time- dependence of the local potential. (in Hungarian with English abstract). Alk.Mat.Lapok 11: 343-347.

      Prigogine and Glansdorff  stated that the local potential monotonically decreases with time near the stationary state. However, "neglecting higher-order" terms- has no rigorous base, since the remaining part is also small. It is an open question: what is the exact condition for the monotonic behaviour of the local potential in the case when the conductivity depends on the temperature.

      22. FARKAS, H. & Z. NOSZTICZIUS 1985. Generalized Lotka- Volterra schemes. Construction of two-dimensional explodator cores and their Liapunov functions via "critical" Hopf bifurcations. J.Chem.Soc. Faraday Trans. 2. 81: 1487-1505.

      The original Lotka-Volterra model contains three reactions. Assuming these reactions with higher-order kinetics, we got a 2D system: dx/dt = x^n(1-y^m) ; dy/dt=Cy^m(x^n - 1). This
      system is conservative, similarly to the original LV model: it performs conservative oscillations around a stationary point. A first integral is explicitly given, and it is shown that this first integral can be used as Liapunov function for some rather general cases when the exponents are changed in the first (x^n* term) and the third reaction (-Cy^m* term).

      23. FARKAS, H., V. KERTÉSZ, & Z. NOSZTICZIUS 1986. Explodator and bistability.  React.Kin.Catal.Lett. 32: 301-306.

      It was stated (McKinnon&Field) that the Explodator model [16] cannot show bistability even in CSTR. Bistability was generally believed to be a necessary property required from an oscillating chemical model. Using the parametric representation method, the bifurcation diagram of multistationarity was constructed and a numerical example was given for bistability.

      27. DANCSÓ, A. & H. FARKAS 1989. On the "simplest" oscillating chemical  system. Periodica Polytechnica CE 33: 275-285.

      From a mathematical point of view, the simplest oscillator is the 2D linear harmonic oscillator. This simplest model is not realistic from the chemical point of view. A method is shown which enables us to construct a chemically realistic model from any given unrealistic polynomial model by addition of new reactions and components. Using this method, we constructed a 4D chemical model which reduces to the harmonic linear oscillator in a limit case. The model at certain parameters yields a stable limit cycle, as numeric calculation shows.

      29. FARKAS H., GYÖKÉR S. , & WITTMANN M. 1989. Investigation of globally equilibrium bifurcations with the aid of the parametric representation method. Alk.Mat. Lapok 14: 335-364. (In Hungarian) Alk.Mat. Lapok 14: 335-364.

      The equilibria of a homogeneous chemical dynamical system is determined from the real solutions of a system of algebraic equation. The first step is to reduce the number of equations by eliminating some state variables. After a suitable elimination we got a single algebraic equation with a single state variable and several real parameters. (For elimination Euclidean algorithm is suggested.)
          The parametric representation method gives easy-to-see results when the algebraic equation contains two real control parameters, both parameters occur linearly. Then the tangent bifurcation diagram in the parameter plane is determined by the condition: discriminant=0. This diagram is expressed in a parametric form (the parameter of this curve is the state variable). Using this diagram, the number of the real solutions can be given. Moreover, the location of the solution is given by the "tangential property", namely, the value of the state variable at the tangential point is just the value of the real solution.

      30. FARKAS, H., Z. NOSZTICZIUS, C.R. SAVAGE, & Z.A. SCHELLY 1989. Two- dimensional explodators 2. Global analysis of the Lotka-Volterra-Brusselator (LVB) model. Acta Phys. Hung. 66: 203-220.

      The Lotka-Volterra-Brusselator model is a modified Lotka- Volterra model, the modification is: instead of the second reaction (X+Y yields 2Y) the cubic reaction of the Brusselator (x+2Y yields 3Y) is used. This is a special case of the Generalized LV scheme in [22]. The unique positive stationary point is always unstable, and a global qualitative investigation revealed that the behaviour after long time maybe monotonous or spirallizing depending on the rate constants.

      33. DANCSÓ, A., H. FARKAS, M. FARKAS, & GY. SZABÓ 1990. Investigations on a  class of generalized two-dimensional Lotka-Volterra schemes.     Acta Applicandae Mathematica 23: 103-127.

      The generalized Lotka-Volterra scheme (with higher-order reactions; [22]) is investigated. The local investigation produces the Hopf-bifurcation diagram and the condition for supercritically. The global investigation involves Liapunov- function (what is just the first integral of the conservative system) for globally stable (dissipative) and globally unstable (explosive) systems. The special case when the stationary points are not isolated is also investigated ("zip bifurcation"). All the integrating factors of the monomial type (product of powers of the state variables) are listed, and this integrable cases are classified.

      34. FARKAS, H., S. GYÖKÉR, & M. WITTMANN 1990.  Use of the parametric  representation method in bifurcation problems. XIIth ICNO, Cracow. In: "Nonlinear Vibration Problems 25, PWN Polish Scientific Publishers,  Warszawa 1993.

      The "simplest" chemically reasonable quasi-harmonic oscillatory model (see [14]) is investigated by the parametric representation method; Hopf-bifurcation diagram is constructed. The eigenvalue problem leads to an algebraic equation of the fourth degree. The parametric representation method is applied to the case of nonreal solutions.

      36. FARKAS, H. & Z. NOSZTICZIUS 1992. Explosive, conservative and dissipative systems and chemical oscillators. In: Advances in Thermodynamics,  Vol.6. Flow, Diffusion, and Rate Processes. Eds.: S. Sieniutycz and P. Salamon.  Taylor and Francis, New York. pp. 303-339.

      The results relating to the oscillating chemical models developed in (16],[22],[27], and [33] are detailed. The parametric representation method and some of its applications are detailed.
          Some general formulations concerning the Second Law of Thermodynamics are also mentioned as conjectures (There is no periodic solution for an isolated system. An open thermodynamic system always has an attractor.)

      37. FARKAS H. & P. SIMON 1992. Use of the parametric representation method in revealing the root structure and Hopf bifurcation. J. Mathematical Chemistry             9: 323-339.

      The parametric representation method is applied to the equation of the type f(x)+ax+b=0, where x is the state variable, a and b are control parameters. All the solutions (real and nonreal) are taken into consideration. Two families of curves are introduced in the parameter plane (a,b) R- curves, where the difference of two solutions is constant, and the I-curves where the sum of two solutions is constant. I=0 gives the condition for Hopf bifurcation, and R=0 gives the condition for tangent bifurcation. The roots of cubic equation and their spectacular relationship is detailed. For quartic equation an "ear-eye transition" occurs: the tangent bifurcation diagram shows a swallow-tail form which reduces into a parabola when the third parameter changes, while a nonreal double-root appears which becomes an isolated point in the interior of the parabola.
          The Routh-Hurwitz criterion for stability is also analysed and it is shown that the (n-1)th RH criterion is just the condition I=0, that is I=0 is a necessary condition for Hopf bifurcation.

      40. LÁZÁR, A., Z. NOSZTICZIUS, H. FARKAS, & H.D. FÖRSTERLING 1995. Involutes: the geometry of chemical waves rotating in annular membranes. Chaos      5: 443-447.

      It is experimentally shown that the rotating stationary wave forms in an annular region (chemical pinwheels) are involutes of the annulus. This is explained by Fermat s principle.

      41. P.L. Simon, H. Farkas, Geometric theory of trigger waves. A dynamical system approach. J. Math. Chem. 19 (1996) 301-315.

      Fermat's principle of minimal propagation time serves a base for a simple geometric theory of waves. Rays are defined as extremals, and wave fronts as orthogonal trajectories of rays. It is proved that the process represents in itself a dynamical system. This enables us to determine the evolutions of fronts. Two kinds of reactors are considered: tube and annular. Long- term wave fronts are found. It is shown that the stationary wave profiles in a homogeneous annular region are involutes of the convex hull of the annulus. This property had been proved previously by Wiener and Rosenblueth.

      42. P.L. Simon, Nguyen Bich Thuy, H. Farkas, & Z. Noszticzius 1996. Application of  the parametric representation method to construct bifurcation diagrams for highly non-linear chemical dynamical systems. J. Chem. Soc., Faraday Trans., 1996, 92 (16), 2865-2871.

      We investigated the catalytic reactions with the equations
      x'=a(1-x-y)-bx-c(x^n)(1-x-y)^2
      y'=d(1-x-y)-y
      This is a generalization of some known catalytic model. We constructed bifurcational diagram by the parametric representation method [37]. The diagram of Hopf-bifurcation makes a loop around a cusp of the tangent bifurcation (saddle- node bifurcation) diagram.

      55. Conditions of Appearance of Mixed-Mode Oscillations and Deterministic Chaos
      in Nonlinear Chemical Systems, V. O. Khavrus, H. Farkas*, P. E. Strizhak 2002. Teoret. i experiment. khimija 2002. T. 38. No 5,  293—298 /in Russian/
      An approach for determining conditions of existence of mixed-mode oscillations and deterministic chaos
      in kinetic scheme after its reduction to simple system of equations is suggested. Simple conditions of existence
      mixed-mode oscillations and deterministic chaos are formulated on the basis of analysis of its steady states position.
      Boundaries of existence of monostability, bistability and oscillations were also found. The results obtained
      are confirmed by numerical modeling.